Quantum Field Theories Coupled to Supergravity: AdS/CFT and Local Couplings
aa r X i v : . [ h e p - t h ] N ov Quantum Field Theories Coupled to Supergravity:AdS/CFT Correspondence and Local Couplings
Johannes Große ∗ Max-Planck-Institut f¨ur Physik,F¨ohringer Ring 6, D-80805 M¨unchen, GermanyArnold-Sommerfeld-Center for Theoretical Physics,Department f¨ur Physik,Ludwig-Maximilians-Universit¨at M¨unchen,Theresienstraße 37, D-80333 M¨unchen, GermanyThis article is based on my PhD thesis [1] and covers the following topics: Holographic mesonspectra in a dilaton flow background, the mixed Coulomb–Higgs branch in terms of instantonson D7 branes, and a dual description of heavy-light mesons.Moreover, in a second part the conformal anomaly of four dimensional supersymmetric quan-tum field theories coupled to classical N = 1 supergravity is explored in a superfield formulation.The complete basis for the anomaly and consistency conditions, which arise from cohomologicalconsiderations, are given. Possible implications for an extension of Zamolodchikov’s c -theoremto four dimensional supersymmetric quantum field theories are discussed. Contents
I Generalizations of AdS/CFT 11
QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 N = 4 Super-Yang–Mills Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 Type IIB
Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.1 p -brane Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.4.1 Abelian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 Non-Abelian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4.3 Quadratic Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 AdS / CFT
Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 ∗ E-mail: [email protected]
J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings
DBI to Quadratic Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.3 Quadratic Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4.1 Vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4.2 Pseudoscalar Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4.3 Scalar Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.4.4 Vector Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.5 Backgrounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5.1 AdS × S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.5.2 GKS
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.6 Chiral Symmetry Breaking in
GKS . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.7 Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.8 Highly Excited Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 × S . . . . . . . . . . . . . . . . . . . . . . . . . . . 556.2 Dilaton Flow Geometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.2.1 GKS
Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 596.2.2 Constable–Myers’ Background . . . . . . . . . . . . . . . . . . . . . . . . . 636.3 Bottom Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
II Space-time Dependent Couplings 67 d > c -Theorem in Two Dimensions . . . . . . . . . . . . . . . . . . . . 868.3 Conformal Anomaly in Four Dimensions . . . . . . . . . . . . . . . . . . . . . . . . 888.4 Local RG Equation and the c -Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 888.4.1 a -Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 SUSY
Local RG Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 919.2 Basis for the Trace Anomaly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 929.3 Wess–Zumino Consistency Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 959.4 Local Counterterms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 969.5 S-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 979.6 Towards a Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 989.7 Superfield Riegert Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 999.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
10 Conclusions 103Acknowledgements 106A Weyl Variation of the Basis 106B Wess–Zumino Consistency Condition 108
Weyl Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Beta Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
C Coefficient Consistency Equations 112D Minimal Algebra on Chiral Fields 115References 116
An important goal of theoretical physics is the algorithmic compression of nature to a set offundamental laws. This means that a minimal description is sought that encodes a maximum ofinformation about our universe. At the current state of knowledge, this description is in termsof the standard model of elementary particles and Einstein gravity, as well as initial conditionsand parameters. Although many models used in other areas of physics are not derived from thosefundamental theories, in principle such a derivation should nevertheless be possible.The standard model is a quantum field theory that describes electromagnetism, the weak andthe strong force, organised by the principle of gauge invariance. The latter arises from makingthe formulation manifestly Lorentz invariant which requires the introduction of extra non-physicaldegrees of freedom. Consequently there are many representations of the same physical state, whichare related by so-called gauge transformations. Gauge transformations can be identified with Liegroups having space-time dependent parameters and form the internal symmetry group of thestandard model, the group U(1) × SU(2) × SU(3), corresponding to quantum electrodynamics(
QED ) describing photons, the weak interaction, whose gauge fields are the W and Z bosons
J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings responsible for the β decay, and quantum chromodynamics ( QCD ), the theory of the strong force,which describes the constituents of hadrons like the proton and the neutron.We shall first have a closer look at
QED , which is a remarkably successful theory, confirmed to anincredible accuracy of up to 10 − over the past decades. Since a rigorous treatment of interactingquantum field theories is difficult, an important reason for this success is the possibility to treat QED perturbatively. In perturbation theory a theory is effectively split into a solvable part; e.g. afree theory, and the remainder that renders the theory unsolvable; e.g. the interaction terms.Assuming that the solutions of the free theory are only slightly modified by the presence of theadditional interaction terms allows an expansion in the coupling constant. However this expansionis not a true series expansion since the coupling constants themselves need to be modified duringthe expansion by a procedure called renormalisation to absorb infinite contributions arising fromthe interplay of the quantisation procedure and perturbation theory. Theories allowing to absorbthese infinities in a finite number of parameters are called renormalisable and can be treatedperturbatively in a well defined manner.There are basically two points where this strategy can fail and interestingly both have a con-nection to string theory as will be seen later.The first problem arises when trying to tackle non-renormalisable theories like gravity. Eachorder of perturbation theory then produces a growing number of coupling constants that destroythe predictive power of the theory. This can either be interpreted as there being somethingwrong with the quantisation procedure assuming that gravity has some miraculous ultraviolet( UV ) behaviour that is merely poorly understood or that Einstein gravity is just an effective fieldtheory that breaks down when leaving its regime of validity (at the order of the Planck mass m P ≈ GeV) and a more fundamental theory is required.In the spirit of the introductory remarks at the beginning, such a “more fundamental” theory,from which also the standard model of elementary particles should be derived, is a natural goal,which unfortunately seems to be currently out of reach. However there exists at least a candidatetheory that consistently quantises gravity and at the same time incorporates gauge theories similarto the standard model, namely superstring theory . Entertainingly this extremely remarkablefeature was not what led to its discovery and it is also not the feature central to this thesis, whichshall be explicated in the followings.The second problem of perturbation theory arises from the phenomenon of running gaugecouplings , a result—though not a consequence—of renormalisation. It is the statement that thestrength of the interaction and thus the validity of perturbation theory depends on the energyscale. While the electroweak force has small coupling constants at low energies, which becomelarge when going to higher energies, the opposite is true for
QCD , which is asymptotically free .For small energies
QCD exhibits a phase transition, the confinement , that effectively screens thetheory’s fundamental particles, the quarks and gluons, from the dynamics by creating bound statesof vanishing colour charge: hadrons. In that sense
QCD is an accelerator theory that can only beobserved at high energies, although there is very strong evidence from lattice calculations that
QCD is also the correct theory for low energies where ordinary perturbation theory is not applicableand the dominating degrees of freedom are better recast in an effective field theory. However abetter understanding of the low-energy dynamics of
QCD and confinement is still sought after.Before the break-through of
QCD there was another candidate theory for the strong interaction,which could reproduce certain relations in the spectra of low energy hadron physics: string theory.String theory describes particles as oscillation modes of strings that propagate through space-time, joining and splitting along their way, thus sweeping out a two-dimensional surface, the world-sheet . The action of a string is that of an idealised soap film; i.e. proportional to the areaof the world sheet. Another interesting feature of the low energy dynamics of hadrons is theformation of flux tubes between quarks, which are also string like and even though nowadaysperfectly understandable from a pure
QCD point of view seemed to hint at a connection betweenstring theory and hadron physics. As will be seen later this connection does indeed exist in the form of the ’t Hooft large N c expansion [2], which was born in an attempt to find a small parameterfor perturbative calculations in the strong coupling regime. The basic idea is to look at SU( N c )Yang–Mills theories, where N c is the number of colours, and perform an expansion in N c . Thisimplies at leading order the ’t Hooft limit N c → ∞ , where additionally λ := g Y M N c is kept fixed,with g Y M the Yang–Mills coupling constant. This particular choice is motivated by keeping thestrong coupling scale Λ
QCD constant in a perturbative calculation of the β function. In a doubleline notation, the diagrams associated to each order in N c can be seen to give rise to a topologicalexpansion, which can be interpreted as a triangulation of two dimensional manifolds, the stringworld sheets in a genus expansion. While this triangulation is not understood in detail—see [3]for recent approaches to this important point—there is nevertheless a map between a particulargauge theory and string theory in a certain background.This map, tested by a large number of highly non-trivial checks, is Maldacena ’s conjecture [4]of
AdS / CFT correspondence. In its boldest form, it is the statement that N = 4 super-Yang–Mills( SYM ) theory, which is a conformal field theory ( CFT ) is dual to (quantised) type
IIB string theoryon AdS × S . By “dual” the existence of a map is meant that identifies correlation functionsof both theories, thus rendering them actually two different pictures of the same theory. Thedetails will be reviewed in Chapter 2. For now it is sufficient to remark that string theory in thatparticular background is still ill-understood, but that there are limits in which things are betterunder control. In the string loop expansion, each hole in the world sheet comes with a factor of g s , while in a similar gauge theory Feynman diagram each hole corresponds to a closed loop and istherefore accompanied by a factor of g Y M . This na¨ıve analysis allows to identify g Y M = g s , whichtherefore go to zero simultaneously in the ’t Hooft limit, demonstrating that the N c expansioncorresponds to a genus expansion of the string world sheet.From the construction of the AdS × S background in type IIB supergravity (
SUGRA ) theory,which is the small curvature, low energy limit of type
IIB superstring theory, it is possible toderive the relation (cid:0) Lℓ s (cid:1) ∼ λ , where L is the respective curvature radius of the anti-de Sitter space(AdS ) and the five-sphere (S ), and ℓ s = √ α ′ is the string length.Therefore, the limit of small curvature L ≫ ℓ s , where type IIB supergravity on AdS × S isa good approximation of the corresponding string theory, is dual to taking λ large in the fieldtheory. Because λ takes over the rˆole of the coupling constant in the large N c limit, with λ ≪ strongly coupled N = 4 SYM theory in the large N c limit. Since the discovery of the actual mapping prescription betweencorrelators on both sides of the correspondence [5, 6], a plethora of non-trivial checks have beenperformed [7, 8, 9, 10, 11], that did not only extend the correspondence to less symmetric regimesbut also provided overwhelming evidence that the conjecture actually holds true.This thesis is devoted to studying the coupling between supergravity ( SUGRA ) theories andquantum field theories. Although the idea was revived by the discovery of
AdS / CFT duality, wherethis coupling is realised holographically, that is between a four and a five dimensional theory, it hasalso been considered earlier in the context of space-time dependent coupling constants [12, 13, 14].In the first part of this thesis several aspects of
AdS / CFT correspondence will be discussed,while the second part uses the idea of space-time dependent couplings to analyse the conformalanomaly in super-Yang–Mills theories coupled to minimal supergravity.Since at a first glance these two subjects seem rather unrelated, I would like to linger on abit on the question of what the two topics have in common before continuing the introduction tothose two parts.The idea of space-time dependent couplings is to promote coupling constants to (external)fields. Generically the coupling takes the form R d x J O , where J acts as a source for theoperator O . A particularly important example for such a source/operator pair is the metric and For N c = 3 this describes the pure glue part of QCD . J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings the energy-momentum tensor, which couple according to S S + Z d x g mn T mn , such that allowing coordinate dependence g mn = g mn ( x ) amounts to coupling the quantum fieldtheory to a (classical) gravity background—or a supergravity background for supersymmetricquantum field theories. Invariance of the action under diffeomorphisms δg mn = L v g mn im-plies ∇ m T mn = 0, while from Weyl invariance ( δg mn = 2 σg mn ) one may conclude T mm = 0.When quantum effects destroy the Weyl symmetry of a classical theory, the trace of the energy-momentum tensor does not vanish anymore. It is said to have an anomaly : the Weyl or traceanomaly, which is a standard example of a quantum anomaly. More will be said about it below.For now let us have a look at the coupling of quantum field field theories to supergravity fromthe AdS / CFT point of view. In the
AdS / CFT correspondence, the prescription for the calculationof
CFT correlators in terms of
SUGRA fields is given by (cid:10) exp Z d x φ (0) O (cid:11) CFT = exp (cid:8) − S SUGRA [ φ ] (cid:9)(cid:12)(cid:12)(cid:12)(cid:12) φ ( ∂ AdS)= φ (0) , where the right hand side is the generating functional of the classical supergravity theory, whichis evaluated with its fields φ determined by their equations of motion and their boundary values φ (0) that appear as sources for field theory operators in the CFT .Much of the excitement about the
AdS / CFT duality came from the prospect of gaining insightinto the strong coupling regime of Yang–Mills ( YM ) theories and QCD . Both N = 4 SYM and type
IIB SUGRA are (almost) entirely determined by their large symmetry group, namely SU(2 , | YM has a much smaller field content and the problem arises how to get rid of the extra fields.Furthermore to describe QCD quarks are needed but N = 4 SYM contains only one hypermultipletwhose gauge field forces its adjoint representation on all other fields.The conformal group SO(2 ,
4) of the
CFT corresponds to the isometry group of AdS . Similarlythe SU(4) R group is matched by the SO(6) isometry group of the S . Therefore a less super-symmetric CFT will be dual to a
SUGRA on AdS × M , where M is a suitable less symmetricmanifold. Unfortunately the operator map relies heavily upon the field theory operators beinguniquely determined by their transformational behaviour under the global symmetry groups, suchthat reducing the symmetry implies making the correspondence less precise. This is especiallytrue when also giving up the conformal symmetry in order to obtain discrete mass spectra.Therefore the strategy employed in this thesis will be to describe theories that are very symmet-ric in the UV but are relevantly deformed and flow to a less symmetric, phenomenologically moreinteresting non-conformal infrared ( IR ) theory. This allows to still use the established AdS / CFT correspondence while at the same time capturing interesting IR physics.Such a renormalisation group ( RG ) flow is represented by a supergravity solution that ap-proaches an AdS geometry only towards the boundary, it is asymptotically
AdS . The interior of thedeformed space corresponds to the field theoretic IR . The interpretation of the radial direction ofthe (deformed) AdS space as the energy scale can be easily seen from considering dilations of theboundary theory. Since the boundary theory is conformal such a dilation should leave the actioninvariant. To achieve the same in the
SUGRA theory, the radial direction has to transform as anenergy to cancel in the metric the transformation of the coordinates parallel to the boundary. Theinterpretation of the radial direction as the renormalisation scale was introduced in [15, 16] andhas been used for a number of checks of the
AdS / CFT duality, for example calculation of the ratioof the conformal anomaly at the fixed points of holographic RG flows [7, 17], which coincides withfield theory predictions. An important step towards a holographic description of
QCD is the introduction of fundamentalfields into the correspondence. The first realisation of such a theory was a string theory in anAdS × S (cid:14) Z background where a number of D7 branes wrapped the Z orientifold plane withgeometry AdS × S [18, 19], which is dual to an N = 2 Sp( N c ) gauge theory. As was realisedby [20], a similar scenario of probe D7-branes wrapping a contractible S in AdS × S leads toa consistent description of an N = 2 SU( N c ) theory, since a contractible S does not give riseto a tadpole requiring cancellation, nor to an unstable tachyonic mode due to the Breitenlohner–Freedman bound [21]. (Further extensions of AdS / CFT using D7 branes to include quarks havebeen presented in [22, 23, 24, 25, 26, 27, 28, 29]. ) The full string picture is that of a D3-branestack, whose near horizon geometry gives rise to an AdS × S space, probed by parallel D7-braneswrapping and completely filling an AdS × S geometry. The strings connecting the two stacksgive rise to an N = 2 hypermultiplet in the fundamental representation. The resulting field theoryis conformal as long as the two brane stacks coincide. In this case the setup preserves an SO(4) × SO(2) subgroup of the original SO(6) isometry, which is dual to an SU(2) L × SU(2) R × U(1) R subgroup of the SU(4) R .Separating the two stacks introduces a quark mass and breaks conformal symmetry as well asthe SO(2) ≃ U(1) R symmetry. Consequently the induced geometry on the D7-branes becomes onlyasymptotically AdS . At the same time, the S starts to slip of the internal S when approachingthe interior of the AdS and shrinks to zero size. At that point the quarks decouple from the IR dynamics and the D7-brane seems to end from a five dimensional point of view. By solving theDirac–Born–Infeld ( DBI ) equations of motion for the fluctuations of the D7 branes about theirembedding the meson spectrum can be determined [24]. The setup is reviewed in more detail inChapter 3.In Chapter 4, I discuss how to combine the ideas laid out above, that is to consider probeD7-branes in background geometries that only approach AdS × S asymptotically. The specificgeometry under consideration is that of a dilaton flow by Kehagias–Sfetsos and Gubser [39, 40],which preserves an SO(1 , × SO(6) isometry while breaking conformal invariance and supersym-metry, thereby allowing chiral symmetry breaking by the formation of a bilinear quark condensate.In the framework of
AdS / CFT correspondence all supergravity fields encode two field theoreticquantities, a source and a vacuum expectation value (
VEV ). The embedding of a probe D7-brane isdetermined by a scalar field arising from the pullback of the ambient metric to the world volume ofthe brane. Solving the equation of motion for this scalar field Φ yields the following UV behaviour,Φ ∼ m q + h ¯ ψψ i ρ , where ρ is the radial coordinate of the AdS space, whose boundary is approached for ρ → ∞ .Extending the solution to the interior of the space, it turns out that generic combinations of thequark mass m q and the chiral condensate h ¯ ψψ i do not produce solutions that have a reasonableinterpretation as a field theoretic flow; i.e. are expressible as a function of the energy scale ρ . Idemonstrate that this requirement is sufficient to completely fix the condensate as a function ofthe quark mass. In the limit of vanishing quark mass there is a non-vanishing bilinear quark VEV indicating that the background is indeed a holographic description of spontaneous chiral symmetrybreaking.I then determine the mass of the lowest scalar, pseudoscalar and vector meson by calculating thefluctuations about the embedding solutions. Since the equations of motion for the D7 embedding inthe deformed background could only be solved numerically, the same holds true for the fluctuationsabout these vacuum solutions. Still the spectrum is well understood because it approaches theanalytic solutions of the supersymmetric case in the limit of large quark mass. This is to beexpected since for larger quark mass, the corresponding mesons decouple from the dynamics at Related models involving other brane setups may be found in [30, 31, 32, 33, 34, 35, 36, 37, 38].
J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings high energies where supersymmetry is restored. I show that in the limit of vanishing quark mass,where chiral symmetry is broken spontaneously, the pseudoscalar meson becomes massless and istherefore a Goldstone boson for the axial symmetry. For small quark mass m q , the mass of theGoldstone mode essentially behaves like √ m q in accordance with predictions from effective fieldtheory.Moreover I discuss the spectrum of highly radially excited mesons (as opposed to excitations onthe S , which are not in mutually same representations of SU(2) L × SU(2) R ). It is explained whyin this holographic setup (as in many others [41]) the field theoretic expectation [42, 43] of chiralsymmetry restoration cannot be met. The reason is the infrared being probed more densely inthe limit of large radial excitations, which also has an interesting effect on the heavy-light spectradiscussed below.In Chapter 5 instead of considering a non-trivial geometry, I discuss the effects of a non-trivial gauge field configuration on the brane. The spectrum of N f ≪ N c coincident D7-branesis described by a non-Abelian DBI action plus Wess–Zumino term C ∧ F ∧ F . Both scalar andvector fields on the brane are now matrix valued. Assuming that the branes are coincident onemay diagonalise and obtain effectively N f copies of the spectrum of a single brane—unless there isa contribution from the Wess–Zumino term. This requires to choose a background configurationwith non-trivial second Chern class; i.e. an instanton solution, which I demonstrate to indeedminimise the D7-brane action.The string connecting the D7 and D3-branes separated by a distance (2 πα ′ ) m q introduces amassive N = 2 hypermultiplet in the fundamental representation, which contributes the term˜ Q i ( m q + Φ ) Q i to the superpotential. ˜ Q i and Q i form the fundamental hypermultiplet and Φ isthe chiral field that is part of the adjoint N = 2 gauge multiplet. The scalar component of Φ is an N c × N c matrix. If some of its elements acquire a VEV such that m q + Φ is zero, then thecorresponding components of the fundamental field may also get a VEV and the theory is on themixed Coulomb–Higgs branch. I show that this Higgs
VEV corresponds to the instanton size ofabove background and calculate the spectrum of scalar and vector mesons as a function of theHiggs
VEV . In the limit of vanishing Higgs
VEV
I reproduce the analytic spectrum of the SU( N c )gauge theory. Not surprisingly there is a sense in which the spectrum of an infinitely large Higgs VEV is equivalent since it belongs to an SU( N c −
1) gauge theory. I show that this equivalenceholds only up to a non-trivial rearrangement of the spectrum by a singular gauge transformation. In Chapter 6 mesons consisting of a light and a heavy quark are discussed. A na¨ıve approachwould be to use the non-Abelian
DBI action, where the diagonal elements of the matrix valuedscalar field now encode a mass and bilinear condensate for each of the corresponding N f quarks.Off-diagonal elements of the embedding solution would contain mass-mixing terms and mixedcondensates, which one could set to zero for phenomenological reasons. Fluctuations about theseembeddings would correspond to the ordinary same-quark meson for the diagonal elements and toheavy-light mesons for the off-diagonal entries. However the latter are not small with respect tothe corresponding light quark and expansion of the DBI action to quadratic order is not possibleanymore. This step however is crucial to obtain an eigenvalue equation for the meson mass.The approach chosen here is to find an effective description for heavy-light mesons from thePolyakov action of the string stretched between two
D7-branes with different separation fromthe D3 branes corresponding to two different quark masses. The separation is assumed to belarge (that is only one quark is heavy, the light quark is taken massless), such that a semi-classicaldescription of this long string is possible. I take the ansatz of a rigid string spanned in the directionof the separation of the two branes. The string is not allowed to oscillate or bend but only tomove along the world volume of the D7s. Then integration over the string length can be carriedout to obtain an effective point-particle-like action. Its equation of motion is a generalisation of After the submission of the thesis, the results obtained in Chapter 5 have been generalized and extended by[44] beyond the approximation employed here; see [45] for related earlier works on defect conformal field theories. the Klein–Gordon equation which can be quantised. I evaluate the resulting eigenvalue equationfor the undeformed
AdS background as well as dilaton deformed backgrounds by Gubser andKehagias–Sfetsos [40, 39] and Constable–Myers [46].The heavy-light meson spectrum for both deformed geometries approximates the
AdS heavy-light spectrum for large quark mass. This behaviour is expected because a large quark masscorresponds to the string probing larger parts of the space-time that are approximately
AdS .At the same time, it can be observed that highly excited mesons converge more slowly to their
AdS values. Again this is in accordance with previous results of Section 4.8, where it has beendemonstrated that highly excited mesons probe the IR region of the space time more densely,where the deviation from the AdS geometry is large.These heavy-light spectra can be used to determine the mass of the B meson by using theresults of Chapter 4 as well as the experimental values of the Rho and Upsilon meson mass to fixthe confinement scale and heavy quark mass. The prediction for the B mesons is 20% above theexperimental value. Since the B mesons are far in the supersymmetric regime of this holographicmodel while at the same time the field theory is strongly coupled at that scale, this level ofagreement is surprisingly good.The
AdS / CFT models I considered here describe chiral symmetry breaking, highly excited mes-ons, the Higgs branch and heavy-light mesons, respectively. They have in common that theyare not focused on building a perfect
QCD dual, but instead are used to investigate particularfeatures of YM theory with matter. The strategy of keeping a connection to standard AdS / CFT with flavours worked out and the results show either the qualitative behaviour expected from fieldtheoretic and
SUGRA considerations or could even be matched quantitatively to analytic resultsin certain limits.
As already mentioned this thesis consists of two parts. In the first part presented so far variousaspects of
AdS / CFT correspondence have been discussed and a number of models extending the
AdS / CFT correspondence to theories with fundamental quarks have been developed and explored. The secondpart is devoted to an analysis of the conformal anomaly in super-Yang–Mills theories coupled to minimalsupergravity in four space-time dimensions. This analysis is aimed at providing building blocks for afuture generalisation of the two dimensional c -theorem, see below, to four-dimensional supersymmetricfield theories. The conformal anomaly expresses the breaking of conformal invariance in a classically conformalfield theory by quantum effects. It arises as the trace of the energy-momentum tensor, which—asmentioned above—vanishes in a conformally invariant theory, and is also called trace anomaly ,hence.An investigation of the trace anomaly is interesting because of its potential relation to a fourdimensional version of Zamolodchikov’s c -theorem [47]. The c -theorem is a statement about theirreversibility of renormalisation group flows connecting two fixed points of a quantum field theoryin two space-time dimensions. To be more precise the theorem states the existence of a monotonicfunction that at the fixed points, where the β functions vanish, coincides with the trace anomalycoefficient c defined by (cid:10) T mm (cid:11) = c π R , where R is the scalar curvature. Moreover the coefficient c turns up as the central charge of theVirasoro algebra and in the two point function of the energy-momentum tensor. The c -theorem is also interesting from a philosophical point of view, because the c -functionis interpreted to measure the number of degrees of freedom along the RG flow. Suppose thatone believes that in the real world this number should be non-increasing when going to lowerenergies, a future “theory of everything” should certainly incorporate a function that measuresthese degrees of freedom and is monotonic hereby. While it is not clear that such an irreversibilitytheorem should be realised in terms of a c -theorem, the questions remains if there is a class oftheories in four dimensions where an analogous statement to the two dimensional c -theorem can bemade. Such a generalisation is not straight forward since conformal symmetry in four dimensionsis far less powerful because the conformal algebra contains only a finite number of generators.In four dimensions the trace anomaly reads (cid:10) T mm (cid:11) = c C − a ˜ R + b R + f (cid:3) R , ( ⋆ )with C , ˜ R and R respectively the square of the Weyl tensor, the Euler density and the squareof the Ricci scalar R . The first question that arises is which of these coefficients is to take overthe rˆole of the two dimensional c . While f can be removed by adding a local counterterm tothe quantum effective action, c is known to be increasing in some theories and decreasing inothers and b is eliminated by Wess–Zumino consistency conditions. For the remaining coefficient,conventionally denoted “ a ”, there is no known counterexample to a UV > a IR , though explicitchecks can only be performed in certain classes of supersymmetric field theories [48, 49]. Thismight be an indication that supersymmetry is a necessary ingredient for such an a -theorem. Theprospect of an a -theorem [50] has attracted some interest in the recent past under the name a -maximisation [51].In this thesis a different approach inspired by an alternative proof of the c -theorem in twodimensions is chosen [52]. The author of [52] couples a quantum field theory that is conformalto a classical gravity background and investigates the anomaly arising from that coupling bypromoting the coupling constants λ to external fields λ ( x ).This trick yields well-defined operator insertions from functional derivations of the generatingfunctional with respect to the couplings. A generalisation of the Callan–Symanzik equation toWeyl rescalings is found, which becomes anomalous when Weyl symmetry is broken upon quanti-sation. The structure of this equation is ∆ σ W = A , where ∆ σ contains a Weyl scaling part and a β function part in analogy to the case of constant couplings and constant scale transformations.The shape of the anomaly A is determined by dimensional analysis, yielding an ansatz that isa linear combination between a set of coefficient functions, which only depend on the couplings,and a set of basis terms, which depend on the curvature and derivatives of the couplings. There isonly a finite number of possible basis terms and their coefficient functions can be perturbativelydetermined for a particular theory.Without resorting to a particular theory, one may nevertheless find constraints between thecoefficients arising from a Wess–Zumino consistency condition (cid:2) ∆ σ , ∆ σ ′ (cid:3) W = 0 . In two dimensions this consistency condition implies β i ∂ i ( c + w i β i ) = χ ij β i β j , where c isthe central charge and w i ( λ k ) and χ ij ( λ k ) are above mentioned coefficient functions. χ ij can berelated to the positive definite Zamolodchikov metric, which is the key ingredient for the definitionof a monotonic c -function.In the four-dimensional case it is such a relation to a positive definite object that is missing. Inparticular the analogous consistency condition for the a coefficient in the four dimensional traceanomaly ( ⋆ ) reads β i ∂ i ( a + w i β i ) = χ gij β i β j , where χ gij ( λ k ) is one of the (many) coefficients in the four-dimensional anomaly ansatz. There is a relation to a positive definite coefficient χ a , χ gij = 2 χ aij + (other terms), but it is spoiled by theoccurrence of extra terms.In supersymmetric theories, some of these extra terms are known to vanish and there might behope that additional constraints arise from a local RG equation incorporating super-Weyl trans-formations that allow the construction of a monotonic a -function. Before tackling this ambitioustask, a first step is to analyse the trace anomaly in a supersymmetric framework, which is whathas been pursued in the second part of this thesis.In Chapters 7 and 8 respectively, I give an introduction to minimal supergravity in an N =1 superfield formulation and to the non-supersymmetric local renormalisation group techniqueoutlined above.In Chapter 9, I present superfield versions of the local RG equation, give a complete ansatzfor the trace anomaly, and determine the full set of consistency equations. I then discuss the N = 4 case, which gives rise to an interesting puzzle: In [53] by a component approach a one-loop result for the trace anomaly of N = 4 SYM was found to contain a conformally covariantoperator of fourth order, the Riegert operator [54], which is reviewed in Section 8.1.4. In [55]a supersymmetric version of this operator is given in components, but I was not able to find asatisfactory superfield version of this operator. A superfield Riegert operator is known to existin new-minimal supergravity [56], which however in general is known to be inconsistent on thequantum level [57, 58]. I discuss the possible origin of that problem, which I suspect to arise fromthe impossibility to separate local U(1) R transformations from super-Weyl transformations in theminimal supergravity formulation such that a too strong symmetry requirement is imposed onthe ansatz. Nevertheless the extended calculations presented here should provide a good startingpoint for further exploration of this fascinating topic. In the conclusions possible future steps arediscussed.
Part I
Generalizations of AdS/CFT
QCD
The gauge theory of the strong interaction, quantum chromodynamics (
QCD ), is based on thesuccess of the parton model [59, 60], which describes the high-energy behaviour of hadrons asbound states of localised but essentially free particles, to describe the high-energy hadron spec-trum. The other key ingredient was to realise that an additional hidden three-valued quantumnumber, colour , is needed.The former means that the theory should be asymptotically free; i.e. the coupling constantbecomes small in the ultraviolet regime ( UV ). This requirement is only met by Yang–Mills theories,that means non-Abelian gauge theories.The latter (hiding the colour) makes plausible a colour dependent force to form colour singletsonly, such that one may assume the colour symmetry (as opposed to the flavour symmetry) to begauged. Indeed lattice calculations demonstrated that QCD is confining , such that the formationof colour singlets is a consequence of the dynamics.The QCD
Lagrangean describes an SU( N c ) Yang–Mills theory with N c = 3 the number ofcolours and N f = 6 the number of quarks, with a global SU( N f ) L × SU( N f ) R × U(1) V × U(1) A In new-minimal supergravity this problem does not arise because U(1) R is indeed a local symmetry of thetheory. Quark Masses
Type Q Generations u c t up . .
15 to 1 .
35 GeV 169 to 179 GeV d s b down − . . Table 1
Quark masses (Particle Data Group [61]) symmetry that is partly broken by the different mass of the six quarks, cf. Table 1. It is given by L QCD = − Tr F mn F mn + N f X i ¯ q i ( iγ m D m − m i ) q i (2.1) F mn = ∂ m A n − ∂ n A m + i g (cid:2) A m , A n (cid:3) D m q i = ( ∂ m − i gA m ) q i A m = A ma T q (cid:2) T a , T b (cid:3) = i f abc T c The N c − A ma are called gluons , the N f = 6 quark fields q i are the Dirac fermions u, d, s, c, b, t . The global flavour symmetry is explicitly broken by (the inequality of) the masses m i , though they can be assumed to be realised approximately for the isospin group SU(2) f oreven (including the strange quark) SU(3) f . The corresponding transformation and algebra as wellas Noether current and charge read δq i = iα a t aij q j , (cid:2) t a , t b (cid:3) = if abc t c ,J aµ = ¯ q i γ µ t aij q j , (2.2) Q a = Z d xJ a , (cid:2) Q a , Q b (cid:3) = if abc Q c , where for SU (3) f the generators t a = λ a are usually expressed by the eight Gell-Mann matrices λ a .Furthermore the Lagrangean is invariant under an overall U(1) V vector symmetry q e iα q ,often also referred to by baryon number symmetry. The massless version of (2.1) is in additioninvariant under the U(1) A axial transformations q e iβγ q giving rise to a second copy of theflavour symmetry group, δq i = iα a t aij q j , J aµ = ¯ q i γ µ t aij q j , (2.3) Q a = Z d xJ a , (cid:2) Q a , Q b (cid:3) = if abc Q c . (2.4)Together they form the chiral symmetry group SU( N f ) L × SU( N f ) R , whose generators and cor-responding algebra are given by Q aL = ( Q a − Q a ) , Q aR = ( Q a + Q a ) , (cid:2) Q aL , Q bL (cid:3) = if abc Q cL , (cid:2) Q aR , Q bR (cid:3) = if abc Q cR , (2.5) (cid:2) Q aL , Q bR (cid:3) = 0 . When switching on mass terms this symmetry is not exact anymore and the associated charges,while still obeying the algebra, are not conserved; i.e. become time dependent.2.2 N = 4 Super-Yang–Mills TheoryWhile classically Yang–Mills theories are conformally invariant, this is no longer true upon quan-tisation and the conformal symmetry becomes anomalous. It turns out that it is actually quitehard to find a field theory that is conformally invariant on the quantum level and it comes as asurprise that N = 4 SYM , whose formulation was first achieved by compactifying ten dimensional N = 1 SYM on a six dimensional torus, actually preserves a larger symmetry group than its higherdimensional ancestor and has vanishing β functions to all orders in perturbation theory [62].Consequently from the commutators of supercharges and the generator of special conformaltransformation, an additional set of (so-called conformal) supercharges is generated. From theperspective of AdS / CFT correspondence this doubling of supercharges is quite important since N =4 has therefore the same number of supercharges as five dimensional maximally supersymmetricsuper gravity . The full superconformal algebra is SU(2 , | , ≃ SO(2 , R , the R-symmetry group.Being maximally supersymmetric, N = 4 SYM consists entirely of one multiplet, the N = 4gauge multiplet. In N = 1 language, this corresponds to one gauge multiplet plus three chiralmultiplets. So the field content is one vector, four chiral fermions and three complex scalars. Asthe gauge and
SUSY generators commute, all fields are in the adjoint representation. Two of thechiral superfields form an N = 2 hypermultiplet, while the other chiral superfield together withthe N = 1 gauge multiplet forms an N = 2 gauge multiplet.In N = 1 superfield language the Lagrangean reads L = Z d θ Tr (cid:0) ¯Φ i e V Φ i e − V (cid:1) + (cid:20) g Z d θ W α W α + Z d θ W + c.c. (cid:21) , (2.6)where the gauge field strength is given by W α = − ¯ D (e − V D α e V ) and the superpotential is W = Tr Φ (cid:2) Φ , Φ (cid:3) . (2.7)2.3 Type IIB
SupergravityThere are only two maximally supersymmetric supergravity theories in ten dimensions, called type
IIA and type
IIB . Both are N = 2 SUGRA s and contain (among others) two chiral gravitini, but
IIA is non-chiral in the sense that these fermions have opposite chirality while
IIB has gravitini ofthe same chirality. The particle content of the latter is given by Table 2.
IIB contains a self-dual five-form field ˜ F := F − C ∧ H + B ∧ F , F := dC , which makesit hard to write down an action from which all equations of motion may be derived. Often in the literature [66, 63], the following action is used, augmented by the self-dualitycondition ˜ F = ∗ ˜ F , which has to be imposed additionally on the equations of motion and where In an attempt to embrace both naming conventions used in
SUSY , multiplets are denoted chiral, gauge orhyper in conjunction with the number of supersymmetries. Super fields on the other hand shall always mean N = 1language and will be distinguished by their constraint (none, chiral, real, linear) and transformation behaviour ofthe lowest component (scalar, spinor, vector, tensor, density). See [64, 65] for recent attempts to improve this situation. The conventions employed here are: A p = p ! A A ...A p , ( dA p +1 ) A ...A p +1 = ( p + 1) ∂ [ A A A ...A p +1 ] , and | F p | = p ! F A ...A p F A ...A p . IIB SUGRA
Particle Content
Symbol
DOF
Field G AB B metric — graviton C + iϕ B axion — dilaton B AB + iC AB B rank 2 antisymmetric C ABCD B antisymmetric rank 4 ψ , Aα F two Majorana–Weyl gravitini λ , α F two Majorana–Weyl dilatini Table 2
IIB SUGRA
Particle Content [63] ∗ denotes the Hodge dual. S IIB = 12 κ Z d x p G E (cid:8) R E − ∂ A ¯ τ ∂ A τ τ ) − | F | − | G | − | ˜ F | (cid:9) − iκ Z C ∧ ¯ G ∧ G , (2.8)where the expressions in order of appearance are the determinant of the metric, the Ricci scalar R E , axion–dilaton field τ := C + i e − ϕ composed of the axion C and the dilaton ϕ , field strength F := dC and G := √ Im τ ( F − iH ) with F := dC and H = dB . The complex objects havebeen introduced to make manifest an additional rigid SL(2 , R ) symmetry of type IIB SUGRA ,which transforms τ aτ + bcτ + d , det (cid:18) a bc d (cid:19) = 1 , (2.9) G c ¯ τ + d | cτ + d | G , (2.10)and leaves invariant the other fields.Many also prefer to follow the historic approach [67, 68, 69, 70, 71] of writing down the equationsof motion only, which restricted to the graviton, axion, dilaton, and four-form Ramond–Ramondpotential read: R AB = e ϕ ∂ A C ∂ B C + ∂ A ϕ ∂ B ϕ + · ˜ F AC ...C ˜ F BC ...C , ∇ A ∇ A C = − ∇ A C )( ∇ A ϕ ) , (2.11) ∇ A ∇ A ϕ = e ϕ ( ∇ A C )( ∇ A C ) ,∂ [ A ( C ) A ...A ] = ε A ...A A ...A ∂ A ( C ) A ...A , where by convention the total anti-symmetric Levi-Civita symbol takes values ±√− det G E forall indices lowered (and accordingly ±√− det G E − for all indices raised).2.3.1 p -brane SolutionsThere is a particular class of solutions to the supergravity equations of motion (2.11) that preservehalf of the supersymmetry and the subgroup SO(1 , p ) × SO(9 − p ) of the ten dimensional Lorentz group. Additionally they have a non-trivial C p +1 charge coupled to the supergravity action by S p ∼ Z dC p +1 . (2.12)These solutions are called p -branes. They are determined by the ansatz ds = H ( y ) α η µν dx µ dx ν + H ( y ) β ( dy + y d Ω ) (2.13)with η µν the ( p +1)-dimensional Minkowski metric, d Ω − p the line element of the (8 − p )-dimensionalunit sphere and constants α , β to be determined by the equations of motion. The directions x arereferred to as world-volume or longitudinal coordinates, while y are called transversal.Since to this thesis, the most relevant p -branes are 3-branes, their full solution in terms ofbosonic supergravity fields is given, ds = H ( y ) − / η µν dx µ dx ν + H ( y ) / ( dy + y d Ω − p ) , Φ = Φ = const , C = const ,B AB = C ,AB = 0 , (2.14) C = H ( y ) − dx ∧ · · · ∧ dx ,H ( y ) = 1 + X i L | ~y − ~y i | , L = 4 πg s N α ′ , for a distribution of 3-branes at positions y i . Close to the origin of a single brane | ~y − ~y i | ≪ L , the1 in the warp factor can be neglected such that the geometry becomes approximately AdS × S .2.4 D-branesA D p -brane is a ( p + 1)-dimensional hypersurface in the target space of string theory, whereopen strings can end [72, 73]. Their discovery integrates some features of superstring theory andsupergravity that would have been puzzling without them. Firstly, the open string admits twokinds of boundary conditions, Dirichlet X i ( τ, σ ) = const , Neumann ∂ σ X i ( τ, σ ) = 0 . However from a na¨ıve point of view, Dirichlet boundary conditions have to be consideredunphysical as they break Lorentz invariance and—worse—make the open strings loose momentumtrough their endpoints. With the discovery of T-duality [74, 75, 76, 77] it became apparent thatone could transform from one kind of boundary condition to the other and it was no longer possibleto exclude Dirichlet boundary conditions a priori. In the D-brane picture, momentum conservationcan be restored by assuming the D-branes as dynamical objects can absorb the above mentionedmomentum flow.Secondly, p -brane solutions of SUGRA are interpreted as the low energy effective objects cor-responding to D p -branes.Thirdly, it was realised early [78], that it is possible to attach gauge group factors to the endpoints of open strings. These Chan–Paton factors have a natural explanation as encoding whichbrane in a stack of coincident branes the string is attached to. p -branes are domain wall solutions of SUGRA , see Section 2.3.1 for details.
Index Conventions longitudinal transversal X a,b,... X i,j X A,B,...
Table 3
Index conventions for ambient space, world volume and transversal coordinates p -brane this factor is a U(1) in accordance with the fact, that the massless modes ofopen string theory form a ( p + 1)-dimensional U(1) SYM with one vector, 9 − p real scalars, whose VEV s describe the position of the brane, and fermionic superpartners, which shall be ignored inthe following. For constant field strengths F ab , F = F ab dX a ∧ dX b , by resummation it is possibleto determine the action to all orders in α ′ [79] to be the first (Dirac–Born–Infeld, DBI ) part of S D p = − T p Z d p +1 ξ e − ϕ p − det P [ G + B ] ab + 2 πα ′ F ab ± T p Z P (cid:2)P C n e B (cid:3) e πα ′ F , (2.15)which couples the brane to the massless Neveu–Schwarz ( NS ) sector of closed string theory whilethe second (Wess–Zumino, WZ ) part determines the coupling of the brane to the massless Ramond–Ramond ( RR ) sector. The index conventions are depicted in Table 3, while the fields are explainedin Section 2.3.The prefactor T p is given by T p = 2 πg s (2 πℓ s ) p +1 , (2.16)with g s the string coupling and ℓ s the string length.Throughout this thesis, for explicit calculations the Kalb–Ramond field will be assumed tovanish. As will be commented on below, the Wess–Zumino term allows coupling to—with respectto the brane’s world volume—lower dimensional RR potentials if the gauge field has a non-trivialChern class . The only RR potential in the backgrounds discussed here, will be C associatedto the five-form flux always present in the AdS / CFT correspondence. In the particular case of aD7-brane, the Wess–Zumino term then reads S D − W Z = T p Z d ξ P [ C ] ∧ F ∧ F. (2.17)2.4.2 Non-Abelian N parallel D-branes describe a U(1) N gauge theory. When these branes approach one another,strings stretched between different branes become light and the gauge symmetry is promoted toU( N ). Generalising to the case of U( N ) is straight forward in the case of D9-branes, which doesnot require a generalised pull-back and thus requires merely an additional trace over gauge indices.The action of D p -branes of arbitrary world volume dimension p + 1 can then be determined by Apart from the additional complication of finding the correct series expansion, which is non-trivial due toordering ambiguities. T-duality, which transforms the T-dualized direction from longitudinal to transversal and viceversa. The result [80] in string frame is S D p = − T p Z d p +1 ξ STr (cid:20) e − ϕ p det Q q − det P [ ˜ E ] ab + 2 πα ′ F ab (cid:21) ± T p Z STr h P [e i (2 πα ′ )i Φ i Φ P C n e B ] e πα ′ F i , (2.18)where “STr” is a trace operation that shall also take care of any ordering ambiguities in theexpansion of the non-linear action. Its name (“symmetrised trace”) is reminiscent of an orderingprescription suggested by [81], which however is not valid beyond fifth order. Throughout thisthesis, an expansion to second order will be sufficient and no ordering ambiguities appear at all.The following abbreviations have been introduced:˜ E AB := E AB + E Ai ( Q − − δ ) ij E jB (2.19a) E AB := G AB + B AB , (2.19b) Q ij := δ ij + iγ (cid:2) Φ i , Φ k (cid:3) E kj , (2.19c)( Q − − δ ) ij := (cid:2) ( Q − ) ik − δ ik (cid:3) E kj , (2.19d) γ := 2 πα ′ , (2.19e)i Φ i Φ f ( n ) := 12( n − (cid:2) Φ i , Φ j (cid:3) f ( n ) jiA ...A n dx A ∧ · · · ∧ dx A n , (2.19f)where f ( n ) is an arbitrary n -form field acted upon by i Φ , the interior product with Φ i . E ij is theinverse of E ij (as opposed to the transversal components of E AB ).In particular static gauge is chosen, X a = ξ a , X i = γ Φ i ( ξ a ) , (2.20)which means transversal coordinates X i are in one-to-one correspondence to the scalar fields Φ i .Then the pull-back of an arbitrary ambient space tensor T A ...A n can recursively be defined by P [ T A ...A n ] a ...a n := P [ T a A ...A n ] a ...a n + γ ( D a Φ i ) P [ T iA ...A n ] a ...a n , (2.21)which yields for the combined metric/Kalb–Ramond field P (cid:2) ˜ E (cid:3) := ˜ E ab + γ ˜ E ai D b Φ i + γ ˜ E ib D a Φ i + γ ˜ E ij D a Φ i D b Φ j . (2.22) D a denotes the gauge covariant derivative.Finally E ab still may contain a functional dependence on the non-commutative scalars Φ andis to be understood as being defined by a non-Abelian Taylor expansion [82] E ab ( ξ a ) = exp[ γ Φ i ∂ X i ] E ab ( ξ a , X i ) (cid:12)(cid:12) X i =0 . (2.23)Again the Wess–Zumino part shall be given for the eight dimensional case; i.e. a stack ofD7-branes, S W Z = T Z STr (cid:26) P [ C ] + γP [ iγ i Φ i Φ C + C ] ∧ F + γ P [( iγ i Φ i Φ ) C + iγ i Φ i Φ C + C ] ∧ F ∧ F + γ P [( iγ i Φ i Φ ) C + ( iγ i Φ i Φ ) C + iγ i Φ i Φ C + C ] ∧ F ∧ F ∧ F (cid:27) , (2.24) where B has been assumed to vanish. For a 3-brane background, there is only a four-form potentialand accordingly the Wess–Zumino part is given by S W Z = T Z STr γ P [ C ] ∧ F ∧ F + iγ P [i Φ i Φ C ] ∧ F ∧ F ∧ F. (2.25)While (2.18) encodes the high non-linearity of a D-brane action in a compact manner, it isoften not suited for explicit calculations and needs to be expanded.2.4.3 Quadratic ActionAs both the non-Abelian scalars and the field strength carry γ as a prefactor, it is tempting tothink of it as an expansion parameter, keeping track of the order. However in equation (2.19c) infront of the commutator there is a factor of γ where following this logic a factor of γ should beexpected. To avoid these pitfalls and unambiguously define what is meant by “quadratic order”, a pa-rameter ε shall be thought to accompany γ in each of the equations of the last Section with thesole exception of (2.19c), where an ε is included in front of the commutator. Then, the order ε n denotes a total of n fields of Φ or F ab in a term.Pulling out a factor E ab ( ε = 0) (which shall also not depend on transverse directions X i asthey come with an ε ) from the DBI part of the D-brane action defines a matrix M ( γ ) according to S DBI = − T p Z d p +1 ξ STr h e − ϕ p det Q p − det E ab (0) p det M ( ε ) i , (2.26)which has the property M (0) = and is given by M ( ε ) ab = E ac ( ε = 0) (cid:16) P [ ˜ E ( γ )] cb + εγF cb (cid:17) . (2.27) E ac is the inverse of E ac . An expansion in ε is performed according to p det M ( ε ) = 1 + ε M ′ (0)) + ε (cid:20) Tr ( M ′′ (0)) − Tr (cid:0) M ′ (0) (cid:1) + Tr ( M ′ (0)) (cid:21) + O ( ε ) , (2.28)where M ′ (0) = γE ac Φ i ∂ X i E cb + E ac ( γE kb D c Φ k + γE ck D b Φ k ) + γE ac F cb , (2.29) M ′′ (0) = γ E ac Φ i Φ j ∂ X i ∂ X j E cb +2 γ E ac Φ i ∂ X i ( E kb D c Φ k + E ck D b Φ k )+ E ac [ E ci (2 iγ (cid:2) Φ i , Φ j (cid:3) − E ij ) E jb + 2 γ E ij D c Φ i D b Φ j ] . (2.30)All quantities on the right hand sides of (2.29) and (2.30) are to be understood as having ε setto zero. In particular this means the right hand sides are evaluated at vanishing transversalcoordinates X i = 0. Furthermore some authors prefer to use factors of α ′ to obtain D3-transversal coordinates with mass dimen-sion 1, thus modifying the manifest α ′ dependence even though in physical observables such redefinitions cancel ofcourse. NN Fig. 1
Double Line Representation: Non-planar diagrams are suppressed by powers of N c [83] For a diagonal metric and vanishing Kalb–Ramond field, the
DBI part of the action up toquadratic order simplifies dramatically, S DBI = − T p Z d p +1 ξ STr e − ϕ p − det G ab (cid:20) γ G ab G ij D a Φ i D b Φ j + γ G ac G bd F ab F cd + γ ( G ab ∂ X i ∂ X j G ab )Φ i Φ j (cid:21) , (2.31)where the following terms vanish unless the transversal coordinates enter the metric linearly,(lin.) := γ Tr M − γ Tr M + γ Tr M , M ac := G ab Φ i ∂ X i G bc . (2.32)2.5 AdS / CFT
CorrespondenceThe
AdS / CFT correspondence (Anti-de Sitter/Conformal Field Theory) is the statement of twoseemingly different theories to be equivalent. These theories are ten dimensional Type
IIB stringtheory on an AdS × S space-time background and four dimensional N = 4 extended supersym-metric SU ( N c ) Yang–Mills theory. The latter is a (super)conformal field theory with couplingconstant g Y M = g s , where g s is the string coupling. The string theory has N c units of five-form flux through the S , which is related to the equal curvature radii L of the AdS and S by L = 4 πℓ s g s N c , where ℓ s = √ α ′ is the string length. This equivalence is supposed to hold forarbitrary values of N c and the coupling constants, but since string theory on AdS × S is notwell-understood, it is usual to take two consecutive limits that make a supergravity descriptionvalid but still leave the duality non-trivial.The first limit to take is the ’t Hooft large N c limit, with N c → ∞ while λ := g Y M N c is keptfixed, in which the field theory reorganises itself in a topological expansion. This can be seenby using a double line representation for Feynman diagrams assigning a line to each gauge index,such that fields in the adjoint are equipped with two indices, while fields in a vector representationcarry a single line. The diagrams, see Figure 1, then correspond to polyhedrons, which contribute with a power of N c that is suppressed by the diagram’s genus and the polyhedrons are interpretedas triangulating the string world sheet, though the exact nature of this triangulation is still tobe understood. Due to g s = λ/N c the strict ’t Hooft limit corresponds to considering classicalstring theory on AdS × S . At the same time the ’t Hooft coupling takes over the rˆole of the fieldtheoretic coupling constant.In the second limit ℓ s →
0, the curvature radius is assumed to be large compared to the stringlength ℓ s ≪ L . This corresponds to the low energy limit where supergravity becomes an effectivedescription. On the field theory side this implies a large ’t Hooft coupling1 ≪ L ℓ = 4 πλ (2.33)and a strongly coupled theory therefore, indicating that AdS / CFT is a weak-strong duality. Thismeans that one theory in its perturbative regime is dual to the other theory in the strong couplingregime, which renders the duality both extremely useful and hard to proof.While on the one hand the supergravity version is the weakest form of the
AdS / CFT conjecture,it is the most useful version for practical calculations on the other hand. The equivalence of boththeories to be expressed by (cid:10) exp Z d x φ (0) O (cid:11) CFT = exp (cid:8) − S SUGRA [ φ ] (cid:9)(cid:12)(cid:12)(cid:12)(cid:12) φ ( ∂ AdS)= φ (0) , (2.34)where the field theoretic operator O is coupled to the boundary value φ of an associated super-gravity field φ , which is determined by the supergravity equations of motion and the boundarycondition.This implicitly introduces the notion of the conformal field theory being defined on the boundaryof AdS , where one may imagine the AdS space being build up from slices of Minkowski spacesparallel to the boundary and fibred over a fifth (“radial”) direction y . The line element reads ds AdS × S = y L dx , + L y dy + L d Ω . (2.35)For the metric to be invariant under rescalings of the coordinates on the boundary x , the radialdirection has to transform reciprocal, which means that y transforms as an energy and is inter-preted as the renormalisation scale of the boundary theory. Considering domain wall solutionsit is actually possible to represent field theoretic renormalisation group flows on the supergravityside [7, 17], establishing the fact that the interior of the AdS space may be interpreted as theinfrared ( IR ) and the boundary as the ultraviolet ( UV ) of the field theory.By the standard AdS / CFT dictionary supergravity fields, φ being solutions to differential equa-tions of second order, encode actually two field theoretic objects, whose conformal dimension canbe read off from the asymptotic behaviour, φ ( y → ∞ ) ∼ J y ∆ − + (cid:10) O (cid:11) y − ∆ , (2.36)where the radial direction is interpreted as the renormalisation scale. The first, non-normalisablepart corresponds to a field theoretic source and has conformal dimension 4 − ∆; the normalisablepart yields the corresponding VEV of mass dimension ∆. A simple example shall illustrate this.For the bilinear operator ¯ ψψ , the dual supergravity field has the asymptotic behaviour φ ( y → ∞ ) ∼ my + cy , (2.37)where m is the mass term of field ψ and c the bilinear quark condensate (cid:10) ¯ ψψ (cid:11) . The difficultpart is to find out which supergravity fields correspond to which field theoretic operators. For - BPS states, which correspond to superconformal chiral primary operators, the situation is simpler because they are determined by their transformational behaviour under the large global symmetrygroup SU(2 , | , × SU(4) ≃ SO(2 , × SO(6)is realised as the conformal and R-symmetry group, while it corresponds to the isometry groupon the supergravity side.From a string theoretical perspective, the correspondence can be understood as two differenteffective descriptions of a D3-brane stack, namely as a Yang–Mills theory from an open stringperspective and a p -brane solution from a closed string perspective. In the latter case, the AdS × S geometry arises from a near-horizon limit. The picture of AdS / CFT being two descriptions of aD3-brane stack turns out to be particularly useful when adding additional branes to includefundamental fields into the duality. This shall be the topic of the next Chapter.
While the
AdS / CFT correspondence has been a remarkable progress in the understanding of the’t Hooft large N c limit [2], a need to extend the Maldacena conjecture beyond N = 4 super-Yang–Mills ( SYM ) theory was soon felt, see [84] for a most prominent example. Since N = 4 SYM contains only one multiplet, the gauge field forces its representation on all other fields in thetheory. As a consequence, also the fermions transform under the adjoint representation, and thusdo not describe quarks.There have been early attempts to augment the boundary theory with fundamental fields byincluding D7-branes in an AdS × S (cid:14) Z geometry [19, 18]. The orientifold was introduced tosatisfy a tadpole cancellation condition, but the dual N = 2 boundary theory had gauge groupSp( N ). In order to obtain an SU( N c ) gauge theory for the description of large N c cousins ofquantum chromodynamics ( QCD ), [20] dropped the orientifold from the setup. This was justifiedby the fact, that the probe D7-brane wraps a contractible S cycle on the S and does not lead to atadpole, hence. In [20] it was shown that the string mode corresponding to the direction in whichthe S slips from the S has negative mass square, but satisfies (saturates) the Breitenlohner–Freedman bound and does not introduce an instability.In this Chapter, the main ideas of [20] will be reviewed, before calculating the meson spectrumof a field theory dual to a more general geometry in the next Chapter.3.1 MotivationConventional AdS / CFT correspondence can be understood as two different limits (see the introduc-tory Chapter) of the same object, namely a stack of N c coincident D3-branes in string theory. Thechoice on which of those N c branes an open string may end, is reflected by the SU( N c ) symmetryof the dual field theory. The number of ways to attach both ends to the stack is N c − N c , indi-cating that the field describing the open string is in the adjoint representation. When includinganother, non-coincident brane in this setup, a string connecting it to the stack has N c choices andthus describes a field transforming under the vector representation of the gauge group. Anotherperhaps less heuristic way to understand this scenario, is to return to the ’t Hooft expansion. Ifone takes the intuition about the field theory’s reorganisation into a triangulation of the closedstring world sheet serious, then apparently, fundamental fields will provide boundaries that lead toa triangulation of the open string world sheet. In this sense, augmenting the AdS / CFT correspon-dence by additional branes, which exactly provide these open strings, extends the correspondencefrom an open-closed duality to a full string duality.While the inclusion of D3 or D5-branes leads to fundamental fields on the boundary of
AdS thatare confined to a lower dimensional defect (so-called “defect
CFT s”), the addition of D7-branesprovides space-time filling fields in the fundamental representation. Furthermore it breaks super-symmetry by a factor of two; from N = 4 to N = 2 on the four-dimensional field theory side by Coordinates0 1 2 3 4 5 6 7 8 9D3 D7 x µ,ν,... y m,n,... z i,j,... ryX a,b,... X A,B,...
Table 4
D3- and D7-brane embedding in the AdS × S geometry. The D7-branes (asymptotically)wrap an AdS × S . The Table also summarises the index conventions used throughout this part of thethesis. inclusion of an N = 2 fundamental hypermultiplet given rise to by the light modes of strings withone end on the D3s and one on the D7s.3.2 Probe BraneIn order to maintain the framework of conventional AdS / CFT correspondence, [20] neglected thegravitational backreaction of the D7-branes on the geometry, which was justified by requiring thenumber N f of D7-branes to be sufficiently small. The contribution of the N c D3-branes and the N f D7-branes to the background fields is of order g s times their respective number. So as longas N c ≫ N f , the geometry is dominated by the D3-branes and the D7-branes are approximatelyprobe branes. In the strict N c → ∞ limit, which comes with the supergravity description of AdS / CFT , this approximation becomes exact. This is analogous to the so-called quenched approximation in lattice
QCD , where the action ofthe gauge bosons on the matter field is included, while the action of the matter on the bosons isneglected.The metric of AdS × S can be written as ds = r L η µν dx µ dx ν + L r ( d~y + d~z )= r L η µν dx µ dx ν + L r dr + L d Ω , (3.1)where the index conventions as well as the embedding of the D7-branes have been summarisedin Table 4. The multiplication of vectors is supposed to denote contraction with a Euclideanmetric, that means d~y = P , , , dy m dy m , d~z = P , dz i dz i . There are three qualitativelydifferent types of directions: x denote the world volume coordinates of the D3s, y the coordinatestransversal to the D3s and longitudinal to the D7s, and z the coordinates transversal to bothkinds of branes. Since y and z are on the same footing in the metric, assigning z to the 8 , ≃ SU(4) R isometry group to SO(4) × SO(2) ≃ SU(2) L × SU(2) R × U(1) R , where the orthogonal groups represent rotational invariance in the It should be noted that meanwhile there are supergravity solutions that include the backreaction of theD7-branes [85]. coordinates y and z , respectively. In the case of coincident D3 and D7 branes, the hypermultipletstemming from the strings stretched between the two stacks is massless, such that there is noclassical scale introduced into the setup and conformal symmetry is maintained in the strict probelimit. Then the R-symmetry of the field theory is SU(2) × U(1) R .When separating the stacks in the z -plane, the SO(2) , ≃ U(1) R group is explicitly broken,though one may use the underlying symmetry to parametrise this breaking as z = 0 , z = ˜ m q . (3.2)Since this introduces a scale into the setup, namely a hypermultiplet mass m q = ˜ m q / (2 πα ′ ), it isnot to be expected that conformal symmetry, and hence AdS isometry, can be maintained. TheR-symmetry of the field theory becomes SU(2) R only, which is in accordance with the geometricsymmetry breaking above.Indeed, the induced metric on the D7s reads ds = y + ˜ m q L η µν dx µ dx ν + L y y + ˜ m q d~y = y + ˜ m q L η µν dx µ dx ν + L y + ˜ m q dy + L y y + ˜ m q d Ω , (3.3)which towards the boundary at | y | → ∞ , with y ≡ | y | := ~y~y , approximates AdS × S , reflectingthe fact that a quark mass term is a relevant deformation that is suppressed in the ultraviolet.This is in accordance with the usual picture of the radial direction r = q y + ˜ m q of the AdS space describing the energy scale of the field theory, where approaching the interior of
AdS fromthe boundary corresponds to following a renormalisation group flow from the ultraviolet ( UV ) tothe infrared ( IR ).When the renormalisation scale is lowered below the quark mass, the quarks should drop outof the dynamics. This happens when reaching the radius r = ˜ m q in the ambient space, whichcorresponds to the interior of the D7s at y = 0, where the D7-branes stop from a five dimensionalperspective, although as depicted in Figure 2 there is no boundary associated to this ending.When ˜ m q = 0, the U(1) R and SO(2 , AdS symmetry are restored and the D7s fill the wholeof the ambient AdS , which suggests that conformal symmetry is restored. However, this is onlytrue in the strict probe limit, as otherwise contributions to the beta function of order N f /N c occur[20, 24]. 3.3 Analytic Spectrum3.3.1 Fluctuations of the ScalarsThe spectrum of the undeformed D3/D7 system described above admits analytic treatment atquadratic order [24] and therefore sets the baseline for the numerical determination of mesonspectra in the more complicated setups of the following Chapters.From equations (2.14), (2.15) and (2.17) the D7-brane action in a background of D3-branesreads S D = − T Z d ξ p − det( P [ G ] ab + (2 πα ′ ) F ab )+ 2 πα ′ T Z P [ C ] ∧ F ∧ F, (3.4) C = r L dx ∧ · · · ∧ dx , (3.5) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) PSfrag replacements AdS D7 S rr = ˜ m q Fig. 2
The D7-brane wraps an S on the internal S which slips towards a pole and shrinks to zerosize. From the five dimensional point of view, the brane terminates at a certain radius, but there is noboundary associated to this ending. (Figure taken from [86]) where P is the pullback to the world-volume of the D7-branes and r = y + z .For fluctuations of the scalars, the Wess–Zumino term contributes only at fourth order (with(scalar) · F ). From the action and for an embedding according to z = 0 + (2 πα ′ ) δz ( ξ ) , z = ˜ m q + (2 πα ′ ) δz ( ξ ) . (3.6)the expansion of the action to quadratic order (2.31) yields L = p − det g ab (1 + (2 πα ′ ) g ij g ab ∂ a z i ∂ b z j ) , (3.7)where the fact that metric admits a diagonal form has been used. For the induced D7 metric(3.3), the Lagrangean (3.7) reads at quadratic order2(2 πα ′ ) − L = y p det(ˆ g )[ η µν ( ∂ µ δz )( ∂ ν δz ) (3.8)+ (cid:18) L y + ˜ m q (cid:19) ( ∂ y δz ) + ˆ g αβ ( ∂ α z )( ∂ β z ) + ( z ↔ z )] , with ˆ g αβ the metric on the three sphere and the equation of motion L ( y + ˜ m q ) ∂ µ ∂ µ δz i + y − ∂ y ( y ∂ y δz i ) + y − ˆ ∇ α ˆ ∇ α δz i = 0 , i = 8 , , (3.9)where ˆ ∇ α is the covariant derivative on the unit S . An ansatz for separation of variables δz i ( x µ , y, S ) = ζ i ( y ) e ik · x Y ℓ (S ), with ˆ ∇ α ˆ ∇ α Y ℓ = − ℓ ( ℓ + 2) Y ℓ , ℓ ∈ N yields (cid:20) ∂ y + 3˜ y ∂ ˜ y + ˜ M s (1 + ˜ y ) − ℓ ( ℓ + 2)˜ y (cid:21) ζ i (˜ y ) = 0 , (3.10)˜ y = y ˜ m q , ˜ M s = − k L ˜ m q , (3.11) where a rescaling has removed all explicit scale dependencies. Requiring regularity at the origin,the radial equation (3.10) can be solved uniquely in terms of a hypergeometric function, ζ i ( y ) = y ℓ ( y + ˜ m q ) n + ℓ +1 2 F (cid:0) − ( n + ℓ + 1) , − n ; ℓ + 2; − y / ˜ m q (cid:1) ,M s = − k = 4 ˜ m q L ( n + ℓ + 1)( n + ℓ + 2) , (3.12)with the discretisation condition n ∈ N from normalisability. Note that the spectrum becomesdegenerate in the conformal ˜ m q → UV behaviour is determined from (3.12), but one may instead simplydiscuss the radial equation (3.10), which for large ˜ y becomes approximately (cid:20) ∂ y + 3˜ y ∂ ˜ y − ℓ ( ℓ + 2)˜ y (cid:21) ζ i (˜ y ) = 0 . (3.13)Its solutions are of the form ζ i (˜ y ) = A ˜ y ℓ + B ˜ y − ℓ − , which contradicts the na¨ıve AdS / CFT expec-tation of ˜ y ∆ − + ˜ y − ∆ as can be seen from taking the sum of the exponents. This is due to theappearance of a determinant factor √− det g ab ∼ ˜ y , which imposes a non-canonical normalisa-tion on the kinetic term. So the generic behaviour should be ˜ y p +∆ − + ˜ y p − ∆ and subtracting theexponent of the non-normalisable solution, which corresponds to a field theory source, from thatof the normalisable one, which corresponds to a vacuum expectation value, it can be seen that − ( ℓ + 2) − ℓ = ( p − ∆) − ( p + ∆ −
4) = −
2∆ + 4= ⇒ ∆ = ℓ + 3 . (3.14)3.3.2 Fluctuations of the Gauge FieldsThe equations of motion for the gauge fields read ∂ a ( p − det g cd F ab ) − ρ ( ρ + ˜ m q ) L ε bβγ ∂ β A γ = 0 , (3.15)with ε αβγ taking values ±
1, and 0 when the free index b is none of the angular S directions.Expanding the equation of motion yields (cid:20) ( g xx ) − ∂ µ ∂ µ + y − ∂ y ( y ( g yy ) − ∂ y ) + ˜ ∇ α ˜ ∇ α (cid:21) A ν − ∂ ν (cid:20) ( g xx ) − ∂ µ A µ + y − ∂ y ( y ( g yy ) − A y ) + ˜ ∇ α A α (cid:21) = 0 , (3.16) (cid:20) ( g xx ) − ∂ µ ∂ µ + ˜ ∇ α ˜ ∇ α (cid:21) A y − ∂ y (cid:20) ( g xx ) − ∂ µ A µ + ˜ ∇ α A α (cid:21) = 0 , (3.17) (cid:20) ( g xx ) − ∂ µ ∂ µ + y − ∂ y ( y ( g yy ) − ∂ y ) + ˜ ∇ α ˜ ∇ α (cid:21) A δ − ∂ δ (cid:20) ( g xx ) − ∂ µ A µ + y − ∂ y ( y ( g yy ) − A y ) + ˜ ∇ α A α (cid:21) − C ′ ˜ g δα ε αβγ ∂ β A γ = 0 , (3.18)each of which has to be satisfied for a particular ansatz. For the components ( A µ , A y , A α ), thefirst two should transform under SO(4) as scalars, while the last should transform as a vector and accordingly be built up from vector spherical harmonics. The simplest choice is ˜ ∇ α Y ℓ , whichtransforms in the ( ℓ , ℓ ). The other two possibilities are Y ℓ, ± α , which transform in the ( ℓ ± , ℓ ∓ )and obey ˜ ∇ Y ℓ, ± α − δ βα Y ℓ, ± β = − ( ℓ + 1) Y ℓ, ± α , (3.19) ε αβγ ˜ ∇ β Y ℓ, ± γ = ± ( ℓ ± Y ℓ, ± α , (3.20)˜ ∇ α Y ℓ, ± α = 0 . (3.21)The modes containing Y ℓ, ± should not mix with the others since they are in a different represen-tations. The following types of solutions can be obtained:Type I ± A α = φ ± I ( y ) e ikx Y ℓ, ± α , A µ = A y = 0 , (3.22a)Type II A µ = ξ µ φ II ( y ) e ikx Y ℓ , A y = A α = 0 , k µ ξ µ = 0 , (3.22b)Type III A y = φ III ( y ) e ikx Y ℓ , A α = ˜ φ III ( y ) e ikx ˜ ∇ α Y ℓ . (3.22c)Type II and III come from recognising that in the gauge ∂ µ A µ = 0, A µ does not appear in (3.17)and (3.18), and can therefore be treated independently. Kruczenski et al. argue that modes notsatisfying the gauge condition are either irregular or have a polarisation parallel to the wave vector k ; i.e. can be brought to the gauge ∂ µ A µ = 0.The simplest radial equation arises from the ansatz II, (cid:20) ∂ y + 3˜ y ∂ ˜ y + ˜ M II (1 + ˜ y ) − ℓ ( ℓ + 2)˜ y (cid:21) A a = 0 . (3.23)Up to the polarisation vector, this is the same equation as (3.9) and therefore produces a degen-eracy of the mass spectrum,˜ M II = ˜ M s = 4( n + ℓ + 1)( n + ℓ + 2) , n, ℓ ≥ , (3.24)with the same conformal dimension ∆ = ℓ + 3.For type III and I ± , an analogous calculation yields the mass formulae and conformal dimen-sions of the corresponding UV operators,˜ M I + = 4( n + ℓ + 2)( n + ℓ + 3) , ∆ = ℓ + 5 ℓ ≥ , (3.25)˜ M I − = 4( n + ℓ )( n + ℓ + 1) , ∆ = ℓ + 1 ℓ ≥ , (3.26)˜ M III = 4( n + ℓ + 1)( n + ℓ + 2) , ∆ = ℓ + 3 ℓ ≥ , (3.27)with n ≥ N = 2 supermultiplets have been added. Since the SU(2) L group commuteswith the supercharges, all states in the same supermultiplet should be in the same representationwith respect to the left quantum number. Indeed redefining ℓ in such a manner that the SU(2) L representations are the same also makes the mass coincide. This argument cannot be applied to theright quantum number, for the supercharges are not singlets under the R -symmetry. (Althoughthe spectrum is symmetric under swapping the rˆoles of the left and right group, which correspondsto considering an anti-D7-brane, that is the opposite sign in front of the Wess–Zumino term.)3.4 Operator MapAs has been seen, the fluctuation modes of the D7-brane organise themselves in N = 2 multiplets,which are made of a chiral primary field and descendants. The mode with highest SU(2) R quantum IIB SUGRA
Particle Content
Type SU(2) R U(1) R ∆ − ℓ ℓ +22 ℓ ≥ ℓ ℓ ≥ ℓ ℓ ≥ ℓ ℓ ≥ ℓ − ℓ ≥ ℓ +12 ℓ ≥ ℓ − ℓ ≥ Table 5
Mesonic Spectrum in AdS × S . The Dirac fermions are deduced from Supersymmetry. ∆ isthe conformal dimension of the corresponding UV operator and the representations have been shifted tohave the same SU(2) L spin ℓ and therefore the same mass ˜ M = 4( n + ℓ + 1)( n + ℓ + 2), n ≥ number is the scalar of type (I-). The choice of the corresponding primary operator is restricted bythe requirement of containing exact two hypermultiplet fields in the fundamental representation,being in the same representation ( ℓ , ℓ +22 ) and having conformal dimension ∆ = ℓ + 2. For ℓ = 0this merely admits the unique combination O I = ψ α σ Iα ˙ β ¯ ψ ˙ β , (3.28)with the Pauli matrices σ I . The higher chiral primary in the Kaluza–Klein tower, can be obtainedby including the adjoint operators obtained a the subset Y , , , of the six adjoint scalars of the N = 4 multiplet by traceless symmetrisation, χ ℓ = Y ( i , . . . Y i ℓ ) . (3.29)The operators χ ℓ transforms under SU(2) L × SU(2) R × U(1) R as ( ℓ , ℓ ) , which in the combination O Iℓ = ψχ ℓ σ I ¯ ψ, (3.30)gives a ( ℓ , ℓ +22 ) of conformal dimension ∆ = ℓ + 2. The other operators can be obtained fromacting with supercharges on those chiral primaries. AdS / CFT correspondence, it has proveddifficult to find a realistic holographic dual of
QCD . There are many reasons, which range frompractical—working with ten-dimensional supergravity equations—to principle: The ultraviolet( UV ) regime is weakly coupled, which corresponds to strong coupling (large curvature) on the AdS side and hence the requirement of quantising string theory on that background. Furthermoremodels discussed so far contain only one scale and cannot provide a separation of supersymmetry(
SUSY ) breaking and confinement scale Λ
QCD . The four scalars belong to the N = 2 hypermultiplet. Despite those obstacles
AdS / CFT correspondence has been remarkably successful in capturingmany aspects of
QCD . In this Chapter, such an aspect will be the important feature of chiral sym-metry breaking, which shall be described holographically. Since supersymmetry prohibits chiralsymmetry breaking as a non-vanishing chiral
VEV violates D-flatness, the background geometryhas to be deformed in such a way that
SUSY is broken. At the same time it is desirable not to loosecontact to the well tested framework of
AdS / CFT . It is therefore crucial to look at a geometrythat in the ultraviolet approaches AdS × S .Here this will be achieved by preserving in the whole space time an SO(1 , × SO(6) isometry.There are two
IIB supergravity backgrounds in the literature [46, 40, 39], which satisfy this condi-tion. The implications of the background by Constable–Myers [46] have been studied in [86]. Herethe focus shall be on the background found by Kehagias–Sfetsos and (independently) Gubser.In analogy to the undeformed case of the previous Chapter, a D7-brane embedding parallel tothe D3s will be considered and its scalar and vector fluctuations be studied. By diagonalising thefields, the discussion of multiple D7-branes reduces to several identical copies of the single branecase and has therefore no impact on the mass spectrum. There is however the important differencethat a D7-brane stack admits non-trivial gauge configurations such that the Wess–Zumino term C ∧ F ∧ F can contribute. The effect of non-trivial F ∧ F will be studied in the next Chapter, theWess–Zumino term will be assumed to vanish for now and an Abelian Dirac–Born–Infeld action( DBI ) can safely be considered therefore.As has been explained in Section 2.5, the quark mass m q and chiral quark condensate c formthe source/ VEV pair that is described by the UV values of scalar fields on the brane. (Which inthe string picture describe the transversal position of the brane.) In the supersymmetric scenario,the only solutions that have a field theoretic interpretation require c = 0 for all m q . In particular,this implies that there is no chiral quark condensate in the limit m q → AdS . Sincethe radial direction of the
AdS space corresponds to the energy scale in the field theory, sucha bending means that the field theory flows to the IR and comes back as is shown (“Bad”) inFigure 3. Clearly this is an unphysical behaviour. The effect of the deformed background is thatthe D7-brane experiences attraction from the singularity and bends inward compensating the effectof the boundary value c . This compensating is highly sensitive to the exact value of the chiralcondensate as a function of the quark mass, which completely fixes the functional dependence.In the previous Chapter, it was explained how adding D7-branes to the AdS / CFT correspondencebreaks the SO(6) ... ≃ SU(4) R isometry of the six D3 transversal coordinates to an SO(4) × SO(2) isometry, which corresponds to SU(2) L × SU(2) R × U(1) R , with SU(2) R × U(1) R theR-symmetry group of the N = 2 superconformal Yang–Mills theory. Giving a mass to the N = 2 hypermultiplet corresponds to separating the two brane stacks and breaking the conformalsymmetry. This has two effects: On the field theory side, the breaking of conformal symmetryreduces the R-symmetry to SU(2) R , on the supergravity side it breaks the rotational invariance inthe 8 , R . Now this breaking acquires an additional interpretation inthe limit m q →
0, where this U(1) is present in the UV , but is broken dynamically by the branesbending away from the symmetry axis, cf. Figure 4: The symmetry spontaneously broken by thechiral condensate is the U(1) A axial symmetry.Since determining the chiral symmetry breaking behaviour is equivalent to finding correctD7-brane embeddings, one may go one step further and also find fluctuations about these embed-dings, which corresponds to meson excitations in the correct field theoretic vacuum. For vanishing To be precise, the SO(2) corresponds to the U(1) A axial symmetry, while the U(1) R R-symmetryis diag[SO(2) × SO(2) × SO(2) ]. Breaking of SO(2) implies breaking of the axial and R-symmetrysimultaneously. z 0 Singularity yGoodBadUgly Fig. 3
Possible solutions for the D7 embeddings. The half circles correspond to constant energy scale µ . The “bad” solution cannot have an interpretation as a field theoretic flow. The “ugly” solution hitsthe singularity (filled circle at the centre) and can thus not be relied on. (Plot taken from [86]) Fig. 4
Spontaneous breaking of the U(1) A symmetry, which rotates (circle) the 8 , y -axis (large axis) vanishes. Non-vanishing quark mass means explicit and therebynot spontaneous symmetry breaking.0 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings quark mass, the bilinear quark condensate breaks the axial symmetry spontaneously and the as-sociated meson becomes massless providing a holographic version of the Goldstone theorem.It should be noted that the explicit breaking of the U(1) A by an instantonic anomaly, whichin QCD is responsible for the η ′ to be heavy, is suppressed in the large N c limit. In that sensethe holographic η ′ is more similar to a Pion even though it is not related to the breaking of thechiral SU( N f ) L × SU( N f ) R to its diagonal subgroup. Therefore in particular for comparison withexperimental data the Pion mass is a more appropriate choice.This Chapter is organised as follows. First, the Dirac–Born–Infeld ( DBI ) action and the equa-tions of motion describing the D7-brane embedding and fluctuations about the vacuum solutionwill be derived. Then the background by Gubser, Kehagias–Sfetsos (
GKS ) will be shortly reviewedand transformed into a convenient coordinate system. The undeformed supersymmetric scenariowill be compared with a numerical evaluation of the chiral symmetry breaking and meson spec-trum in the dilaton deformed background. Additionally, the behaviour of strongly radially excitedmesons will be discussed.4.2
DBI to Quadratic OrderConsider the diagonal background metric ds = g xx ( y, z ) dx , + g yy ( y, z )( dy + y d Ω ) + ˆ g zz ( y, z )( dz + z dθ ) , (4.1)which may be written as g (10) = diag( g xx , , g yy , g yy y ˜ g αβ , ˆ g zz , ˆ g θθ ) , (4.2)where ˜ g αβ is the metric on the unit three sphere, and it holdsˆ g θθ = z ˆ g zz . (4.3)In the case g yy = ˆ g zz , the radial direction of the warped AdS space can be expressed as r = y + z = ~y + ~z with y, Ω y , . . . , y and z, θ z = z sin θ, z = z cos θ a transformationfrom respectively spherical or polar to Cartesian coordinates.Choosing static gauge, x ,..., = ξ ,..., , y ,..., = ξ ,..., , z ,..., = φ , ( ξ ,..., ) , (4.4)the DBI action in Einstein frame for a D7-brane in this background is given by S = Z d x dy d Ω √− g e ϕ (cid:20) g zz g ab ( ∂ a Φ)( ∂ b ¯Φ)+ e − ϕ F ab F ab (cid:21) (4.5)where expansion to second order in the scalar fields Φ , ¯Φ = φ ± iφ and field strength has beenperformed. The remaining determinant is √− g = y p ˜ g ( g xx g yy ) . (4.6)4.3 Quadratic FluctuationsExpanding an action S = Z d ξ L ( φ i , ∂ a φ i ) (4.7) into small fluctuations δφ i around a solution φ of the Euler–Lagrange-equations yields φ i = φ i + εδφ i , (4.8) S = Z d ξ L + 12 ε (cid:20) ∂ L ∂ ( ∂ a φ i ) ∂ ( ∂ b φ j ) (cid:21) ε =0 ( ∂ a δφ i )( ∂ b δφ j ) . (4.9)Note that the above statement is merely the Legendre criterion for an extremal solution of avariational principle, which is a minimum if the parenthesised expression above is positive definite.In accordance with the previous Chapter, where dependence on x was associated to massiveexcitations and dependence on the spherical coordinates Ω gave rise to Kaluza–Klein states, theembedding of the D7 that forms a ground state should only depend on the radial direction y . Forfluctuations about a vacuum solution φ = φ ( y ), F ab ≡
0, the quadratic expansion in scalar andvector fluctuations yields S = Z d x dy d Ω √− g e ϕ q | φ ′ ( y ) | (cid:20) (cid:18) g ab ˆ g ij | φ ′ ( y ) | (cid:19) ( ∂ a δφ i )( ∂ b δφ j ) (4.10) − (cid:18) g yy ˆ g ij ( ∂ y φ i )( ∂ y δφ j )1 + | φ ′ ( y ) | (cid:19) + 14 F ab F ab | φ ′ ( y ) | (cid:21) , with √− g = y ( g xx g yy ) p ˜ g, | φ ′ ( y ) | := ˆ g ij g ab ( ∂ a φ i )( ∂ b φ j ) , (4.11)and F ab F ab expressed solely in terms of fluctuations δA m about the trivial background A m ≡ z , z has someadvantages. From the field theoretic point of view, expressing the fluctuations in polar coor-dinates z e iθ = z + iz is more natural, because the fluctuations of the pseudo-Goldstone modecorrespond then exactly to rotations of the U(1) A . Since both approaches yield the same resultsdue to the infinitesimal nature of the fluctuations, the polar coordinate formulation will be chosenhere.For Φ = z e iθ , z = z ( y )+ δσ ( ~x, ~y ) and θ = 0+ δπ ( ~x, ~y ), expansion of the DBI action to quadraticorder in the fluctuations yields S = Z d x dy d Ω p ˜ g e ϕ y ( g xx g yy ) q z ′ ( y ) (cid:20) g ab ˆ g θθ ( ∂ a δπ )( ∂ b δπ ) + g ab ˆ g zz ( ∂ a δσ )( ∂ b δσ )1 + z ′ ( y ) − (cid:18) g yy ˆ g zz ( ∂ y z )( ∂ y δσ )1 + z ′ ( y ) (cid:19) + 14 F ab F ab | z ′ ( y ) | (cid:21) , (4.12)where ( z ′ ) = ˆ g zz g yy ( ∂ y z ) . In particular, the excitation number n of the meson tower (3.12) corresponds to the number of zeros of thesolution to the radial equation (3.10), which provides a good check whether a meson solution was accidentallyskipped. δσ = δπ = 0, the equation describing the D7 embedding in terms of z ( y )is obtained, ddy (cid:20) g yy g zz F ( y, z ) p z ′ ( y ) z ′ ( y ) (cid:21) = g yy g zz q z ′ ( y ) ∂∂z F ( y, z ) , F = e ϕ y ( g xx g yy ) . (4.13)4.4.2 Pseudoscalar MesonsThe pseudoscalar mesons correspond to fluctuations along the U(1) A and—as shall be seen below—become massless for vanishing quark mass. They are thus (pseudo-) Goldstone bosons, whichbecome true Goldstones for m q →
0. Their equations of motion are ∂ a (cid:20) p | ˜ g |F ( y, z ) p z ′ ( y ) ˆ g θθ g ab ∂ b δπ (cid:21) = 0 , (4.14)which for the ansatz δπ = δπ ( y ) e ik · x Y ℓ (S ) and M π = − k read p z ′ ( y ) F ∂ y (cid:20) F g yy ˆ g θθ p z ′ ( y ) ∂ y δπ (cid:21) (4.15)+ (cid:20) M π ˆ g θθ g xx − ℓ ( ℓ + 2) ˆ g θθ g yy y (cid:21) δπ = 0 , with the same shorthand F as in (4.13).4.4.3 Scalar MesonsThese correspond to fluctuations in the radial direction transverse to the U(1) A . The equationsof motion for the scalar mesons are ∂ a (cid:20) p | ˜ g |F ( y, z ) p z ′ ( y ) ˆ g zz g ab ∂ b δσ (cid:21) = ∂ y (cid:20) p | ˜ g |F ( y, z ) p z ′ ( y ) (ˆ g zz g yy ) ( ∂ y z ) ∂ y δσ (cid:21) , (4.16)which for the ansatz δσ = δσ ( y ) e ikx Y ℓ (S ) and M σ = − k become p z ′ ( y ) F ∂ y (cid:20) F ˆ g zz g yy p z ′ ( y ) (cid:18) − ˆ g zz g yy z ′ ( y ) z ′ ( y ) (cid:19) ∂ y δσ (cid:21) (4.17)+ (cid:20) ˆ g zz g xx M σ − ℓ ( ℓ + 2) ˆ g zz g yy y (cid:21) δσ = 0 . Again it holds F ( y, z ) = e ϕ y ( g xx g yy ) .4.4.4 Vector MesonsIn accordance with Section 3.3.2, vector mesons can be obtained from the D7-brane gauge fieldswhose equations of motion are ∂ a (cid:20) √ ˜ g y ( g xx g yy ) p z ′ ( y ) F ab (cid:21) = 0 (4.18) for solutions with no components on the S , δA α = 0. The ansatz δA ν = ξ ν δρ ( y ) e ik · x Y ℓ (S ),where the polarisation vector ξ ν satisfies k µ ξ µ = 0, yields p z ′ ( y ) y ( g xx g yy ) ∂ y (cid:20) y g xx g yy p z ′ ( y ) ∂ y δρ (cid:21) (4.19)+ (cid:20) ( g xx ) M ρ − ℓ ( ℓ + 2) y (cid:21) δρ = 0 . × S In this Section, it is demonstrated that the holographic description of the undeformed, supersym-metric case [24] shows no chiral symmetry breaking. To describe the field theoretic vacuum, theembedding should neither depend on x , which gives rise to a massive excitation, nor on the coor-dinates of the internal S , which gives rise to Kaluza–Klein states. Using the SO(2) symmetry,one may choose the coordinate system such that the embedding is simply z = z ( y ).Then the linearised equation of motion (3.10) is given by (cid:20) ∂ y + 3˜ y ∂ ˜ y (cid:21) z ( y ) = 0 (4.20)with ˜ M = ℓ = 0. The full (as opposed to only asymptotic) solutions are of the form z ( y ) = m + c y − , (4.21)with the conformal dimension of the dual operator ¯ ψψ given by ∆ = ℓ + 3 = 3. For c = 0, this isthe constant embedding chosen by Kruczenski et al. [24] and presented in the previous Chapter.For c = 0, the solution diverges when approaching the centre y → r = y + z ( y ) , whichcorresponds to the energy scale. This is also depicted as the “bad” solution in Figure 3.4.5.2 GKS
GeometryA particular solution to the type
IIB supergravity equations of motion (4.1) that preserves SO(1 , × SO(6) isometry was found by Gubser [40] (and independently by Kehagias–Sfetsos [39]), who chosean appropriate warped diagonal ansatz for the metric, a Freund–Rubin ansatz for the five-formflux and took only the dilaton as a non-constant supergravity field with a radial dependence.The solution presented in [40, 39] takes the form ds = e σ dx , + L dσ B e − σ + L d Ω , (4.22) ϕ − ϕ = r
32 arcoth p B − e σ , (4.23) Keep in mind that these are only solutions expanded to quadratic order. For the Abelian case one can dobetter, expand the determinant to all orders and keep the square root unexpanded. However, the outcome doesnot change. In the original publication B /
24 is used instead of B to parametrise the deformation. where due to arcoth x = ln x +1 x − the dilaton ϕ may be written as ϕ − ϕ = r
38 ln √ B − e σ + 1 √ B − e σ − . (4.24)These coordinates are such that IR σ → −∞ singularity in the far interior, UV σ → + ∞ boundary,where there is a naked singularity in the infrared.For calculating the meson spectrum in a background, it is more convenient to work in a coor-dinate system that brings the metric exactly to the SO(1 , × SO(6) manifest form (4.1). Thiscan be achieved by the coordinate transformatione σ = s B r r r − r r , (4.25)which yields ds = g xx ( r ) dx , + g yy ( r )( dr + r d Ω ) ,g xx ( r ) = r L r − r r ,g yy ( r ) = g zz = L r ,ϕ − ϕ = r
32 ln r + r r − r . (4.26)Note that additionally x has been rescaled such that g xx reproduces the canonical normalisationof the asymptotic AdS that is approached for r → ∞ and r is the minimum value of r where theinfrared singularity resides.For computations it is convenient to rescale the coordinates by r such that effectively r r . In this frame the quark mass is measured in units of r T , with T the string tension, and the meson mass in units L − r . As will be shown below, forlarge quark masses the supersymmetric results of the undeformed AdS × S are reproduced, suchthat M ∼ m q . Due to M L r = const. · (2 πα ′ ) m q r (4.27)the supersymmetric limit r → GKS
For (4.26), the equation of motion (4.13) for the vacuum solution z = z ( y ) is given by ddy (cid:20) y f p z ′ ( y ) z ′ ( y ) (cid:21) = y q z ′ ( y ) ∂∂z f, (4.28) f = ( r + 1) (1+∆ / ( r − (1 − ∆ / r , r = y + z ( y ) , ∆ = √ . PSfrag replacements0 . . . . . . . . . . m q = 0 . c = 1 . m q = 0 . c = 1 . m q = 0 . c = 1 . m q = 1 . c = 0 . m q = 1 . c = 0 . m q = 2 . c = 0 . z ( y ) r y/r Fig. 5
The Figure shows regular D7 embeddings with different quark mass. The embedding coordinate z ( y ) is a radial coordinate in the 8 , A symmetry inthat plane, the zero quark mass solution (dashed) does so spontaneously . The constant ∆ has been defined for convenient comparison to a background by Constable–Myers,cf. Chapter 6, and should not be mixed up with the conformal dimension.Since the background (4.26) approaches AdS × S towards the boundary, it does not come asa surprise that the UV behaviour of z ( y ) is given by m q + cy − with m q the quark mass and c the bilinear quark condensate as in the supersymmetric case. In the infrared there are still twosolutions of qualitatively different behaviour: One is divergent and cannot correspond to fieldtheoretic vacuum therefore, the other approaches a constant. However, the infrared dynamics ismodified such that the pair in the UV mixes while going to the IR . Whereas in the supersymmetriccase the UV solution with c = 0 corresponded one-to-one to the regular behaviour in the IR , nowfor each value of m q there is only one value of c such that the combined solution mixes into aregular one in the IR . Such regular solution have been determined numerically and are plottedin Figure 5. Each of the solutions is determined by a pair of quark mass and quark condensate.These pairs also determine the quark condensate as a function of the quark mass as is shown inFigure 6.The possible outcomes for arbitrary combinations of m q and c are depicted in Figure 3: Thesolution can hit the singularity (denoted “ugly”, since the supergravity approximation fails whencoming to close to the singularity), the solution may diverge (denoted “bad”, because it cannotcorrespond to a field theoretic flow), or the solution may reach a constant value for the embeddingcoordinate z ( y ) at y = 0, denoted “good”. In terms of the ambient space radial coordinate r = y + z the D7-brane “ends” at r = z (0) by the S slipping from S of the backgroundand shrinking to zero size at a pole of the S , cf. Figure 2. There is a tachyon associated to thisslipping mode, but its mass obeys (saturates) the Breitenlohner–Freedman bound [21] and doesnot lead to an instability hence.One might however worry about regular solutions reaching the singularity. For the discussionof whether this may happen, it is advantageous to shift the point of view to the infrared.Starting at a finite value in the IR , there has to be a unique flow to the UV , which fixes thecorrect combination of m q and c , since one also needs the IR - divergent solution to create arbitrarycombinations of m q and c . As has been explained above, z (0) sets the scale were the quarks PSfrag replacements0 . . .
75 101 . . . .
00 2 4 6 8 cr (2 πα ′ ) m q r (a) PSfrag replacements 1056 67 78 8 9 r c (2 πα ′ ) m q r (b) Fig. 6
The first plot shows the chiral condensate h ¯ ψψ i as a function of the quark mass m q as determinedby regularity requirements for the D7 embedding. For large quark mass m q the chiral condensate behaveslike c ∼ m q in accordance with predictions from effective field theory.7 PSfrag replacements 0 . − . . − . z ( y ) r yr Fig. 7
Two solutions of the same quark mass and the zero quark mass solution (dashed) are depicted.The zero mass solution exactly avoids the region between the inner circle, which is the singularity, anddashed outer “shielding” circle. Of the two massive solutions, only the one with the larger action entersthe shielded region, cf. fig. 8. drop out of the dynamics. So one generically may expect that a large quark mass correspondsto a large value of z (0). Starting at distances closer to the singularity generates solutions withsmaller quark mass till one reaches a limiting solution at z (0) ≈ .
38 that corresponds to vanishingquarks mass. Going even closer to the singularity gives rise to a spurious negative quark mass.Due to the SO(2) present, these solutions are in fact positive mass solutions with negativequark condensate, as can be seen by rotating around the y -axis, see Figure 4. This assigns twopotentially valid solutions to each positive quark mass. However solutions that do not come closerto the singularity than the zero quark mass solution have a smaller potential energy V = − L ,cf. Figure 8, and are therefore physical. This realises some sort of screening mechanism preventingsolutions from entering the region between the zero-quark mass solution and the singularity,cf. Figure 7. The physical solutions outside have a positive quark condensate.Having established the conditions that determine the chiral condensate as a function of thequark mass—the result is plotted in Figure 5—the case m q = 0 will be discussed in more detailnow. z ( y ) ≡ IR , thus forcing the solution to behave as desired, or tostart with an infinitesimal deviation from z ≡ m q = 0 in the UV . This situation is analogous tocalculations of the magnetisation in solid state physics, where spontaneous symmetry breaking isinitiated by an arbitrarily small but non-vanishing external B field.The conclusion is that indeed spontaneous chiral symmetry breaking is observed in this geom-etry and one may wonder about the appearance of an associated Goldstone mode. The situation is to some extent analogous to asking which is the shortest route connecting two points on asphere. The answer is a grand circle, which however also provides the longest straight route.
PSfrag replacements 0 . . . . − . . . . − . . − . πα ′ ) m q r E ( m q ) − E (0) h ¯ ψψ i > h ¯ ψψ i < Fig. 8
Potential Energy of D7-brane embeddings as function of the quark mass: Since the energy itselfis infinite what is actually plotted is the finite difference to the action of the zero quark mass solutiondefined as follows, E ( m q ) − E (0) = − ∆ S = − lim Y →∞ Y Z L ( m q ) − L ( m q = 0) dy. The physical solutions have smaller energy and are farther from the singularity than the zero-mass solution. c + c /y . In contrast to the embedding solutions, where regularityfixed c as a function of c , the fluctuations should always be normalisable, such that the solutionsbehave as y − towards the boundary. The remaining overall factor c is undetermined becausethe equations of motion are linear. The requirement of regularity in the infrared can then only besatisfied by a discrete set of values for the meson mass M , which determines the spectrum. Theresult for the lowest lying meson modes is depicted in Figure 9.With these results it is possible to return to the question of a holographic realisation of Gold-stone’s theorem. For the following discussion, it is important to keep in mind that the supergravityapproximation in AdS / CFT correspondence implies being in the N c → ∞ limit, where the U(1) A axial symmetry is non-anomalous in the field theory. A look at the large N c limit of QCD , wherethe η ′ becomes massless and thus a true Goldstone boson, inspires to look for the corresponding(pseudo-)Goldstone meson in this geometric setup.A massless embedding with UV behaviour z ( y ) ∼ c y − restores the U(1) A ≃ SO(2) symmetryin the UV and therefore shows spontaneous symmetry breaking. In particular that means that theembedding solution z e iθ , has an undefined angle θ at the boundary, which acquires an arbitraryvalue along the flow, picked out spontaneously by the dynamics. Clearly any fluctuation in the θ angle simply corresponds to a rotation into an—because of the presence of the U(1)—equivalent but different value of θ . Since these values are all equivalent, the fluctuation in the θ directionshould be a flat direction in the potential and correspond to a massless meson.When the U(1) A symmetry is explicitly broken in the UV by the quark mass ( z ∼ m q + c ( m q ) y − ), fluctuations in the angular direction do not rotate into an equivalent embedding andare therefore expected to become massive. Figure 9 shows that this holographic version of theGoldstone theorem is indeed realised. Furthermore beyond a certain quark mass, supersymmetry isrestored and the meson masses become degenerate. For small quark mass, Figure 9(b), accordancewith a prediction from effective field theory is found, the Gell-Mann–Oakes–Renner ( GOR ) relation[87] M π = m q (cid:10) ¯ ψψ (cid:11) N f f π . (4.29)4.8 Highly Excited MesonsIn this Section inspired by a similar analysis in [41], the highly excited meson spectrum in thepresent background shall be investigated. In AdS / CFT this corresponds to considering mesons withlarge radial excitation number n . According to [42] the semi-classical approximation becomingvalid in this limit gives rise to a restoration of chiral symmetry, because its breaking resulted fromquantum effects at one-loop order which are suppressed for S ≫ ~ .[41] found the rather generic behaviour M n ∼ n, n ≫ , (4.30)for holographic duals of QCD -like theories. This is not in accordance with field theoretic expec-tations [43], which can be derived from simple scaling arguments: The length of the flux tubespanned between two ultra-relativistic quarks of energy E = p = M n / L ∼ M n Λ QCD , (4.31)such that from the quasi-classical quantisation condition Z p dx ∼ p L ∼ M n Λ QCD ∼ n, (4.32)one reads off M n ∼ √ n. (4.33)This is in contradiction to the results (4.30) and also the numerical results one obtains for the GKS background shown in Figure 10. However this behaviour was to be expected since it is alsofound in the analytic spectrum of Kruczenski et al.A to some extent related question is whether the difference δM n of the scalar and pseudoscalarmeson mass shows the right field theoretic behaviour, which has been predicted to be | δM n | . n − / Λ QCD with alternating sign for δM n [43].While the analytic supersymmetric case fulfils this requirement trivially δM n = 0, interestinglythis seems not to be the case for the GKS background as can be seen in Figure 10. Actually δM n even could not be shown to vanish at all in the limit n → ∞ implying that neither chiral symmetrynor supersymmetry is restored.Having a closer look at the behaviour of such highly excited mesons, cf. Figure 11, one noticesthat the effect of large radial excitation is that the interior of the holographic space correspondingto the field theory’s infrared is probed more densely. This suggests that for highly excited mesons PSfrag replacements 0 . . . .
52 2 . . πα ′ ) m q r M π,σ,ρ L r pseudoscalarscalarSUSYvector (a) Lightest mesons PSfrag replacements 0 . . . . . . . . . πα ′ ) m q r M π L r (b) Pseudoscalar meson and GOR relation: M π ∼ q m q Fig. 9
Plot (a) shows the lightest vector, scalar and pseudoscalar meson (in order of decreasing mass).While the scalar and vector meson retain a mass, the pseudoscalar meson becomes massless and thereforea true Goldstone boson in the limit m q →
0. Furthermore its mass exhibits a square root behaviour aspredicted from effective field theory, plot (b). For large quark masses, supersymmetry is restored and theanalytic
SUSY result M ( n = 0 , ℓ = 0) = m q L √ PSfrag replacements 10 10 1520 20 2530 304050 56070 M π , M σ , M KMMW n (a) Highly Excited Mesons PSfrag replacements 10 151 − − − M M σ − M KMMW M π − M KMMW n (b) Difference to SUSY
Case
Fig. 10
These plots show highly radially excited mesons for the m q = 0 embedding (with r = L = 1 fornumerics). For the analytically solvable SUSY case, this corresponds to n ≫ M KMMW =2 p ( n + ℓ + 1)( n + ℓ + 2) ∼ n . While the proportionality to n is preserved in the deformed case, theoverall slope of the SUSY case is different and has been adjusted by multiplying M KMMW by 1 .
15 forcomparison. The difference to this rescaled mass as depicted in plot (b) suggests that the mass of thescalar and pseudoscalar mesons is not degenerate in the limit of large excitations.2 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings in such a holographic description infrared effects might indeed not be sufficiently suppressed. Onthe other hand it seems surprising that mass degeneracy is not restored contrary to the caseof large quark mass, where the mesons end up in the supersymmetric regime and do becomedegenerate as has been demonstrated in Figure 9.Currently the method for calculating the meson spectrum inherently requires expansion toquadratic order in fluctuations. It would certainly be interesting to extend this procedure toinclude higher order contributions and reexamine the question of whether at least restoration ofmass degeneracy can be achieved in the limit of highly radial excitation. N = 2 super-Yang–Mills ( SYM ) theory (5.3) that can be described by a D3/D7-brane system [20] in the framework of
AdS / CFT correspondence [4, 5, 6] will be determined. The analogous calculations for the Coulombbranch can be performed analytically [24], see Chapter 3, and can be made contact to in the casesof zero and infinite Higgs vacuum expectation value (
VEV ).The work presented here is intrinsically a generalisation of the D3/D7 system of Chapter 3 tothe case of more than one
D7 brane, which corresponds to having multiple quark flavours. Inparticular, an additional effect that goes beyond simply having multiple copies of the Abeliancase is considered. On the supergravity side it arises from the Wess–Zumino term in the D7-braneaction, allowing four-dimensional instanton configurations to be classical solutions of the D7-branegauge fields. On the field theory side this corresponds to switching on a vacuum expectation value(
VEV ) for the fundamental hypermultiplet. The field theory is therefore on the Higgs branch.In the following Sections, the dual field theory will be presented and the exact notion of “Higgsbranch” (which actually is a mixed Coulomb–Higgs branch) will be clarified. A short review ofthe
BPST instanton solution is given.The equations of motions are derived that determine the vector meson spectrum, which iscalculated numerically and discussed analytically in the limits of small and large Higgs
VEV .Finally the operator dictionary is explained and the fluctuations corresponding to scalar mesonsare shown to fall into the same supermultiplets.5.2 ConventionsThe main difference between this Chapter and the preceding ones is the use of a non-AbelianD7-brane action to extend the analysis of the
SUSY
D3/D7 system to the sector of two flavours( N f = 2). Therefore, the introduction of non-Abelian gauge covariant derivatives D a = ∂ a + gA a ,F ab = ∂ a A b − ∂ b A a + g (cid:2) A a , A b (cid:3) , can no longer be avoided and in addition to the index conventions of Table 6, a few notationshave to be established.The indices M, N, . . . will also be used as SU(2) generator indices, with the convention ε = 1and the Hermitean Pauli matrices( T , T , T ) = (cid:18)(cid:18) (cid:19) , (cid:18) − ii (cid:19) , (cid:18) − (cid:19)(cid:19) ,T M T N = iε MNK T K , Tr T M T N = 2 δ MN , PSfrag replacements 0 . − . . − . . − .
003 10020 40 60 80 yζ ( y ) (a) Strong IR dependence PSfrag replacements 0 .
012 0 .
014 0 .
016 0 .
018 0 . − . − . − . yζ ( y ) (b) IR Regularity
Fig. 11
Pseudoscalar meson solution with excitation number n = 49; i.e. the solution plotted in (a) has49 zeros. Most of them concentrate in the far IR , but the solution is still smooth close to the centre (b).Increasing the excitation number scans the IR in more detail, where scalar and pseudoscalar meson massare different. Therefore it is not expected to find mass degeneracy when increasing n further. (Note thatCartesian fluctuations as opposed to polar fluctuations in z and θ have been plotted. The mass spectrumis independent of this choice.)4 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings Coordinates0 1 2 3 4 5 6 7 8 9D3 D7 x µ,ν,... y m,n,... z i,j,... y M,N,... ryX a,b,... X A,B,...
Table 6
Index Conventions which allows to introduce the (anti-Hermitean) quaternion basis σ , , , = ( iT , , , ) . (5.1)The reader shall be reminded that in this basis SO(4) transformations of y m can be alsowritten as y m σ m y m U L σ m U R , (5.2)with U L and U R elements of SU(2) L and SU(2) R respectively. Since the vector (0 , , , y ) isinvariant under transformations U L = ( U R ) − , rotations in the first three coordinates SO(3) can be identified with the diagonal subgroup diag[SU(2) L × SU(2) R ].5.3 Dual Field TheoryOn the string theory side, the setup discussed here is that of a stack of D3-branes and a parallelstack of D7s. In the decoupling limit, this amounts to considering type IIB supergravity (
SUGRA )on AdS × S with N f probe D7-branes, which is dual to an N = 2 gauge theory obtained fromcoupling N f N = 2 hypermultiplets in the fundamental representation to N = 4 SU( N c ) SYM [20].In N = 1 language the Lagrangean of the dual field theory is L = Z d θ Tr (cid:16) ¯Φ i e V Φ i e − V + Q † i e V Q i + ˜ Q i e V ˜ Q i † (cid:17) + (cid:20) g Z d θ W α W α + Z d θ W + c.c. (cid:21) (5.3)where the chiral fields Φ , , and the gauge field V build up the N = 4 adjoint hypermultiplet,which in turn can be split into an N = 2 adjoint hypermultiplet composed of Φ , and an N = 2adjoint gauge multiplet of V and Φ . Q i and ˜ Q i are the N f chiral fields that build up the N = 2fundamental hypermultiplet, and the superpotential is W = Tr( ǫ IJK Φ I Φ J Φ K ) + ˜ Q i ( m q + Φ ) Q i . (5.4) At finite N c this theory is not asymptotically free, and the corresponding string backgroundsuffers from an uncancelled tadpole. However, in the strict probe limit N f /N c →
0, the contribu-tions to the ’t Hooft couplings β function, which scale like N f /N c , are suppressed. Furthermorethe dual AdS string background has no tadpole problem because the probe D7-branes wrap acontractible S . Although contractible, the background is stable, since the tachyon associatedwith shrinking the S satisfies (saturates) the Breitenlohner–Freedman bound [21]. Moreover theAdS × S embedding has been shown to be supersymmetric [88].5.3.1 Higgs BranchIn terms of N = 2 multiplets, the theory consists of an adjoint gauge and hypermultiplet, whichform the N = 4 hypermultiplet of N = 4 SU( N c ) SYM , and N f fundamental hypermultiplets.When the scalars of the latter acquire a VEV , the theory is on the Higgs branch.While the scalars φ , of the adjoint hypermultiplet independently may also have a VEV , VEV sof the N = 2 gauge multiplet’s scalar φ prohibit a VEV for the fundamental hypermultiplets.Refining the discussion for the components gives rise to the mixed Coulomb–Higgs branch. Thesuperpotential in N = 1 language is W = Tr( ǫ IJK Φ I Φ J Φ K ) + ˜ Q i ( m q + Φ ) Q i , (5.4)with index i enumerating the N f = 2 hypermultiplets.Assume that a small number k of the components of φ obtain a VEV ,( φ ) N c × N c = − v . . . − v , (5.5)which is dual to separating out k D3-branes from the stack, and moreover that these
VEV s areexactly such that some of the components of m + h φ i vanish, v = m , which is dual to the singled outD3-branes coinciding with the D7-branes. Then F-flatness conditions ˜ q i ( φ + m ) = ( φ + m ) q i = 0permit the corresponding 2 k components of the fundamental hypermultiplet to also acquire anon-vanishing VEV ( q i ) N c × = α i ... α ik , (˜ q i ) × N c = (cid:0) · · · β i · · · β ki (cid:1) . (5.6)These VEV s, which are further constrained by additional F- and D-flatness conditions, are thestring theory dual of the D3-branes that coincide with the D7-branes to be dissolved [89] in theD7-branes and form instantons in the gauge fields of the D7s. This process is caused by theWess–Zumino coupling S W Z ∼ R P [ C (4) ] ∧ F ∧ F . Note that the backreaction of the dissolved There are three adjoint N = 1 chiral fields Φ , , with lowest components φ , , and one real field V , whichforms an N = 2 gauge multiplet with Φ . The N f chiral fields Q i and ˜ Q i make up the N = 2 fundamentalhypermultiplet and have lowest components q i and ˜ q i . D3-branes can only be neglected when their number k is small in comparison to N c . Specificallyin this Chapter k = 1 will be assumed.Taking into account the breaking of SU(2) R × SU(2) f to its diagonal subgroup, which is mediatedby the instanton configuration on the supergravity side, the structure of the VEV s is as follows( φ ) N c × N c = − m , ( q iα ) = ε iα Λ , (5.7)with α = 1 , R index and q = q , q = ˜ q .5.4 Supergravity5.4.1 InstantonsIn Yang–Mills ( YM ) theories, instantons arise as finite action solutions from the semi-classicalapproximation to path integrals, which requires to find all solutions that minimise the Euclidianaction. These solutions, (anti-)self dual gauge field configuration of arbitrary topological charge k , can be found from a set of algebraic equations, the so-called ADHM constraints due to Atiyah,Drinfeld, Hitchin and Manin. These equations are non-linear and cannot be solved in generalbecause of their complex structure, though there has been recent progress in
AdS / CFT inspiredlarge N c considerations [90]. In particular the four dimensional ADHM constraints arise from Dand F-flatness conditions of D( p + 4)-branes probed by D p -branes [91, 92].Although the ADHM formalism works for all non-exceptional groups, the focus here will be onSU( N ) theories in Euclidian space. Consider the following action, S = − Z d x Tr F mn + iθk, (5.8)with the topological charge and field strength k := − g π Z d y Tr F mn ∗ F mn , k ∈ Z , (5.9) F mn := ∂ m A n − ∂ n A m + g (cid:2) A m , A n (cid:3) , (5.10) ∗ F mn := ε mnkl F kl (5.11)and anti-Hermitean gauge field A m such that the covariant derivative reads D m = ∂ m + gA m .Instantons with negative topological charge, also known as anti-instantons, will not be consid-ered here. The action is minimised by self dual solutions ∗ F mn = ± F mn , = ⇒ S = − πikτ k > , (5.12)with the complex coupling τ = πig + θ π .The self-dual SU(2) instanton solution, also known as the Belavin–Polyakov–Shvarts–Tyupkin( BPST ) instanton [93], is given by A inst n = g − y m − Y m ) σ mn ( y − Y ) + Λ , F inst mn = g − ρ σ mn (( y − Y ) + Λ ) , (5.13)with the instanton moduli Λ (size) and Y m (position). The Lorentz generators are given by σ mn = 14 ( σ m ¯ σ n − σ n ¯ σ m ) , ¯ σ mn = 14 (¯ σ m σ n − ¯ σ n σ m ) , (5.14) and it holds σ mn = 12 ε mnkl σ kl , ¯ σ mn = − ε mnkl ¯ σ kl . (5.15)The above identification of gauge indices with vector indices expresses the instanton breaking theSU(2) L × SU(2) to its diagonal subgroup, with SU(2) L from the double covering group of theEuclidian Lorentz group SO(4) and SU(2) the gauge group.The BPST instanton falls off slowly for large distances, which creates convergence problems ofvarious integrals. A well known solution in the instanton literature is the use of a singular gaugetransformation U ( y ) := σ m ( y − Y ) m | y − Y | , (5.16)which transforms the non-singular instanton solution to a singular one, A n = g − ( y − Y ) m ¯ σ mn ( y − Y ) [( y − Y ) + Λ ] , (5.17)that has better large distance behaviour. This particular gauge transformation also associatesSU(2) R with the gauge group, such that (5.17) breaks the SU(2) L × SU(2) R × SU(2) to SU(2) L × diag[SU(2) R × SU(2)]. Note that also in the instanton literature a known consequence of (5.16) is the modification of boundary terms . Therefore consequences for the
AdS / CFT dictionary are alsoto be expected.5.4.2 D7-brane ActionAs a reminder the
AdS × S background as given in (3.1), (3.5) is ds = H − / ( r ) η µν dx µ dx ν + H / ( r ) ( d~y + d~z ) , (5.18)with H ( r ) = L r , r = ~y + ~z , (5.19) L = 4 πg s N c ( α ′ ) , ~y = X m =4 y m y m , (5.20) C (4)0123 = H − , ~z = ( z ) + ( z ) , (5.21)e ϕ = e ϕ ∞ = g s . (5.22)The constant embedding z = 0 , z = ˜ m q (5.23)defines the distance ˜ m q = (2 πα ′ ) m q between the D3 and D7-branes and therefore determines themass m q of the fundamental hypermultiplet.Moreover it yields the induced metric (3.3) ds = H − / ( r ) η µν dx µ dx ν + H / ( r ) d~y ,r = y + (2 πα ′ ) m q , y ≡ y m y m (5.24) on the D7-brane.At quadratic order, the non-Abelian DBI action (2.31) and the Wess–Zumino term (2.25) arerespectively S DBI = − µ Z d p +1 ξ STr e − ϕ p − det G ab (cid:20) λ D a Φ i D a Φ i + λ F ab F ab (cid:21) = − T γ Z d x d y Tr h − H ( r ) D µ Φ D µ ¯Φ + 2 D m Φ D m ¯Φ+ H ( r ) F µν F µν + 2 F mν F mν + H − ( r ) F mn F mn i , (5.25) S W Z = T Z STr γ P [ C (4) ] ∧ F ∧ F = T γ Z Tr H − ( r ) F mn F rs dx ∧ . . . dx ∧ dy m ∧ dy n ∧ dy r ∧ dy s | {z } = ε mnrs dy ∧ dy ∧ dy ∧ dy = T γ Z d x d y H − ( r ) Tr F mn ∗ F mn , (5.26)where Φ , ¯Φ = Φ ± i Φ , γ = 2 πα ′ and the Hodge dual is ∗ F mn := ε mnrs F rs , with the epsilonsymbol ε = 1. All indices have been lowered and are now contracted with a Minkowski metric η ab = ( η µν , δ mn ). This will be true for all subsequent expressions in this Chapter, providing aconvenient framework for the discussion of solutions that are self-dual with respect to the flatmetric δ mn .These solutions arise because there is a (known, cf. [91, 92]) correspondence between instantonsand the Higgs branch. The discussion in this thesis will be confined to quadratic order, wherethe DBI and Wess–Zumino term due to F mn ( F mn − ∗ F mn ) = 2 F − mn F − mn complement one anotherto yield S = − T γ Z d x d y Tr h − H ( r ) D µ Φ D µ ¯Φ + 2 D m Φ D m ¯Φ+ H ( r ) F µν F µν + 2 F mν F mν +2 H − ( r ) F − mn F − mn i . (5.27)This action is extremised by the configuration F − mn = 0 , Φ = ˜ m q , F µν = F mn = 0 , (5.28)which is manifestly self-dual with respect to the D3-transversal flat metric δ mn . The particularbackground configuration that will be investigated here, A m = 2Λ ¯ σ mn y n y ( y + Λ ) , A µ = 0 , Φ = ˜ m q , (5.29)takes the singular gauge instanton (5.17) as an ansatz for (5.28) that brings the correct boundarybehaviour for the AdS / CFT dictionary as will be seen below. The explicit expanded form of the non-Abelian
DBI action is only known to finite order, cf. [94] for terms atsixth order. The existence of instanton solutions puts constraints on unknown higher order terms [95, 96]. y m = 0).These will be ignored and concentration will be instead on the more interesting fluctuations ofthe gauge fields and scalars. The simplest modes are vector fluctuations of type II, cf. eq. (3.22b),and scalar fluctuations, both in the same supermultiplet and in the the lowest representation ofSU(2) L × diag[SU(2) R × SU(2) f ]. In particular this means that the fluctuations will be assumedto be independent of angular variables in the D3-transversal/D7-longitudinal coordinates; i.e. inthe language of the analytically solvable scenario of Chapter 3: ℓ = 0.5.5.1 Vector FluctuationsIn accordance with the coordinate splitting X a = x µ , y m performed in the action (5.27), fluctua-tions of the form A := A − A inst will be considered. The simplest ansatz for gauge fluctuation,which at the same time is most interesting due to describing vector mesons, is given by “Type II”fluctuation (3.22b) in the language of Kruczenski et al., see Chapter 3. This particular ansatz isnon-trivial in the D3-longitudinal components only, such that the simplest non-Abelian choice isa singlet under SU(2) L and a triplet under diag[SU(2) R × SU(2) f ]. An obvious ansatz is given by A µ ( a ) = ξ µ ( k ) f ( y ) e ik µ x µ T a , y ≡ y m y m , (5.30)and A µ = A µ , A m = A inst m . (5.31)The Euler–Lagrange equations ∂ µ ∂ L ∂∂ µ A Mν + ∂ m ∂ L ∂∂ m A Mν − ∂ L ∂A Mν = 0 , (5.32) ∂ µ ∂ L ∂∂ µ A Mn + ∂ m ∂ L ∂∂ m A Mn − ∂ L ∂A Mn = 0 (5.33)for the action (5.27) are D µ ( HF µν ) + D m F mν = 0 , (5.34) D µ F µn + 2 D m (cid:2) H − F − mn (cid:3) = 0 . (5.35)To linear order the former becomes ∂ µ A µ = 0, which is solved by k µ ξ µ = 0, while the latter reads H∂ µ ∂ µ A ν + ∂ m ∂ m A ν + g ∂ m (cid:2) A inst m , A ν (cid:3) + g (cid:2) A inst m , ∂ m A ν (cid:3) + g (cid:2) A inst m , (cid:2) A inst m , A ν (cid:3)(cid:3) = 0 , (5.36)which for the ansatz (5.30) yields0 = (cid:20) M L ( y + (2 πα ′ ) m q ) − y ( y + Λ ) + 1 y ∂ y ( y ∂ y ) (cid:21) f ( y ) , (5.37)where M = − k µ k µ in accordance with having chosen a Minkowski metric with mostly plusconvention for contraction of flat indices. For numerics it is convenient to join the two parameters quark mass and instanton size byrescaling according to˜ y ≡ y πα ′ m q , ˜Λ ≡ Λ2 πα ′ m q , ˜ M ≡ M L (2 πα ′ m q ) , (5.38)such that equation (5.37) becomes0 = (cid:20) ˜ M (˜ y + 1) − ˜ y (˜ y + ˜Λ ) + 1˜ y ∂ ˜ y (˜ y ∂ ˜ y ) (cid:21) f (˜ y ) . (5.39)At large ˜ y (5.39) has two linear independent solutions whose asymptotics are given by ˜ y − w with w = 0 ,
2. The normalisable solutions corresponding to vector meson states behave as ˜ y − asymptotically. From standard AdS / CFT correspondence, one expects w = ∆ and w = 4 − ∆,where ∆ is the UV conformal dimension of the lowest dimension operator with the quantumnumbers of the vector meson. However, the kinetic term does not have a standard normalisation;i.e. the radial component of the Laplace operator appearing in the equation above is not (only) ∂ y , and consequently an extra factor of ˜ y α , for some α , appears in the expected behaviour; so theexponents actually read w = α + ∆ , α + 4 − ∆. From the difference it is read off that ∆ = 3. Thedimensions and quantum numbers are those of the SU(2) f flavour current, J bµ = − ¯ ψ ± i γ µ σ bij ψ ∓ j + ¯ q αi ↔ D µ σ bij q αj , (5.40)with α the SU(2) R index and i, j the flavour indices. This current has SU(2) R × SU(2) L × U(1)quantum numbers (0 , .The asymptotic behaviour of the supergravity solution is A µb ( a ) = ξ µ ( k ) e ik · x f (˜ y ) δ ab ∼ ˜ y − h a, ξ, k | J µb ( x ) | i , (5.41)where J µ is the SU(2) f flavour current and | a, ξ, k i is a vector meson with polarisation ξ , momen-tum k , and flavour triplet label a . Note that the index b in A µb ( a ) is a Lie algebra index, whereasthe index ( a ) labels the flavour triplet of solutions.The meson spectrum is numerically determined by a shooting technique using interval bisectionto find the values ˜ M that admit solutions to (5.39) that are regular ( c = 0 for IR behaviour c ˜ y + c ˜ y − ) and normalisable ( c = 0 for UV behaviour c + c ˜ y − ). The result for the lowestlying modes is shown in Figure 12.In passing it is noted that the second term in (5.39), which comes from the g term in theequation of motion (5.36), is roughly the instanton squared and up to numerical constants wouldhave been y / ( y + Λ ) for the instanton in non-singular gauge. This contribution would havechanged the UV behaviour of f ( y ) and therefore prohibited to make contact to the SUSY case inthe limit of zero instanton size, where (5.36) can be solved analytically.In the limit of infinite instanton size, one might expect the same spectrum since the fieldstrength vanishes locally. This corresponds to infinite Higgs
VEV in the field theory, which reducesthe gauge group from SU( N c ) to SU( N c − N c limitand one might expect to return to the origin of moduli space. However there is a non-trivial shiftof the spectrum, which makes the flow from zero to infinite Higgs VEV not quite a closed loop ascan be seen in Figure 12(b).Since at both ends the analytic spectrum in reproduced, it should be possible to capture thisbehaviour in the equation of motion (5.36). Indeed a simultaneous treatment of both cases canbe achieved by rewriting (5.36) in the suggestive form0 = (cid:20) ˜ M (˜ y + 1) − ℓ ( ℓ + 2)˜ y + 1˜ y ∂ ˜ y (˜ y ∂ ˜ y ) (cid:21) f (˜ y ) , (5.42) PSfrag replacements 1 2 3 4 5 f ( ρ ) ρ (a) Regular solutions of (5.39) in arbitrary units for Λ = 2 πα ′ m PSfrag replacements 10 − −
10 10
15 10 Λ2 πα ′ m q ML (2 πα ′ ) m q (b) Numerically determined meson masses. Fig. 12
Each dotted line represents a regular solution of the equation of motion, corresponding to avector multiplet of mesons. Plot (a) shows the five regular solutions of (5.39) corresponding to the lightestmeson masses in (b). The units on axis of ordinate in (a) are arbitrary because (5.39) is a linear equation.The vertical axis in (b) is √ λM/m q where M is the meson mass, λ the ’t Hooft coupling and m q the quarkmass. The horizontal axis is v/m q , where v = Λ / πα ′ is the Higgs VEV . In the limits of zero and infiniteinstanton size (Higgs VEV), the spectrum (grey horizontal lines) obtained analytically in Chapter 3 isrecovered.2 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings with ℓ = 0 , A a = 0, but ℓ was given rise to by excitations on the internal manifold. The ansatz was A µ = ξ µ ( k ) e ik µ x µ f ( y ) Y ℓ (S ) , (5.43)with Y ℓ (S ) the scalar spherical harmonics on S transforming under ( ℓ , ℓ ) representations ofSU(2) L × SU(2) R . [24] found that (5.42) can be solved analytically in terms of a hypergeometricfunction (3.12) parametrised by n and ℓ , which by regularity and normalisability become quantisedand yield the discrete spectrum˜ M = 4( n + ℓ + 1)( n + ℓ + 2) , n, ℓ ≥ . (3.24)For intermediate values of the instanton size, a flow connecting the analytically known spectrais expected and could be confirmed numerically, see Figure 12(b).It remains to comment on how it is possible to continuously transform a spherical harmonicin the (0 ,
0) of the unbroken SU(2) L × SU(2) R into one that transforms under the (1 , L is unbroken along the flow. The solution to this puzzle is that the instanton in singulargauge does not vanish in the limit of large instanton size, while in non-singular gauge it does. Sothe spectrum at large instanton size is related to the one at zero instanton size exactly by thesingular gauge transformation (5.16), which reads U = y m σ m | y | . (5.44)This gauge transformation is large . While in the instanton literature it is merely employed as acomputational trick to improve convergence of numerical calculations for large distance from theinstanton core, in this setup it has physically observable consequences because the large distancebehaviour is related to the conformal dimension of boundary operators. It also does not leavethe global charges under SU(2) L × SU(2) R × SU(2) f invariant: Acting on the ansatz (5.30), thesingular gauge transformation (5.16) yields A µ ( a ) = ξ µ ( k ) f ( y ) e ik µ x µ (cid:2) y m y n y σ m T a ¯ σ n (cid:3) . (5.45)The parenthesised expression should be the ℓ = 2 spherical harmonic. Due to σ m T a ¯ σ n being trace-less, there is indeed no singlet contribution. Moreover a spherical harmonic should be independentof | y | as is true for y m y n y . With ˆ g ij the metric on the three sphere it holds ∂ m ∂ m Y ℓ = y − ˆ ∇ i ˆ g ij ˆ ∇ j Y ℓ = − ℓ ( ℓ + 2) y − Y ℓ , (5.46)which is also satisfied by (5.45).5.5.2 Scalar FluctuationsThe mesons arrange themselves in massive N = 2 multiplets, some of which are obtained bydifferent, scalar ans¨atze for the gauge fluctuations (5.30). In addition, there arise mesons fromfluctuations of the scalars in (5.27). For these the equation of motion reads H∂ µ ∂ µ Φ + D m D m Φ = 0 , (5.47) SU(2) R × SU(2) f is broken to diag[SU(2) R × SU(2) f ] except at zero and infinite Higgs VEV . PSfrag replacements 0 . . . . πα ′ m q Λ ML Λ PSfrag replacements 10 πα ′ m q Λ ML Λ Fig. 13
Numerical results for the meson mass spectrum as function of the quark mass. Both for m q / Λ → m q / Λ → ∞ , the curves become linear, however with different slope. The asymptoticslopes correspond to the constant values approached in Figure 12(b).4 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings where D m D m Φ = ∂ m ∂ m φ + (cid:2) A inst m , ∂ m Φ (cid:3) + ∂ m (cid:2) A inst m , Φ (cid:3) + (cid:2) A inst m , (cid:2) A inst m , Φ (cid:3)(cid:3) , (5.48)which coincides with the equation of motion for the gauge field (5.36) except for the vector indexpresent. Therefore the same ansatz up to a polarisation vectorΦ ( a ) = f (˜ y ) e ik µ x µ T a (5.49)yields exactly the same radial differential equation (5.39) and the same mass spectrum, Figure 12.The scalar fluctuations (5.49) are dual to the descendant QQ ( q i ¯ q i ) of the scalar bilinear q i ¯ q i ,which has conformal dimension ∆ = 3. At Λ = 0 the scalar bilinear is in the (0 ,
0) representationof the unbroken SU(2) L × SU(2) R symmetry. This Chapter is similar in spirit to the D3-D7 systems discussed so far, though different in imple-mentation. The reason is that while fundamental fields are still assumed to arise from D7 branesin a—possibly deformed—
AdS space, the requirement to describe quarks of vastly different mass,as needed for heavy-light mesons, makes those quarks arising from a stack of coincident
D7-branesbeing no longer a good approximation. In this regard, heavy-light mesons are intrinsically stringy and cannot be captured by the
DBI techniques discussed in the previous Chapters. Unfortunatelyas full quantised string theory on
AdS is not well understood, the question arises of how to transfersuch features into a supergravity framework.Here idealised heavy-light mesons will be considered, composed of a massless and a very massivequark, such that in an appropriate background, the light quark may exhibit dynamical chiralsymmetry breaking, while the heavy quark does not. For now, let us stick with the
AdS case.Clearly the geometric picture is that of two parallel (probe) D7-branes in a background determinedby a stack of D3-branes. The different quark masses correspond to the two different separations ofthe D7-branes from the D3 stack. Strings describing heavy-light mesons now differ from light-lightand heavy-heavy ones, whose ends are attached to the respective same brane, by being stretchedbetween the two different D7-branes. In the limit where the heavy quark is much heavier thanthe light quark, henceforth called large separation limit , the string becomes very long and admitsa classical description.To obtain a description both simple and similar to the examples studied so far, the ansatz of arigid non-oscillating string is chosen that moves in the
AdS radial direction along the D7-branes,with the essential assumption that oscillations and longitudinal movement are suppressed in thelarge separation limit. Integration of the Polyakov action along the string can then be per-formed, yielding effectively a centre-of-mass movement weighted by a factor from averaging overthe geometry between the two D7s. To obtain a field equation, na¨ıve quantisation is performed,which results in a modified Klein–Gordon equation. (In a Minkowski space, this procedure yieldsthe conventional, unmodified Klein–Gordon equations.) After the
AdS case, the discussion willbe moved on to the dilaton deformed background by
GKS introduced in Chapter 4 and a similarbackground by Constable–Myers. Both exhibit chiral symmetry breaking. While these are knownto be far from perfect
QCD gravity duals, experience shows that even simple holographic modelscan reproduce measured mass values with an accuracy of 10–20%. Assuming the two respectivequark flavours associated to the D7-branes being up and bottom, the mass of the rho ( u ¯ u ) and On the field theory side at large separation; i.e. large quark mass m H , effects distinguishing vector fromscalar mesons are suppressed by m − H . Indeed the formalism described here is not capable of capturing such adifference and meson masses are thus manifestly degenerate. Fig. 14
The geometry of the D3-D7 system under consideration [97]. upsilon ( b ¯ b ) meson can be used to fix all scales in the theory and yield a numerical prediction forthe B meson mass, which indeed is less than 20% from the experimental value.6.1 Heavy-Light Mesons in AdS × S As shown in Chapter 3, quarks can be introduced into the
AdS / CFT correspondence by augmentingthe D3 stack with a stack of probe D7-branes [20]. The backreaction of the N f D7-branes on theAdS × S geometry (3.1) formed by the N c D3-branes may be neglected as long as N f ≪ N c ; i.e. N f is kept fixed in the ’t Hooft limit. ds = r L η µν dx µ dx ν + L r dr + L d Ω , (6.1)This corresponds to the quenched approximation of lattice gauge theory on the field theory side.The D7-branes wrap an AdS × S geometry when coincident with the D3s. When separated thecorresponding N = 2 hypermultiplet acquires a mass and the D7-branes wrap a geometry ds = y + ˜ m q L η µν dx µ dx ν + L y + ˜ m q dy + L y y + ˜ m q d Ω , (6.2)which is only asymptotically AdS × S and does not fill the complete AdS background, butinstead terminates from the five-dimensional point of view and drops from the IR dynamics. Thisconfiguration is shown in Figure 14. The meson spectrum can be determined analytically [24] andthe degenerate mass of the scalar and pseudoscalar meson is given by M s = 4 ˜ m q L ( n + ℓ + 1)( n + ℓ + 2) . (6.3)These mesons are build up from quarks carrying all the same mass; i.e. they form “light-light”or “heavy-heavy” mesons depending on the distance ˜ m q = (2 πα ′ ) m q between the D7-branes andthe D3 stack. When considering two D7-branes with different distances ˜ m L and ˜ m H to the D3stack, there are accordingly two towers of mesons M H and M L whose lightest representatives havea mass ratio of m L m H and which come from strings having attached both ends to the same brane.The configuration is shown in Figure 15. Strings stretched between the two branes should thenform a set of mesons composed of a heavy and a light quark.In the limit m H ≫ m L the string becomes very long and will be assumed to be in the semi-classical limit, where quantum effects to the unexcited string can be neglected. The string de-scribed here will therefore approximate above mesons, which by construction will be degenerate. D3Heavy quarkD7’ braneLight quarkD7 brane LLHHHL
Fig. 15
The brane configuration including both a heavy and a light quark. The 77 and 7 ′ ′ stringsare holographic to light-light and heavy-heavy mesons respectively. Heavy-light mesons are described bystrings between the two D7-branes. The gauge-fixed Polyakov action will be taken as a starting point S P = − T Z dσ dτ G µν ( − ˙ X µ ˙ X ν + X ′ µ X ′ ν ) , (6.4)such that the constraints G µν ˙ X µ X ′ ν = 0 , G µν ( ˙ X µ ˙ X ν + X ′ µ X ′ ν ) = 0 , (6.5)have to be taken into account.The two D7-branes are assumed to be separated from the D3 stack in the same direction θ = 0;i.e. the string connecting them will obey σ = z , where σ is the spatial world sheet coordinate and z e iθ = z + iz . While the string will be allowed to move along the world volume of the D7s, itshall be stiff such that integration over σ can be performed to generate an effective point particleaction. With the embedding X A = ( x µ ( τ ) , y m ( τ ) , z = 0 , z = σ ) , (6.6)which implies ˙ X a X ′ a = 0 automatically, and the AdS × S geometry (6.1), the Polyakov actionreads S P = − T Z dτ ˜ m H Z ˜ m L dσ (cid:20) − y + σ L ˙ x α ˙ x α − L ( y + σ ) ˙ y i ˙ y i + L ( y + σ ) (cid:21) , (6.7)where y ≡ | y | ≡ qP i =4 , , , ( y i ) . Integrating over σ yields S P = − T Z dτ (cid:2) − f ( y ) ˙ x − g ( y ) ˙ y + g ( y ) (cid:3) , (6.8)with (choosing ˜ m L = 0) f ( y ) = 1 L (cid:0) y ˜ m H + 13 ˜ m H (cid:1) , g ( y ) = L y arctan ˜ m H y . (6.9)The remaining constraint equation G µν ( ˙ X µ ˙ X ν + X ′ µ X ′ ν ) = 0 is y + σ L ˙ x α ˙ x α + L ( y + σ ) ˙ y i ˙ y i + L ( y + σ ) = 0 , (6.10) which gives 1 f ( y ) p x + 1 g ( y ) p y + T g ( y ) = 0 , (6.11) p αx := ∂ L ∂ ˙ x α ,p iy := ∂ L ∂ ˙ y i when integrating over σ . The same calculation for Minkowski space gives f ( y ) = g ( y ) = ˜ m H , suchthat one obtains E = m + p . For AdS space the mass m depends on the position of the string y via the factors f ( y ) and g ( y ), which average over the geometry between the two D7-branes.For the quantisation prescription p
7→ − i∂ , the following modified Klein–Gordon equation isobtained (cid:20) (cid:3) x + f ( y ) g ( y ) ∇ y − T g ( y ) f ( y ) (cid:21) φ ( ~x, ~y ) = 0 . (6.12)The usual procedure for this kind of equations is to find the correct background solution, whichby assumption only depends on the radial direction y and find fluctuations about this solution.By a separation ansatz these fluctuations can be seen to be a plain wave in the x direction andspherical harmonics in the angular coordinates Ω ( y , , , ). The remaining equation for the radialcoordinate y often has to be solved numerically.In the UV limit y → ∞ , (6.12) is dominated by the Laplace operator in the y directions due to fg ∼ y and f g →
1, such that ∇ y φ = 0 . (6.13)When φ only depends on y , the solution has the form required to couple to the VEV and sourceof a heavy-light quark bilinear ¯ ψ H ψ L . φ ( y → ∞ ) = ˜ m HL + c HL y + . . . (6.14)However there are no heavy-light mass mixing term and no heavy-light bilinear condensate in QCD , so φ ( y ) ≡ L × SU(2) R , the ansatz for linearised fluctuations about abovevacuum solution reads φ = 0 + h ( y ) e ik · x , M HL = − k , (6.15)where h ( y ) shall be regular in the IR and normalisable h ( y → ∞ ) ∼ y − . Only for a discreteset of values for M HL this requirement can be satisfied. For numerics it is convenient to employrescaled coordinates y = ˜ m H ˜ y , such that equation (6.12) reads " πλ ˜ y + ˜ y arctan y ∇ y + (cid:18) ˜ y + 13˜ y (cid:19) arctan 1˜ y + M L ˜ m H h (˜ y ) = 0 . (6.16)The ’t Hooft coupling λ arises from R / (2 πα ′ ) = g s N c /π . The mass ratios yielding regularnormalisable solutions to (6.16) have been plotted in Figures 16 and 17. It can be read off M H m H = 2 πα ′ L h √ λ + O ( λ − ) i . (6.17)In the large λ limit, ˜ M HL = ˜ m H is approached in agreement with the na¨ıve expectation of themeson mass being equal to the string length times its tension. For comparison in Figure 17 themass ratio is plotted for small values of the ’t Hooft coupling, where supergravity is not a reliableapproximation anymore. PSfrag replacements0 1001 . . . .
81 202 . .
42 40 60 80˜ M HL ˜ m H √ λ PSfrag replacements0 100 10012020 2040 4060 6080 80 √ λ ˜ M HL ˜ m H √ λ Fig. 16
The mass ratio of the heavy-light meson and the heavy quark mass (the light quark is takento be massless) as a function of the ’t Hooft coupling for the
AdS background. In the large λ limit, M HL L / (2 πα ′ m H ) behaves as 1 + const. / √ λ + O ( λ − ). The black line in the second plot corresponds to M HL L = (2 πα ′ ) m H .9 PSfrag replacements0 1015 12025 230 3 45 5 √ λ ˜ M HL ˜ m H √ λ Fig. 17
The heavy-light meson spectrum in
AdS for small ’t Hooft coupling with vanishing mass forthe light quark. The mass ratio behaves as const. / √ λ + O ( λ ). Note however that the supergravityapproximation is not reliable in this regime. N = 2 SYM considered so far provides a basis for studying meson spectra since it givesanalytic expressions for solutions and masses consisting of identical quarks. However it does notcapture a number of phenomenologically relevant features like chiral symmetry breaking sincechiral symmetry breaking requires
SUSY breaking. The setup discussed now improves at least inthat regard by providing a simple geometry that describes a non-supersymmetric dual of a large N c QCD -like theory and thus exhibits dynamical chiral symmetry breaking.The first background discussed is the dilaton deformed background by Gubser, Kehagias–Sfetsos, which has been described in Chapter 4. It is demonstrated that the semi-analytic pre-diction of the
AdS case is reproduced in the large heavy-quark limit. Then the same procedure isapplied to the similar geometry of Constable and Myers, but it turns out that in this setup theheavy-light meson spectrum does not approach the
AdS spectrum in a similar manner.6.2.1
GKS
GeometryLet me remind the reader that the
GKS geometry is given by, cf. (4.26), ds = g xx ( r ) dx , + g yy ( r )( d~y + d~z ) ,g xx ( r ) = r L p − r − ,g yy ( r ) = g zz = L r , e ϕ = e ϕ (cid:18) r + 1 r − (cid:19) √ ,r = ~y + ~z , (6.18) PSfrag replacements0 . . . . . . . . . . m q = 0 . c = 1 . m q = 0 . c = 1 . m q = 0 . c = 1 . m q = 1 . c = 0 . m q = 1 . c = 0 . m q = 2 . c = 0 . z ( y ) r y/r Fig. 18
Chiral symmetry breaking embeddings in the
GKS geometry. where Einstein frame has been used and the coordinates have been rescaled such that infra-red singularity resides at r = 1. The coordinates y , , , and z , are on equal footing andcan be interchanged by SO(6) transformations until probe D7-branes, which break the SO(6)to SO(4) × SO(2), are introduced to obtain quarks. The D7-branes are embedded according to z = | z + iz | = z ( y ), which yields the following equation of motion ddy (cid:20) y f p z ′ ( y ) z ′ ( y ) (cid:21) = y q z ′ ( y ) ∂∂z f, (6.19) f = ( r + 1) (1+∆ / ( r − (1 − ∆ / r , r = y + z ( y ) , ∆ = √ . At large y , solutions to (6.19) take the form z = ˜ m q r + cr y + . . . , (6.20)which by standard AdS / CFT duality corresponds to a source of conformal dimension 1 and a
VEV of conformal dimension 3 in the field theory. The former corresponds to the quark mass m q = ˜ m q / (2 πα ′ ) and describes the asymptotic separation ˜ m q of the D3 and D7-branes, the latteris the bilinear quark condensate c ∼ (cid:10) ¯ ψψ (cid:11) . The factor of r , which gives the position of thesingularity, arises from the coordinate rescaling used to remove r from the metric and equationsof motion.Requiring regularity in the IR by ∂ y z (0) = 0 fixes the quark condensate as a function of thequark mass, see Section 4.6. Some regular solutions to (6.19) are plotted in Figure 18, whichprovide the D7 embeddings that are used as the boundary conditions for the heavy-light string inthe following. The Polyakov action (6.4), which due to being in string frame requires additional factors ofe ϕ/ , reads for this background S P = − T Z dτ z ( m H ) Z z ( m L ) dz h − e ϕ/ g xx ˙ x α ˙ x α − e ϕ/ g yy ˙ y i ˙ y i + e ϕ/ g yy i , (6.21)with the metric factors and dilaton from (6.18).One obtains again an equation of motion of the form (cid:20) (cid:3) x + f ( y ) g ( y ) ∇ y − T g ( y ) f ( y ) (cid:21) φ ( ~x, ~y ) = 0 , (6.22)where the coefficients f ( y ) and g ( y ) this time are given by f ( y ) = z ( m H ) Z z ( m L ) dz e ϕ/ g xx , g ( y ) = z ( m H ) Z z ( m L ) dz e ϕ/ g yy . (6.23)The integration limits in (6.23); i.e. the positions of the D7-branes, are given by the solutionsto (6.19), which are only known numerically, such that f ( y ) and g ( y ) also require numerics.For an ansatz describing a field theoretic vacuum φ ≡ φ ( y ), equation (6.22) has the same UV behaviour as the AdS case, φ ( y → ∞ ) ∼ ˜ m HL + c HL y − , where ˜ m HL corresponds to heavy-lightmass mixing term and c HL to a heavy-light quark condensate. Because both are absent in QCD ,fluctuations about the trivial vacuum φ ( y ) ≡ δφ = φ ( y ) e ik · x (6.24)as it can be obtained from " M HL Λ + πλ ˆ f ( y )ˆ g ( y ) ∇ y − g ( y ) f ( y ) φ ( ~x, ~y ) = 0 (6.25)with Λ = r / (2 πα ′ ) the QCD scale. ˆ f and ˆ g can be obtained from (6.23) by setting L = 1.The light quark mass m L has been set to zero to describe a quark experiencing dynamical chiralsymmetry breaking, while the large quark mass m H is varied.The spectrum obtained is very similar to that of the AdS geometry. To make the deviationscaused by the deformation more visible, the binding energy has been plotted. In Figure 19 itis shown for λ = 100 as a function of the quark mass. It is also shown as a function of the’t Hooft coupling with the (for now arbitrary value of the) heavy quark mass m H = 11 .
50 Λ. Thebinding energy approaches its
AdS values for m H → ∞ , but highly excited mesons converge moreslowly. Both features can be understood from the spectrum of light-light/heavy-heavy mesons inChapter 4. The higher the quark mass, the higher is the energy scale, where the brane “ends”and decouples from the spectrum. At high energies supersymmetry is restored and the light-lightmesons become degenerate. While the effect is the same for the heavy-light mesons, that argumentis not quite true anymore since the light quark has been set to be massless all the time—at leastone end of the string stays close to IR region. However the centre of mass of the heavy-light stringmoves farther away from the interior of the space when the heavy quark mass grows. The effectiveaveraging of the geometry in (6.23) takes into account more and more of the geometry far fromthe centre, which is nearly AdS .At the same time highly excited mesons probe the IR more densely as has been seen in Sec-tion 4.8, so they require the string to be stretched much more to allow neglecting the vicinity ofthe singularity. PSfrag replacements0 10 151 20 252 30345 56 M HL m H − m H Λ b PSfrag replacements0 100101520 202530 405 60 80 √ λ (cid:16) M HL m H − (cid:17) √ λ Fig. 19
The binding energy of the heavy-light meson masses as a function of the heavy quark massfor λ = 100 (first plot) and as a function of the ’t Hooft coupling for m H = 11 .
50 Λ (second plot). Therespective
AdS values are shown as gray lines in the background and are approached in the limit of largevalues of the heavy quark mass, while for small values effects of the chiral symmetry breaking are seen.3
AdS geometry introduced in [46],which has been employed by [86, 98] to describe chiral symmetry breaking in
AdS / CFT . Like thebackground of the previous Section it is a warped AdS × S geometry with a running dilaton thatpreserves SO(1 , × SO(6) isometry.The background is given by ds = H − / X δ/ dx , + H / X (2 − δ ) / Y ( d~y + d~z ) ,H = X δ − , X = r + b r − b , Y = r − b r , e ϕ = e ϕ X ∆ , C (4) = H − dx ∧ · · · ∧ dx ,δ = L b , ∆ = 10 − δ , (6.26)with r = ~y + ~z . R and b are free parameters and will be set to 1 for the numerics, since thatallows to make contact with [86], where the same choice has been made. The authors of [86]embedded the D7-branes according to z = | z + iz | = z ( y ) and obtained the following equationof motion ddy " e ϕ G ( y, z ) p ∂ y z ) ( ∂ y z ) = q ∂ y z ) ∂∂z [e ϕ G ( y, z )] , (6.27)where G ( y, z ) = y (( y + z ) + 1) / (( y + z ) − − ∆ / ( y + z ) . (6.28)This is the same equation as (6.19) albeit with a free parameter ∆, which in the GKS geometryhas the fixed value √
6. The asymptotic behaviour and their field theoretic interpretation are thesame as for the
GKS background and have been reviewed in the previous Section. Note howeverthat only the particular combination e ϕ √− g appearing in the equation for the vacuum embedding(6.27) coincides in both backgrounds. On the level of meson spectra, the results for light-lightmesons are similar but not identical to those in GKS .Expanding the
DBI action (2.15) to quadratic order in fluctuations (4.12) yields (4.19) for avector meson ansatz, that is an ansatz of the form A µ = ξ µ δρ ( y ) e ik · x , M ρ = − k for the D7 gaugefield. The vector meson radial equation (4.19) reads for the Constable–Myers background ∂ y (cid:0) K ( y ) ∂ y δρ ( y ) (cid:1) + M ρ K ( y ) δρ ( y ) = 0 , (6.29)with K = X / y (1 + z ′ ) − / , K = HX − δ/ Y y (1 + z ′ ) − / (6.30)and X = ( y + z ) + 1( y + z ) − , Y = ( y + z ) − y + z ) . (6.31)The Polyakov action S P = − T Z dτ (cid:2) − f ( y ) ˙ x − g ( y ) ˙ y + g ( y ) (cid:3) (6.32) PSfrag replacements0 10 151 20 252 30345 56 M HL m H − m H Λ b PSfrag replacements0 100101520 202530 405 60 80 √ λ (cid:16) M HL m H − (cid:17) √ λ Fig. 20
The binding energy of the heavy-light meson masses as a function of the heavy quark massfor λ = 100 (first plot) and as a function of the ’t Hooft coupling for m H = 12 . / Λ b (second plot). Therespective AdS values are shown as gray lines in the background and are approached in the limit of largevalues of the heavy quark mass, while for small values effects of the chiral symmetry breaking are seen.5 preserves its
AdS form but the coefficients are now f ( y ) = z ( m H ) Z z ( m L ) dz ( X / − − / X ∆+ , (6.33) g ( y ) = z ( m H ) Z z ( m L ) dz Y ( X / − / X ∆+ , (6.34)with X , Y defined in (6.31) and the integration limits are given by the solutions to (6.27).Scalar fluctuations of the form φ = 0 + δφ ( y ) e ik · x yield (cid:20) M HL Λ b + (2 πα ′ ) b f ( y ) g ( y ) ∇ y − g ( y ) f ( y ) (cid:21) φ = 0 , (6.35)with Λ b = b/ (2 πα ′ ) the QCD scale and (2 πα ′ ) /b = 2 πδ/λ . For boundary conditions ∂ y δφ (0) = 0and δφ ( y → ∞ ) ∼ cy − equation (6.35) determines the meson spectrum. Since it is very similar tothe AdS spectrum, the binding energy, which demonstrates the deviations more clearly, has beenplotted in Figure 20 for massless light quark.6.3 Bottom PhenomenologyThere has been a number of attempts to apply holographic methods to phenomenological models[99, 100], even for the Constable–Myers background of the previous Section [101], successfullyreproducing light quark meson data with an accuracy better than 20%. That shall be motivationenough to compare the heavy-light spectra calculated here with the bottom quark sector of
QCD ;i.e. the massless quark in the setup above will be assumed to play the rˆole of an up quark, whilethe heavy quark, which will lie in the
AdS -like region, will be interpreted as a bottom quark.In that regime supersymmetry will be restored and the field theory will be strongly coupledeven though
QCD dynamics should be perturbative at this energy scale. These are respectiveconsequences of the background being too simple (though a background exhibiting separation ofscales is not known yet) and an intrinsic feature of the
SUGRA version of
AdS / CFT that can onlybe overcome by a full string treatment, which is currently out of reach.The scales of the theory will be fixed by identifying the mass of the lowest vector meson modewith the Rho and Upsilon mesons, which are chosen as input data since they are less sensitive tothe light quark mass than the pseudoscalar modes roughly corresponding to the Pion, cf. Figure 21and Section 4.7 for details.From Figure 21 the ρ mass for the GKS background is read off to be M ρ L /r = 2 .
93. Preservingthe physical ratio M Υ /M ρ = 9 . /
770 MeV, (6.36)the Υ mass has to be M Υ L /r = 35 . m b = 12 . λ for which the numerical value of the lowest heavy-lightexcitation satisfies (cid:18) M B M ρ (cid:19) phys = (cid:18) M HL ( λ )Λ (cid:19) num (cid:18) r M ρ L (cid:19) num r λπ . (6.37)Unfortunately this yields a value of the ’t Hooft coupling of λ = 2 .
31. As can be seen in Figure22 the mass ratio of the predicted B and B * meson reaches its physical value of approximately PSfrag replacements 0 . . . .
52 2 . . πα ′ ) m q r M π,σ,ρ L r pseudoscalarscalarSUSYvector Fig. 21
Lightest pseudoscalar, scalar and vector mesons in the dilaton deformed geometry (
GKS ). Thevector mode for the massless quark is interpreted as a Rho meson, while for the heavy quark mass it yieldsthe the Upsilon. See also Section 4.7.
PSfrag replacements0 1001 . . . . .
51 20 40 60 80 M B ∗ M B √ λ Fig. 22
Ratio of the mass of the lowest and first excited heavy-light meson mode for the
GKS andConstable-Myers background. (They really do look exactly the same, since the different units expressingthe different dependence on the respective deformation parameter cancel in the ratio.) For large ’t Hooftparameter the ratio approaches 1, while the physical B/B * ratio (which is 1.01) is reached at λ ≈ λ . Identifying M HL with the physical quark mass M B = 5279 MeV, oneobtains a QCD scale of 225 MeV.With respect to the B mass ratio, the situation is slightly better for the background by Constableand Myers, where the same procedure yields a prediction of λ = 5 .
22. While it is not clear if thisvalue is sufficient for the large λ approximation inherent in the employed formulation of the AdS / CFT correspondence, it gives a prediction for M B ∗ = 6403 MeV, which is 20% larger thanthe measured value of 5325 MeV. Again a much larger value of the ’t Hooft coupling would berequired to achieve a better agreement. For the QCD scale on obtains Λ b = 340 MeV, which is alittle too high. With m H = 12 .
63 Λ b the physical b quark mass is predicted to be 4294 MeV.The overall agreement with experiment is far from perfect. However this does not come as asurprise since the b quark mass ( m b ≈
12 Λ in both backgrounds) is far in the supersymmetricregime: Restoration of supersymmetry takes place approximately at m q ≈ . AdS . The only way to improve this situation would be to use a (yet unknown) background thatallows to separate the
SUSY breaking scale from the
QCD scale.
Part II
Space-time Dependent Couplings
The second part of this thesis is devoted to the discussion of the conformal anomaly in supersym-metric field theories, in particular supersymmetric Yang–Mills theories.The approach chosen is an extension to superfields of the space-time dependent coupling tech-niques Osborn [52] applied to non-supersymmetric theories coupled to a gravity background inorder to give an alternative proof of Zamolodchikov’s c -theorem, cf. Chapter 8. Consequently acoupling to supergravity will have to be considered and its superfield formulation shall be reviewedin this Chapter.In Chapter 9 a discussion of the supersymmetric conformal anomaly will be given.7.1 ConventionsTo establish notations, a few basic ingredients for supersymmetry are reviewed in the shortestpossible manner. Throughout this part, a dotted/undotted Weyl spinor notation is being used.The simplest double covering representation of the Lorentz group can be constructed as follows.An arbitrary vector v α ˙ α transforms under a Lorentz transformation Λ ab ∈ SO(1 ,
3) according to x a x ′ a = Λ ab x b . (7.1)The double covering group SL(2 , C ) transforms the same vector according to σ aα ˙ α x a ( U αβ σ aβ ˙ β U † ˙ α ˙ β ) x a ≡ σ aα ˙ α x ′ a , (7.2)with U the element of the double covering group chosen such that x ′ a coincides with the definition(7.1). The matrices σ a := ( , ~σ ) are the Pauli matrices augmented by the unity matrix. As anaside, the “1 to 2” relation of the two representations can be easily seen from the fact that forany U being a solution to ( U αβ σ β ˙ βa U † ˙ α ˙ β ) = σ bα ˙ α Λ ba , − U is also a solution. The group SL(2 , C )leaves invariant the antisymmetric tensors ε αβ and ε ˙ α ˙ β , defined by ε = ε ˙1˙2 = − , ε = ε ˙1˙2 = 1 , (7.3) where the epsilon symbols with raised indices constitute the respective inverse matrices by ε αβ ε βγ = δ αγ . Since for any element U of SL(2 , C ) it holds the relation ε αβ = ε γδ U γα U δβ , the combination ε αβ ψ α ψ β is invariant under ψ α U αβ ψ β and therefore a Lorentz scalar. In other words, theepsilon matrices can be used to obtain contragradiently transforming representations according to ψ α = ε αβ ψ β , ψ α = ε αβ ψ β , (7.4)¯ ψ ˙ α = ε ˙ α ˙ β ¯ ψ ˙ β , ¯ ψ ˙ α = ε ˙ α ˙ β ¯ ψ ˙ β . (7.5)For the sake of brevity, an indexless notation is often employed for contracted adjacent objects,where different conventions are being used for dotted and undotted indices, ψχ := ψ α χ α , ¯ ψ ¯ χ = ¯ ψ ˙ α ¯ χ ˙ α . (7.6)This particular choice has the advantage that ψχ = ¯ ψ ¯ χ .It is common to introduce x α ˙ α := ˜ σ aα ˙ α x a , (7.7)with ˜ σ α ˙ αa = ε αβ ε ˙ α ˙ β ( σ a ) βδ , and convert back and forth between the two representations using therelations ( σ a ) α ˙ γ (˜ σ b ) β ˙ γ + ( σ b ) α ˙ γ (˜ σ a ) β ˙ γ = − η ab δ αβ , (7.8)(˜ σ a ) γ ˙ α ( σ b ) γ ˙ β + (˜ σ b ) γ ˙ α ( σ a ) γ ˙ β = − η ab δ ˙ β ˙ α , (7.9)which imply x a = − ( σ a ) α ˙ α x α ˙ α , x a x a = − x α ˙ α x α ˙ α . (7.10)A superspace is defined to be a space with coordinates x α ˙ α of even Graßmann parity and θ α ,¯ θ ˙ α = ( θ α ) † of odd Graßmann parity; i.e. anticommuting. The Graßmann parity of a quantity q is symbolised by q and capital Latin letters are used to denote collective indices; e.g. thesupercoordinates are labelled z A = ( x α ˙ α , θ α , ¯ θ ˙ α ) and transform under the ( , ), ( ,
0) and(0 , ) representations, respectively. Arbitrary irreducible representations ( m , n ) are given bysymmetric tensors ψ α ,...,α m , ˙ β ,..., ˙ β n ≡ ψ { α ,...,α m } , { ˙ β ,..., ˙ β n } , (7.11)where the weight is chosen such that (anti-)symmetrisation is idempotent, ψ { α ,...,α N } = 1 N ! X ψ π ( α ) ,...,π ( α N ) , (7.12) ψ [ α ,...,α N ] = 1 N ! X sign( π ) ψ π ( α ) ,...,π ( α N ) , (7.13)and (anti-)symmetrisation is performed over only those indices enclosed in braces that are notadditionally enclosed in a pair of vertical bars | | . From the spin-statistics theorem follows thatany physical field ψ α ,...,α m , ˙ β ,..., ˙ β n has Graßmann parity m + n (mod 2).Partial superderivatives ∂ A = ( ∂ α ˙ α , ∂ α , ¯ ∂ ˙ α ) are defined by (cid:2) ∂ A , z B (cid:9) = ( ∂ A z B ) := δ AB (7.14) This convention implies that components of a tensorial object t A ...A n have a varying number of indices.Commas will be used to separate index pairs α ˙ α, β ˙ γ whenever this disambiguation is necessary. The latter are (complex) Weyl spinors as opposed to Dirac spinors, which are composed of two Weyl spinors. where the ( Z -) graded commutator is defined by (cid:2) A, B (cid:9) := AB − ( − A B BA (7.15)and obeys the graded Leibniz rule and Jacobi identity (cid:2) A, B C (cid:9) = (cid:2) A, B (cid:9) C + ( − A B B (cid:2) A, C (cid:9) , (7.16)( − A C (cid:2) A, (cid:2) B, C (cid:9)(cid:9) + (cyclic A B C ) = 0 . (7.17)The partial derivatives in a flat superspace satisfy (cid:2) ∂ A , ∂ B (cid:9) = 0 . (7.18)A superfield f ( x, θ, ¯ θ ) on R | can be defined by a Taylor expansion in the non-commutingcoordinates according to f ( z A ) = A ( x ) + θ α ψ α ( x ) + ¯ θ ˙ α ¯ ψ ˙ α ( x )+ θ F ( x ) + ¯ θ ¯ F ( x ) + θσ a ¯ θV a ( x )+¯ θ θ α λ α ( x ) + θ ¯ θ ˙ α ¯ λ ˙ α ( x ) + θ ¯ θ G ( x ) , (7.19)where the respective coefficients are called components . Mass dimension and Graßmann parityof the superfield are by definition given by the respective property of the lowest component A .This definition of a superfield can be extended to include tensorial fields by simply promoting thecomponents to tensors.In a similar manner a superfield can be defined on C | , which is build up from four complex( y a ) and two anticommuting ( θ α ) coordinates. For the remaining part of this introduction, thesetwo superspaces will be referred to as the real ( R | ) and complex ( C | ) superspace respectively.The real superspace is a subspace of the complex superspace, embedded according to y a = x a + iθσ a ¯ θ. (7.20)By this relation holomorphic superfields can be defined on the real superspace (where they areknown as chiral superfields ) according toΦ( x, θ, ¯ θ ) = Φ( x + iθσ a ¯ θ, θ ) = e iH Φ( x, θ ) H := θσ a ¯ θ∂ a , (7.21)where H has been defined with future generalisations in mind. (The current choice of H has theunique property of making super-Poincar´e transformations on both spaces coincide, thus providingthe only Poincar´e invariant embedding of R | into C | .)The property ¯ ∂ Φ( y ) = 0 can be rewritten as¯ D ˙ α Φ( x, θ, ¯ θ ) = 0 , ¯ D ˙ α := e iH ( − ¯ ∂ ˙ α ) e − iH = − ¯ ∂ ˙ α − iθ α ∂ α ˙ α . (7.22a)Analogously, for an antichiral field it holds D α Φ( x, θ, ¯ θ ) = 0 , D α := e − iH ( ∂ α ) e iH = ∂ α + iθ α ∂ α ˙ α . (7.22b)The set of derivatives D A = ( ∂ a , D α , ¯ D ˙ α ) has the property of commuting with the supersymmetrygenerators and mapping a tensor superfield into a tensor superfield with respect to the Lorentzgroup. Hence, they are called (flat) covariant derivatives. The observant reader has noticed theunusual sign in front of ¯ ∂ ˙ α in definition (7.22), which is related to convenient complex conjugation Conjugations
O O † O ∗ O T O · · · O n O † n · · · O † π F O ∗ · · · O ∗ n π F O T n · · · O T1 ψ α ¯ ψ ˙ α ¯ ψ ˙ α ψ α ψ α ...α m ˙ β ... ˙ β n ¯ ψ ˙ β n ... ˙ β α m ...α π n π m ¯ ψ ˙ β n ... ˙ β α m ...α π n π m ψ α m ...α ˙ β n ... ˙ β ∂ a − ∂ a ∂ a − ∂ a ∂ α ¯ ∂ ˙ α − ¯ ∂ ˙ α − ∂ α D a − D a D a − D a D α − ¯ D ˙ α ¯ D ˙ α − D α Table 7
Definition of the Hermitean and complex conjugate as well as transposition (from left to right).The symbol π m := ( − ⌊ m ⌋ = ( − m ( m − denotes the sign change induced by reversing the order of m anticommuting objects while F is thenumber of fermionic terms in the corresponding expression. properties as will be explained below. While partial derivatives obey trivial (anti-)commutationrules, this is no longer true for covariant derivatives ( (cid:8) D α , ¯ D ˙ α (cid:9) = − i∂ α ˙ α ), and consequently spe-cial attention has to be paid to the reordering upon complex conjugation, in particular Hermiteanand complex conjugation no longer coincide.The Hermitean conjugate O † and transpose O T of an operator O are respectively defined by Z O † χψ := Z ¯ χ O ψ, (7.23) Z ( O T χ ) ψ := ( − χ O Z χ O ψ, (7.24)which additionally allows to define the complex conjugate by O ∗ := ( O † ) T . (7.25)In particular, these definitions imply the following reorderings( O . . . O N ) † = O † N . . . O † , (7.26)( O . . . O N ) T = ( − O O O T N . . . O T1 , (7.27)( O . . . O N ) ∗ = ( − O O O ∗ . . . O ∗ N . (7.28)From (cid:8) ( ¯ ∂ ˙ α ) † , (¯ z ˙ β ) † (cid:9) = (cid:8) ¯ ∂ ˙ α , ¯ z ˙ β (cid:9) † = ( δ ˙ α ˙ β ) † = δ αβ = (cid:8) ∂ α , z β (cid:9) , (7.29) − (cid:2) ( ∂ a ) † , ( z a ) † (cid:3) = (cid:2) ∂ a , z b (cid:3) † = ( δ ab ) † = δ ab = (cid:2) ∂ a , z b (cid:3) (7.30)one may deduce ( ∂ a ) † = − ∂ a , (7.31)( ∂ α ) † = ¯ ∂ ˙ α , (7.32) SUGRA Index Conventions c -coordinates ( x ) a -coordinates ( θ ) m, n, . . . µ, ν, . . . world M, N, . . .a, b, . . . α, β, . . . tangent
A, B, . . .
Table 8
Superfield Supergravity Index Conventions while the transpose ∂ TA = − ∂ A is determined by partial integration. So complex conjugation of aspinor partial derivative involves an additional minus sign compared to other fermionic objects.As complex conjugation is an operation which will be employed quite frequently when workingdirectly with the supergravity algebra, the definition of covariant spinor derivatives (7.22) involvesan additional minus sign for compensation. The conjugation rules are summarised in Table 7. Asone can see, for the case of (anti-)commuting objects—“numbers”—Hermitean conjugation andcomplex conjugation are the same.In the supergravity literature, the use of different notations and conventions is quite common.In particular it crucially depends on the task to be performed, which conventions are the mostsuitable. This thesis follows closely the conventions of [102], which contain the potential trap thatfor an antisymmetric tensor ψ αβ ∼ ε αβ (7.33)the corresponding contragradient tensor reads ψ αβ = ε αγ ε βδ ψ γδ ∼ ε αγ ε βδ ε γδ = − ε αβ (7.34)as a consequence of the conventions used for raising and lowering operators.The other major source of this compilation [103] uses an imaginary symplectic metric, whichintroduce a relative minus sign for complex conjugation of contragradient indices. Additionally,there appears a minus sign in the complex conjugation of spinorial covariant superderivatives D α = ( ¯ D ˙ α ) † = − ( ¯ D ˙ α ) ∗ . Furthermore, quadratic quantities D contain a factor of one half, whichmaterialises upon partial integration.7.2 Superspace SupergravityIn analogy to the non-supersymmetric case, a pseudo-Riemannian supermanifold is defined by anatlas of maps from open sets of points on the supermanifold to coordinates in flat superspace.When there is curvature, in general more than one map is required to cover the whole manifoldand the maps are distorted in the sense, that a non-Minkowski metric is needed to capture thisdistortion in terms of those superspace coordinates, which shall be called world or curved co-ordinates coordinates z M = ( z m , θ µ , ¯ θ ˙ µ ). To each point of the supermanifold one may attach a tangent superspace (also referred to as flat ), whose coordinates are called z A = ( z a , θ α , ¯ θ ˙ α ). Thedistinction of flat vs. curved will also be made in referring to the indices only as indicated inTable 8.Superspace supergravity requires a tangent space formulation, where superspace general coor-dinate transformations, realised as gauged curved superspace translations, are augmented by anadditional set of superlocal Lorentz transformations acting on the tangent space only. The reason is that without this doubling spinors can only be realised non-linearly, which is inconvenient [103,p. 235].A first order differential operator, expressed as K = K M ∂ M + K ab M ab = K M ∂ M + K αβ M αβ + K ˙ α ˙ β ¯ M ˙ α ˙ β , (7.35)therefore allows to define covariant transformation under combined supercoordinate and super-Lorentz transformations according to X e K X e − K . (7.36)The sl (2 , C ) versions M αβ = ( σ ab ) αβ M ab and ¯ M ˙ α ˙ β = (˜ σ ab ) ˙ α ˙ β M ab of the Lorentz generator M ab act on the corresponding indices (i.e. only on indices of the same kind) according to M βγ ψ α ...α n = X i ( ε α i β ψ γα ···6 α i ...α n + ε α i γ ψ βα ···6 α i ...α n ) , (7.37)¯ M ˙ β ˙ γ ψ ˙ α ... ˙ α n = X i ( ε ˙ α i ˙ β ψ ˙ γ ˙ α ···6 ˙ α i ... ˙ α n + ε ˙ α i ˙ γ ψ ˙ β ˙ α ···6 ˙ α i ... ˙ α n ) . (7.38)In particular, it holds M βγ ψ α = ( ε αβ ψ γ + ε αγ ψ γ ) ,M βγ ψ α = ( δ αβ ψ γ + δ αγ ψ β ) ,M αβ ψ β = ψ β . In analogy to ordinary gravity (with torsion) one may define a derivative D A = E A + Ω A (7.39)that transforms covariantly under (7.36) by adding a vierbein field E A := E AM ∂ M and a super-connection Ω A := Ω ABC M BC = Ω Aβγ M βγ + Ω A ˙ β ˙ γ ¯ M ˙ β ˙ γ . (7.40)The vierbein obeys the algebra (cid:2) E A , E B (cid:9) = C ABC E C , (7.41) C ABC = ( E A E BM − ( − A B E B E AM ) E CM , (7.42)where C ABC are the supersymmetric generalisation of anholonomy coefficients. The non-degeneratesupermatrix E AM can be used to convert between world and tangent indices according to V A = E AM V M , (7.43)and the bosonic submatrix E am is the well known vierbein field of gravity obeying η ab = g mn E am E bn . (7.44)The covariant derivatives form an algebra (cid:2) D A , D B (cid:9) = T AB + R AB , (7.45) T AB := T ABC ∂ C , (7.46) R AB := R ABbc M bc = R ABβγ M βγ + R AB ˙ β ˙ γ ¯ M ˙ β ˙ γ , (7.47)with T AB = − ( − A B T BA the supertorsion and R AB = − ( − A B R BA the supercurvature,which may be completely expressed in terms of the supertorsion as a consequence of the Bianchiidentities. The latter are just the Jacobi identities (7.17) for the algebra (7.45). T α ˙ βγ = T α ˙ β ˙ γ = R αβcd = 0 ,T α ˙ βc = − iσ cα ˙ β (cid:27) ⇐⇒ D α ˙ α = − i (cid:8) D α , ¯ D ˙ α (cid:9) , (7.48a) T αβγ = T ˙ α ˙ β ˙ γ = T α,β { ˙ β,βγ } = T α, { β ˙ β,γ } ˙ β = 0 , (7.48b)which are equivalent to redefinitions of the algebra’s constituents, and representation preservingconstraints T αβc = T ˙ α ˙ βc = T αβ ˙ γ = T ˙ α ˙ βγ = 0 , (7.48c)which imply the existence of (anti-)chiral superfields by ensuring the Wess–Zumino consistencycondition ¯ D ˙ α χ = 0 = ⇒ (cid:8) ¯ D ˙ α , ¯ D ˙ β (cid:9) χ = 0 . (7.49)While the Bianchi identities are trivially fulfilled by the unconstrained derivatives, this is no longertrue, when introducing constraints whose consequences for the remaining torsion fields have tobe evaluated. Since this procedure is straight-forward, it will not be reproduced here due to thelength of the calculation and the fact, that it may be found in the literature [103, 102, 104, 105]under the name of “solving the Bianchi identities”.After solving the Bianchi identities, all torsions and curvatures can be expressed in terms of afew basic fields, T α := ( − B T αBB , (7.50) G α ˙ α := iT β,β ˙ α,α + iT ˙ β,α ˙ β, ˙ α , (7.51) R := R ˙ α ˙ β ˙ α ˙ β , (7.52) W αβγ := T { ia ˙ β,β | ˙ β | ,γ } , (7.53)where R and ¯ R are chiral and antichiral superfields, G α ˙ α is real, and T α , W αβγ are complexsuperfields, all of which are subject to a set of Bianchi identities and obey the so-called “non-minimal supergravity algebra”.7.3.1 Algebra and Bianchi identitiesThe non-minimal supergravity algebra is defined by the following three (anti-)commutators, (cid:8) D α , ¯ D ˙ α (cid:9) = − i D α ˙ α , (7.54) (cid:8) D α , D β (cid:9) = − RM αβ , (7.55) (cid:2) ¯ D ˙ α , D β ˙ β (cid:3) = ε ˙ α ˙ β (cid:20) ¯ T ˙ γ D β ˙ γ − i ( R + ¯ D ˙ γ ¯ T ˙ γ − ¯ T ) D β − i ¯ ψ β ˙ γ ¯ D ˙ γ + i ( ¯ D ˙ δ − ¯ T ˙ δ ) ¯ ψ β ˙ γ ¯ M ˙ δ ˙ γ + 2 iX γ M βγ − iW βγδ M γδ (cid:21) − i ( D β R ) ¯ M ˙ α ˙ β . (7.56) The missing relations can be determined from the Bianchi identities (see below) and complexconjugation. (cid:8) ¯ D ˙ α , ¯ D ˙ β (cid:9) = 4 R ¯ M ˙ α ˙ β , (7.57) (cid:2) D α , D β ˙ β (cid:3) = ε αβ (cid:20) T γ D γ ˙ β + i ( ¯ R + D γ T γ − T ) ¯ D ˙ β + iψ γ ˙ β D γ + i ( D δ − T δ ) ψ ˙ βγ M δγ − i ¯ X ˙ γ ¯ M ˙ β ˙ γ + 2 i ¯ W ˙ β ˙ γ ˙ δ M ˙ γ ˙ δ (cid:21) + i ( ¯ D ˙ β ¯ R ) M αβ , (7.58) (cid:2) D α ˙ α , D β ˙ β (cid:3) = i (cid:8)(cid:2) D α , D β ˙ β (cid:3) , ¯ D ˙ α (cid:9) + i (cid:8)(cid:2) ¯ D ˙ α , D β ˙ β (cid:3) , D α (cid:9) , (7.59)with the abbreviations ψ α ˙ α = G α ˙ α − D α ¯ T ˙ α − ¯ D ˙ α T α , (7.60) X α = (cid:20) ( ¯ D ˙ γ − ¯ T ˙ γ )( ¯ D ˙ γ − ¯ T ˙ γ ) − R (cid:21) T α + (cid:20) ψ α ˙ α + ( ¯ D ˙ α − ¯ T ˙ α )( D α − T α )+ ( D α − T α )( ¯ D ˙ α − ¯ T ˙ α ) (cid:21) ¯ T ˙ α . (7.61)The Bianchi identities expressed in terms of the supertorsion components read¯ D ˙ α R = 0 , G a = ¯ G a , W αβγ = W { αβγ } , D α T β + D β T α = 0 , ( ¯ D ˙ α − ¯ T ˙ α ) ψ α ˙ α = D α R, ( ¯ D ˙ α − ¯ T ˙ α ) W αβγ = 0 , ( D γ − T γ ) W αβγ = i ( D α ˙ α − i ( D α ¯ T ˙ α )) ψ β ˙ α + ( α ↔ β ) . (7.62)7.3.2 Partial IntegrationFrom the supergravity algebra (7.45) it can be shown that( − A E − D A V A − ( − B V A T ABB = ( E − V A ) ← E A , (7.63)which implies Z d z E − ( D α ˙ α − ( − B T aBB ) V α ˙ α = 0 , (7.64) Z d z E − ( D α + T α ) V α = 0 , (7.65) Z d z E − ( ¯ D ˙ α + T ˙ α ) V ˙ α = 0 . (7.66) E − := sdet − E AM is the real superspace analogue of √− g mn .Clearly it is a natural alternative to consider the combination D α + T α as the basic covariantderivative. Then T α takes over the rˆole of a U(1) R connection, an approach chosen in [103]. A M N := exp STr ln A M N , (7.67)where the supertrace is STr A M N := ( − M A M M , (7.68)which is cyclic and invariant under a suitably defined supertransposition( A sT ) M N := ( − N + M N A N M . (7.69)For practical calculations, the following theorem is much more important z ′ M = e − K z M , K = K M ∂ M , (7.70)sdet ∂z ′ M ∂z N = (1 · e ← K ) , ← K = K M ← ∂ M . (7.71)The right partial derivative ← ∂ M in ← K acts on the components K M and everything to the left of ← K , such that ← K = ( − M ← ∂ M K M + ( − M ( ∂ M K M ) . (7.72)Additionally the following rule holds(1 · e ← K )(e K Φ) = (Φ · e ← K ) . (7.73)Proofs for any of these statements can be found in the literature, in particular [102].7.3.4 Super-Weyl TransformationsWhile the algebra of the previous Sections is by construction invariant under general supercoor-dinate and superlocal Lorentz transformations, it is in addition invariant under transformationsof the vierbein of the form E α LE α , (7.74)¯ E ˙ α ¯ L ¯ E ˙ α , (7.75) E α ˙ α L ¯ LE α ˙ α , (7.76) E ( L ¯ L ) E, (7.77)which are easily seen to represent Weyl transformation of the bosonic vierbein component, whenrestricting L to (the real part of) its lowest component. The unconstrained complex superfield L = exp( ∆ + i κ ) parametrises mixed superlocal scale transformations (by ∆) and superlocalchiral transformations (by κ ). The latter can also be understood as local U(1) R transformations.The elements of the non-minimal supergravity algebra transform under this symmetry as D α L D α − D β L ) M αβ , (7.78)¯ D ˙ α ¯ L ¯ D ˙ α −
2( ¯ D ˙ β ¯ L ) ¯ M ˙ α ˙ β , (7.79) T α LT α + D ′ α ln( L ¯ L ) , (7.80) R
7→ − ( ¯ D − R ) ¯ L . (7.81) F , W and N αµ , called prepotentials , E α = F N αµ e W ∂ µ e − W , det N αµ = 1 , (7.82)¯ E ˙ α = − ¯ F ¯ N ˙ α ˙ µ e ¯ W ¯ ∂ ˙ µ e − ¯ W . (7.83)Because the “superscale” field F has been introduced to allow the choice det N αµ = 1, it is also theonly prepotential that transforms under super-Weyl transformations: F LF . Under coordinatetransformations induced by K = K M ∂ M = ¯ K , all prepotentials transform covariantly, F ′ = (e K F ) , ( N αµ ) ′ = (e K N αµ ) , W ′ = (e K W ) , (7.84)while only N αµ transforms under superlocal transformations( N αµ ) ′ = (e K ab M ab ) N αµ . (7.85)While all supergravity superfields can be expressed in terms of prepotentials, only the twosimple expressions T α = E α ln[ EF (1 · e ← W )] , (7.86) R = − ˆ¯ E ˙ µ ˆ¯ E ˙ µ ¯ F (7.87)shall be given here with the semi-covariant vierbein ˆ E defined byˆ E α := F − E α , ˆ E α =: N αµ ˆ E µ ˆ¯ E ˙ α := ¯ F − ¯ E ˙ α , (7.88)ˆ E α ˙ α := i (cid:8) ˆ E α , ˆ¯ E ˙ α (cid:9) . There is an additional prepotential ϕ , the chiral compensator, that can be chosen to take over therˆole of F , see Sectionrefsec:superweyltrafos.7.4 Minimal SupergravityFrom the non-minimal supergravity algebra, a formulation containing less auxiliary fields maybe obtained by setting T α = 0. This has a number of consequences: The algebra simplifiesconsiderably, W αβγ becomes a chiral field and super-Weyl transformations can be formulatedusing a chiral parameter field.7.4.1 Algebra and Bianchi IdentitiesThe minimal supergravity algebra is determined by the three (anti-)commutators (cid:8) D α , ¯ D ˙ α (cid:9) , (cid:8) D α , D β (cid:9) , (cid:2) D α , ¯ D β ˙ β (cid:3) , which are listed below with some of their straight-forward implications (cid:8) D α , ¯ D ˙ α (cid:9) = − i D α ˙ α , (7.89a) (cid:8) D α , D β (cid:9) = − RM αβ , (7.89b) (cid:8) ¯ D ˙ α , ¯ D ˙ β (cid:9) = 4 R ¯ M ˙ α ˙ β , (7.89c) D α D β = ε αβ D − RM αβ , (7.89d)¯ D ˙ α ¯ D ˙ β = − ε ˙ α ˙ β ¯ D + 2 R ¯ M ˙ α ˙ β , (7.89e) D α D = 4 ¯ R D β ( ε αβ + M αβ ) , (7.89f) D D α = − R D β ( ε αβ + M αβ ) , (7.89g) (cid:2) D , ¯ D ˙ α (cid:3) = − G α ˙ α + i D α ˙ α ) D α + 4 ¯ R ¯ D ˙ α (7.89h) − D γ G δ ˙ α ) M γδ + 8 ¯ W ˙ α ˙ γ ˙ δ ¯ M ˙ γ ˙ δ , (cid:2) ¯ D , D α (cid:3) = 2 i (cid:2) ¯ D ˙ α , D α ˙ α (cid:3) + 4 i D α ˙ α ¯ D ˙ α (7.89i)= − G α ˙ α − i D α ˙ α ) ¯ D ˙ α + 4 R D α −
4( ¯ D ˙ γ G α ˙ δ ) ¯ M ˙ γ ˙ δ + 8 W αγδ M γδ , (cid:2) ¯ D ˙ α , D β ˙ β (cid:3) = − iε ˙ α ˙ β ( R D β + G β ˙ γ ¯ D ˙ γ ) (7.89j) − i ( D β R ) ¯ M ˙ α ˙ β + iε ˙ α ˙ β ( ¯ D ˙ γ G β ˙ δ ) ¯ M ˙ γ ˙ δ − iε ˙ α ˙ β W β γδ M γδ , (cid:2) ¯ D ˙ β , D β ˙ β (cid:3) = − i ( R D β + G β ˙ γ ¯ D ˙ γ ) + 2 i ( ¯ D ˙ γ G β ˙ δ ) ¯ M ˙ γ ˙ δ − iW βγδ M γδ , (7.89k) (cid:2) D α , D β ˙ β (cid:3) = iε αβ ( ¯ R ¯ D ˙ β + G γ ˙ β D γ ) (7.89l)+ i ( ¯ D ˙ β ¯ R ) M αβ − iε αβ ( D γ G δ ˙ β ) M γδ + 2 iε αβ ¯ W ˙ β ˙ γ ˙ δ ¯ M ˙ γ ˙ δ , (cid:2) D β , D β ˙ β (cid:3) = 2 i ( ¯ R ¯ D ˙ β + G γ ˙ β D γ ) − i ( D γ G δ ˙ β ) M γδ + 4 i ¯ W ˙ β ˙ γ ˙ δ ¯ M ˙ γ ˙ δ , (7.89m) (cid:2) D , ¯ D (cid:3) = (cid:2) D , ¯ D ˙ α (cid:3) ¯ D ˙ α − ¯ D ˙ α (cid:2) D , ¯ D ˙ α (cid:3) (7.89n)= 8 iG α ˙ α D α ˙ α − i D α ˙ α (cid:2) D α , ¯ D ˙ α (cid:3) − D α R ) D α + 4( ¯ D ˙ α ¯ R ) ¯ D ˙ α − R D + 8 ¯ R ¯ D − D γ G δ ˙ α ) ¯ D ˙ α M γδ + 8( ¯ D ˙ γ G α ˙ δ ) D α ¯ M ˙ γ ˙ δ (7.89o) − W αγδ D α M γδ + 16 ¯ W ˙ α ˙ γ ˙ δ ¯ D ˙ α ¯ M ˙ γ ˙ δ − D β W βγδ ) M γδ + 8( ¯ D ˙ α ¯ W ˙ α ˙ γ ˙ δ ) ¯ M ˙ γ ˙ δ . In minimal
SUGRA R and W αβγ are chiral fields, G α ˙ α is real. G a = ¯ G a , (7.90a)¯ D ˙ α R = 0 , (7.90b)¯ D ˙ α W αβγ = 0 , W αβγ = W ( αβγ ) . (7.90c)The remaining identities also simplify dramatically,¯ D ˙ α G α ˙ α = D α R, (7.90d) D α G α ˙ α = ¯ D ˙ α ¯ R, (7.90e) D γ W αβγ = i D α ˙ α G β ˙ α + i D β ˙ α G α ˙ α . (7.90f) Some trivial consequences of the above identities are¯ D ˙ α G α ˙ α = −D α R, (7.91) D α G α ˙ α = − ¯ D ˙ α ¯ R, (7.92) D α ˙ α G α ˙ α = i ( D R − ¯ D ¯ R ) , (7.93)( D λ )( ¯ D ¯ λ ) = 4 G α ˙ α ( D α λ )( ¯ D ˙ α ¯ λ ) + 8( D α ˙ α λ )( D α ˙ α ¯ λ )+(total derivative) . (7.94)7.4.2 Chiral Projector and d’AlembertianAs a consequence of (7.89c) as long as R = 0, ¯ D U is no longer chiral ( U being an arbitrarysuperfield). But for tensor superfields carrying no dotted indices the following operator gives acovariantly chiral superfield.¯ D ˙ α ( ¯ D − R ) U α ...α n = 0 ∀ undotted tensor superfield U (7.95)Evidently the flat space limit, R → D U .Since chiral scalar superfields will play an important rˆole in this thesis, the commutators (7.89)acting on chiral scalar fields are worked out explicitly in appendix D. The combination ¯ D − R is also known as the chiral projector .From the chiral projector a generalisation of the d’Alembert operator to the space of (anti-)chiral superfields can be given. The (anti-)chiral d’Alembertian (cid:3) + ( (cid:3) − ) is defined by (cid:3) + := ( D a + iG a ) D a + ( R D α + ( D α R )) D α , (7.96) (cid:3) − := ( D a − iG a ) D a + ( ¯ R ¯ D ˙ α + ( ¯ D ˙ α ¯ R )) ¯ D ˙ α , (7.97)and maps to (anti-)chiral fields as long as it acts on (anti-)chiral fields. In this case (cid:3) + ( (cid:3) − ) maybe rewritten in the following manner, (cid:3) + λ = ( ¯ D − R ) D λ, (7.98) (cid:3) − ¯ λ = ( D − R ) ¯ D ¯ λ, (7.99)which makes manifest the (anti-)chirality property.Also note that ¯ D D λ = 16( (cid:3) + + R D ) λ .7.4.3 Super-Weyl TransformationsThe condition T α = 0 is only invariant under a subset of the mixed super-Weyl/local U(1) R trans-formations discussed in Section 7.3.4. To ensure that 0 maps to 0 under those transformations,from 0 = T α LT α + L D α ln( L ¯ L ) = 0 , (7.100)the condition D α ln( L ¯ L ) = 0 is read off. Consequently the parameter L is restricted to be ofthe form L = exp( σ − ¯ σ ) , ¯ D ˙ α σ = D α ¯ σ = 0 , ¯ L = exp( ¯ σ − σ ) . (7.101) The minimal supergravity fields transform according to D ′ α = L D α − D β L ) M αβ , (7.102) R ′ = − ( ¯ D − R ) ¯ L , (7.103) G ′ α ˙ α = L ¯ LG α ˙ α + ¯ L D α ¯ D ˙ α L − L ¯ D ˙ α D α ¯ L (7.104) W ′ αβγ = L ¯ LW αβγ , (7.105)or in terms of σ and ¯ σ , D ′ α = e σ − ¯ σ ( D α − ( D β σ ) M αβ ) , (7.106) R ′ = − e − σ [( ¯ D − R ) e ¯ σ ] , (7.107) G ′ α ˙ α = e − ( σ +¯ σ ) / (cid:2) G α ˙ α + ( D α σ )( ¯ D ˙ α ¯ σ ) + i (cid:0) D α ˙ α (¯ σ − σ ) (cid:1)(cid:3) , (7.108) W ′ αβγ = e − σ/ W αβγ . (7.109)Formulating ¯ T ˙ α = 0 in terms of prepotentials (7.86) yields the important equation¯ E ˙ α ϕ = 0 , ϕ := E − ¯ F − (1 · e ← ¯ W ) − , (7.110)where the exponent of “3” is for convenience as is seen in the next equation. Since for any scalar¯ D ˙ α ≡ ¯ E ˙ α , the field ϕ is chiral and transforms under generalised super-Weyl transformations into ϕ [( L ¯ L ) − E − ][ ¯ L − ¯ F − ](1 · e ← ¯ W ) − = L − ¯ L − ϕ = (e σ ϕ ) . (7.111)This makes ϕ the compensating field for super-Weyl transformations. Accordingly it is called chiral compensator .7.4.4 Chiral Representation and Integration RulePerforming the picture changing operation˜ V = e − ¯ W V, (7.112)˜ D A = e − ¯ W D A e ¯ W = ˜ E AM ∂ M + ˜Ω bcA M bc , (7.113)and additionally going to the gauge N αµ = δ αµ introduces the so-called chiral representation . Theimportant feature of the chiral representation is that the spinorial vielbein ˜¯ E ˙ α = − ¯ F ¯ ∂ ˙ α takes amost simple form, while ˜ E α and complex conjugation are more complicated than in the vectorrepresentation used so far. The determinant of the vierbein becomes˜ E − = ( E − e − ← W ) , (7.114)such that Z d z ˜ E − ˜ L = Z d z ( E − e − ← W ) e − W L (7.73) = Z d z E − L . (7.115)In chiral representation, equations (7.110) and (7.87) read˜ ϕ ¯ F = ˜ E − , (7.116)˜ R = ¯ ∂ ˙ µ ¯ ∂ ˙ µ ¯ F , (7.117) which combined yield ˜ ϕ ˜ R = ¯ ∂ ˙ µ ¯ ∂ ˙ µ ˜ E − , (7.118)= ⇒ ˜ ϕ ˜ L c = ¯ ∂ ˙ µ ¯ ∂ ˙ µ (cid:18) ˜ E − ˜ R ˜ L c (cid:19) . (7.119)This gives the important chiral integration rule Z d z ˜ ϕ ˜ L c = Z d z ˜ E − ˜ R ˜ L c (7.115) = Z d z E − R L , (7.120)due to d ¯ θ = ¯ ∂ ˙ µ ¯ ∂ ˙ µ .7.5 Component Expansion7.5.1 Superfields and First Order OperatorsIn supergravity as opposed to flat supersymmetry, the (non-linearised) components of a superfieldare given in terms of covariant derivatives D α and ¯ D ˙ α and are in one-to-one correspondence tothe coefficients in the usual θ, ¯ θ expansion of a superfield. f (cid:12)(cid:12) D α f (cid:12)(cid:12) ¯ D ˙ α f (cid:12)(cid:12) − D f (cid:12)(cid:12) − ¯ D f (cid:12)(cid:12) (cid:2) D α , ¯ D ˙ α (cid:3) f (cid:12)(cid:12) (7.121) − D α ¯ D f (cid:12)(cid:12) − ¯ D α D f (cid:12)(cid:12) − (cid:8) D , ¯ D (cid:9) f (cid:12)(cid:12) Here, the notation f (cid:12)(cid:12) := f ( x, θ = 0 , ¯ θ = 0) (7.122)has been introduced.For arbitrary super fields f and f , it holds( f f ) (cid:12)(cid:12) = f (cid:12)(cid:12) f (cid:12)(cid:12) , (7.123)which obviously can no longer be true when f is an operator containing derivatives on anticom-muting coordinates.The space projection of a general first order differential operator O = O M ( z ) ∂ M + O ab ( z ) M ab (7.124)is defined to be O (cid:12)(cid:12) = O M (cid:12)(cid:12) ∂ M + O ab (cid:12)(cid:12) M ab . (7.125)Acting with such an operator on an arbitrary superfield (with Lorentz indices of f suppressed),one immediately sees that ( O f ) (cid:12)(cid:12) = ( O M ∂ M f ) (cid:12)(cid:12) + ( O ab M ab f ) (cid:12)(cid:12) = O M (cid:12)(cid:12) ∂ M f (cid:12)(cid:12) + O ab (cid:12)(cid:12) M ab f (cid:12)(cid:12) (7.126)= ( O (cid:12)(cid:12) f ) (cid:12)(cid:12) is different from O (cid:12)(cid:12) f (cid:12)(cid:12) = O m (cid:12)(cid:12) ∂ m f (cid:12)(cid:12) + O ab (cid:12)(cid:12) M ab f (cid:12)(cid:12) . (7.127) Using pure superspace methods, it is possible (though tedious) to show, that in Wess–Zuminogauge the vector derivative has the following expansion, D α ˙ α (cid:12)(cid:12) = ∇ α ˙ α (cid:12)(cid:12) + Ψ α ˙ α,β D β (cid:12)(cid:12) + ¯Ψ α ˙ α, ˙ β ¯ D ˙ β (cid:12)(cid:12) , (7.128)with Ψ the gaugino field strength. As a simple example, the expansion of D α ˙ α f is given,( D α ˙ α f ) (cid:12)(cid:12) = ( D α ˙ α (cid:12)(cid:12) f ) (cid:12)(cid:12) = ∇ α ˙ α ( f (cid:12)(cid:12) ) + Ψ α ˙ α,β (cid:0) ( D β f ) (cid:12)(cid:12) (cid:1) + ¯Ψ α ˙ α,β (cid:0) ( ¯ D β f ) (cid:12)(cid:12) (cid:1) . (7.129)More complicated combination of the derivatives D α , ¯ D ˙ α and D α ˙ α acting on a field require re-arrangement such that the leftmost derivative is of vector type. Then the above rule (with f containing the remaining derivatives) can be used to recursively reduce the superspace derivativesto space-time covariant derivatives ∇ α ˙ α until only expressions containing component combinations(7.121) of the spinorial derivatives are left over. Due to the three-folding caused by each applica-tion of (7.129), let alone the required rearrangement of vector derivatives to the left, even termswith a relatively small number of derivatives may grow dramatically. The situation is (slightly)better when one is not interested in terms containing the gaugino field strength. Therefore, theoperator (cid:12)(cid:12) b shall denote space-time projection while simultaneously neglecting all gravitationalfermionic and auxiliary fields.7.5.2 Supergravity FieldsThe derivation of the component expansion in Wess–Zumino gauge is rather involved and onlythe final expression shall be reproduced here. The real part of the prepotential W can be gaugedaway, but requiring instead the conditionexp( ¯ W n ∂ n ) x m = x m + i H m ( x, θ, ¯ θ ) H m = ¯ H m (7.130)defines the gravitational Wess–Zumino gauge, also called gravitational superfield gauge. In thisgauge, the gravitational degrees of freedom are encoded in the gravitational superfield H m andthe chiral compensator ˆ ϕ ( x, θ ). H m = θσ a ¯ θe am + i ¯ θ θ α ψ mα − iθ ¯ θ ˙ α ¯Ψ m ˙ α + θ ¯ θ A m ˆ ϕ = e − (1 − iθσ a ¯Ψ a + θ B ) ˆ ϕ = e − ¯ W ϕ (7.131)ˆ¯ ϕ = e − (1 − i ¯ θ ˜ σ a Ψ a + ¯ θ ¯ B )In Wess–Zumino gauge, the spinorial semi-covariant vierbein fields (7.88) coincide with thepartial derivatives and can therefore be used to extract the components of the above gravitationalsuperfields just as in flat supersymmetry.The spinorial semi-covariant vierbein fields ˆ E α , ˆ¯ E ˙ α were defined by just pulling out a factor of F from the covariant spinorial derivatives D α , ¯ D ˙ α . In addition without proof, for the prepotential F it holds F (cid:12)(cid:12) = 1 , ˆ E α F = − i ¯Ψ α ˙ β ˙ β , (7.132)such that D α O (cid:12)(cid:12) = ˆ E α O (cid:12)(cid:12) , − D O (cid:12)(cid:12) = − ˆ E O (cid:12)(cid:12) + i ¯Ψ α ˙ β, ˙ β D α O (cid:12)(cid:12) . (7.133) This allows to write down the chiral compensator’s components in terms of covariant derivatives ϕ (cid:12)(cid:12) = e − , (7.134a) D α ϕ (cid:12)(cid:12) = − ie − ( σ a ¯Ψ a ) α , (7.134b) − D ϕ (cid:12)(cid:12) = e − ( B − ¯Ψ˜ σσ ¯Ψ) , (7.134c)where ¯Ψ˜ σσ ¯Ψ = − ¯Ψ α ˙ β, ˙ β ¯Ψ α ˙ γ, ˙ γ . In other words ϕ (cid:12)(cid:12) = e − / (7.135a) D α ϕ (cid:12)(cid:12) = − ie − / ( σ a ¯Ψ a ) α (7.135b) − D ϕ (cid:12)(cid:12) = 13 e − / ( B −
13 ¯Ψ˜ σσ ¯Ψ) (7.135c)For the chiral supertorsion component:¯ R (cid:12)(cid:12) = 13 B , B = B + ¯Ψ a ˜ σ a σ b ¯Ψ b + ¯Ψ a ¯Ψ a , (7.136a)¯ D ˙ α ¯ R (cid:12)(cid:12) = 43 ¯Ψ ˙ α ˙ β, ˙ β + i B Ψ β ˙ α,β , (7.136b)¯ D ¯ R (cid:12)(cid:12) = 23 ( R + i ε abcd R abcd ) + 89 ¯ BB − B (Ψ a σ a ˜ σ b Ψ b + Ψ a Ψ a )+ i ¯ D ˙ α ¯ R (cid:12)(cid:12) (˜ σ b Ψ b ) ˙ α + 2 i α ˙ α,β D { α G β } , ˙ α (cid:12)(cid:12) , (7.136c)where R denotes the Ricci scalar, tensor or Riemann tensor, respectively.For the real supertorsion component: G a (cid:12)(cid:12) = 43 A a , (7.137a) A a = A a + 18 ε abcd C bcd −
14 (Ψ a σ b ¯Ψ b + Ψ b σ b ¯Ψ a ) −
12 Ψ b σ a ¯Ψ b + i ε abcd Ψ b σ c ¯Ψ d , (7.137b)¯ D { ˙ α G β ˙ β } (cid:12)(cid:12) = − ˙ α ˙ β,β + i B ¯Ψ β { ˙ α, ˙ β } , (7.137c)¯ D { ˙ α D { γ G δ } ˙ β } (cid:12)(cid:12) = 2 E γδ ˙ α ˙ β + 2 i Ψ { γ { ˙ α,δ } ¯ D ˙ β } ¯ R (cid:12)(cid:12) − i Ψ α { ˙ α,α D { γ G δ } ˙ β } (cid:12)(cid:12) +2 i ¯Ψ α { ˙ α, ˙ β } W αγδ (cid:12)(cid:12) + 23 B (˜ σ ab ) ˙ α ˙ β Ψ aγ Ψ bδ , (7.137d)with E ab := (cid:0) R ab + i ( ε acde R cdeb + ε bcde R cdea − η ab ε cdef R cdef ) (cid:1) , (7.138a)˜ R ab := ( R ab + R ba ) − η ab R = G { ab } + η ab R . (7.138b) S = Z d z E − L = Z d z E − R (cid:0) − (cid:1) ( ¯ D − R ) L | {z } =: L c (7.139)Then the following manipulations, which crucially depend on the semi-covariant vierbein coin-ciding (7.88) with the partial derivatives in Wess–Zumino gauge, lead to the density formula S = Z d z ˆ ϕ ˆ L c = Z d x ∂ α ∂ α ( ˆ ϕ ˆ L c ) = − Z d x ˆ E ( ϕ L c ) (cid:12)(cid:12) = − Z d x ϕ (cid:12)(cid:12) ˆ E L c (cid:12)(cid:12) + 2 D α ϕ (cid:12)(cid:12) D α L c (cid:12)(cid:12) + ˆ E ϕ (cid:12)(cid:12) L c (cid:12)(cid:12) = Z d x ϕ (cid:12)(cid:12) ( − D L c ) (cid:12)(cid:12) − D α ϕ (cid:12)(cid:12) D α L c (cid:12)(cid:12) + B L c (cid:12)(cid:12) . (7.140)where B = B − ¯Ψ a ˜ σ a σ b ¯Ψ b − ¯Ψ a ¯Ψ a = − D ϕ + e − ¯Ψ˜ σσ ¯Ψ. This Chapter is meant to give a short introduction into the space-time dependent couplings tech-nique and its application to a proof of Zamolodchikov’s c -theorem in two dimensions. Additionallythe four dimensional trace anomaly and some of the problems encountered when trying to extendthe theorem to four dimensions are discussed.8.1 Weyl Transformations8.1.1 Conformal Killing EquationA Weyl transformation is a rescaling of the metric by a space-time dependent factor g mn e − σ g mn . (8.1)Upon restriction to flat space these transformations generate the conformal group, which locallypreserves angles.Using δg mn = − σg mn ,δx m = ξ m ,δdx m = ( ∂ n ξ m ) dx n , (8.2)the requirement of invariance of the line element δ ( ds ) ! = 0 = [ − σg mn + ∂ m ξ n + ∂ n ξ m ] dx m dx n (8.3)amounts to the conformal Killing vector equation ∂ m ξ n + ∂ n ξ m = d ∂ k ξ k g mn ,σ = d ∂ k ξ k , (8.4) Conformal Transformations
Name Group Element Generatortranslations x a x a + a a P a (Lorentz ) rotations x a Λ ab x b M ab dilation x a λx a D SCT x a x a + b a x Ω SCR ( x ) K a Table 9
Finite Conformal Transformations where d is the dimension of space time.Under (8.2), the action transforms as follows, δS = Z d d x δSδg mn δg mn = Z d d x (cid:2) − T mn ][ − σg mn ] , (8.5)which demonstrates that for conformal invariance the trace of the energy-momentum tensor hasto vanish.As an aside, in two dimensions after Wick rotation the conformal Killing vector equationbecomes the Cauchy–Riemann system, such that conformal transformations are given by holo-morphic or antiholomorphic functions. Decomposing these functions by a Laurent expansiondemonstrates that the two dimensional conformal group has infinitely many generators, whichform the Witt/Virasoro algebra.The four dimensional case is generic and will be discussed below.8.1.2 Conformal Algebra in d > d > ξ a ( x ) = a a + ω ab x b + λx a + ( x b a − x a x b b b ) (8.6)with the corresponding generators δ C = ia a P a + iω ab M ab + iλD + ib a K a , (8.7)which form the conformal algebra[ M ab , P c ] = − iP [ a η b ] c , [ M ab , K c ] = − iK [ a η b ] c , [ D, P a ] = − iP a , [ D, K a ] = iK a , [ D, M ab ] = 0 , [ P a , K b ] = 2 i ( M ab − η ab D ) , [ M ab , M cd ] = 2 i (cid:0) η a [ c M d ] b − η b [ c M d ] a (cid:1) . (8.8)This can be identified with the algebra so ( d,
2) by defining a suitable ( d + 2) × ( d + 2) matrix M ˆ m ˆ n := M mn ( K m − P m ) ( K m + P m ) − ( K m − P m ) 0 − D − ( K m + P m ) D (8.9) Λ ca η cd Λ db = η ab Special Conformal Transformation and choosing η ˆ m ˆ n = diag( η mn , , −
1) as metric. As an aside, the d -dimensional conformal algebrais identical to the ( d + 1)-dimensional ads algebra cf d ≡ ads d +1 ≡ so (2 , d ) . (8.10)The finite transformations corresponding to the infinitesimal solutions (8.6) are shown in Figure 9,where Ω SCT ( x ) := 1 − ~b · ~x + b ~x is the scale factor Ω of the metric for special conformaltransformations, and ~a · ~b has been used as a short-hand for η mn a m b n .8.1.3 Weyl Transformations of the Riemann TensorSince superspace supergravity is described using a tangent space formulation, which has the addi-tional advantage of a metric δ [ η ab ] = 0 invariant under Weyl transformations, the transformationalbehaviour of the Riemann R abcd and Weyl C abcd tensor, Ricci tensor R ab and scalar R , and co-variant derivative ∇ under δ [ g mn ] = − σg mn shall be given in terms of tangent space objects. δ [ e am ] = σe am , (8.11a) δ [ p − det g ] = δ [det e − ] = − σd p − det g = − σd det e − , (8.11b) δ [ R abcd ] = δ [ a [ c ∇ b ] ∇ d ] σ + 2 σ R abcd , (8.11c) δ [ R abcd ] = η [ c [ a ∇ b ] ∇ d ] σ + 2 σ R abcd , (8.11d) δ [ R ab ] = η ab ∇ σ + 2 ∇ a ∇ b σ + 2 σ R ab , (8.11e) δ [ R ] = 6 ∇ σ + 2 σ R , (8.11f) δ [ G ab ] = δ [ R ab ] − η ab δ [ R ]= − η ab ∇ σ + 2 ∇ a ∇ b σ + 2 σ G ab , (8.11g) δ [ C abcd ] = 2 σC abcd , (8.11h) δ [ ∇ a ] = σ ∇ a − ( ∇ b σ ) M ab , M ab V c = δ ca V b − δ cb V a , (8.11i) δ [ ∇ a λ ] = σ ∇ a λ, (8.11j) δ [ ∇ λ ] = 2 σ ( ∇ λ ) + (2 − d )( ∇ a σ )( ∇ a λ ) , (8.11k)where d is the space-time dimension, which from now on will be assumed to be equal to four.8.1.4 Weyl Covariant Differential OperatorsBy definition a field ψ is denoted conformally covariant if it transforms under Weyl transformationsinto e wσ ψ , that is homogeneously with Weyl weight w . In particular, it is interesting to haveinvariant expressions of the form Z d x e − χ ∗ ∆ − w ψ, (8.12)with ∆ − w a differential operator of order 4 − w and ψ , χ are assumed to be Lorentz scalars.The unique local, Weyl covariant differential operator acting on such fields ψ and χ of Weylweight 1 is given by ∆ = ∇ − R , (8.13)which can be easily verified using relations (8.11). It is however entertaining to derive this expres-sion in a slightly different manner. General relativity is not invariant under Weyl transformations as can be seen from the Einstein–Hilbert action transforming according to Z d x e − R 7→ Z d x e − [e − σ R + 6( ∇ a e − σ )( ∇ a e − σ )] . (8.14)Since Weyl transformations form an Abelian group, a parametrisation may be chosen where twoconsecutive transformations with parameters σ and σ correspond to a single Weyl transformationwith parameter σ + σ . (Evidently e am e σ e am is such a parametrisation.) Replacing theparameter of the first transformation by a field φ = e − σ of Weyl weight 1 yields an invariantexpression as can be seen frome σ e am = φ − e am (e − σ φ − )(e σ e am ) = φ − e am . (8.15)Therefore, the following action is Weyl invariant Z d x e − [ φ R + 6( ∇ a φ )( ∇ a φ )] = 6 Z d x e − φ [ ∇ − R ] φ (8.16)and the operator ∆ has been rederived.In addition the important notion of a compensating field , here φ , has been introduced. Com-pensating fields allow incorporating a symmetry into the formulation of a theory that originallywas not part of it. An analogue procedure is needed to embed Poincar´e supergravity into theWeyl invariant supergravity algebra by use of a so-called chiral compensator .Unfortunately, the elegant method above does not lend itself to generalisations and clearlycannot be used to construct a conformally covariant operator for a field of vanishing Weyl weight.However a dimensional analysis can be used to write down a basis for such an operator anddetermine the prefactors from Weyl variation. The following operator due to Riegert [54] is theunique conformally covariant differential operator of fourth order, which because of its importancefor this work will be given in several equivalent forms,∆ := ∇ + 2 G ab ∇ a ∇ b + ∇ a R∇ a = ∇ + 2 G ab ∇ a ∇ b + ( ∇ a R ) ∇ a + R∇ = ∇ + 2 R ab ∇ a ∇ b + ( ∇ a R ) ∇ a − R∇ = ∇ + 2 ∇ a R ab ∇ b − ( ∇ a R ) ∇ a − R∇ , (8.17)or partially integrated, λ ′ ∆ λ = ( ∇ λ )( ∇ λ ′ ) − G ab ( ∇ a λ )( ∇ b λ ′ ) − R ( ∇ a λ )( ∇ a λ ′ ) + (total deriv.) . (8.18)8.2 Zamolodchikov’s c -Theorem in Two DimensionsIn a classical theory scale invariance is expected at the ultraviolet limit where particle massesmay be neglected and at the infrared limit where massive particles decouple from the theory. Inthis sense the transition from UV to IR is irreversible in a classical theory. For simple theoriesscale invariance (which implies one additional symmetry generator) may be enough to establishconformal symmetry (which in two dimensions implies an infinite set of symmetry generators andis thus a much larger symmetry). At the quantum level, conformal invariance is often broken.Still there are many known examples of two dimensional theories which flow from one conformalfixed point in the UV to another one in the IR . In four dimensions the existence of conformal fixedpoints is much more difficult to establish. PSfrag replacements IR limit cycle λ j λ i UV fixed point Fig. 23
Limit Cycle in the Space of Couplings
The breaking of conformal invariance at the quantum level is induced by the introduction ofa regulator during renormalisation, which creates a scale µ that leads to non-vanishing anomalyterms in the trace of the energy-momentum tensor.Renormalisation group ( RG ) theory describes the change of the effective Hamiltonian of a theoryduring the change of scale. The breaking of scale invariance is described by the RG equation µ ddµ W = µ ∂∂µ W + β i ∂∂λ i W = 0 , (8.19) β i := µ ∂λ i ∂µ , (8.20) W = W ( λ i , µ ) , (8.21)where W is the generating functional of the connected Green’s functions, which due to being aformal series expansion of physical observables is expected to be RG invariant, that is constantwith respect to the scale µ .From a mathematical point of view, there is no reason a theory should not exhibit a complexflow behaviour. In particular the RG flow could approach a limit cycle, see Figure 23, possiblymaking the theory increase and decrease its number of degrees of freedom periodically while goingto lower and lower energies. Since this is certainly an unphysical behaviour, a natural question isunder which conditions such a behaviour cannot be displayed by a quantum field theory.A partial answer to this question was given by Zamolodchikov’s fundamental theorem [47] intwo dimensions, which states the irreversibility of RG flows connecting two fixed points in twodimensions. Theorem 1 (Zamolodochikov 1986) . “There exists a function c ( g ) of the coupling constant g ina 2D renormalisable field theory which decreases monotonically under the influence of a renormal-isation group transformation. This function has constant values only at fixed points, where c isthe same as the central charge of a Virasoro algebra of the corresponding conformal field theory.”Therefore, it holds c UV ≥ c IR , (8.22)where c is the respective value of central charge at the infrared and ultraviolet. c -theorem was hoped to soon be generalisedto four dimensions, but an accepted proof is outstanding for 20 years.The first obstacle that arises is the question of which quantity is to take over the rˆole of thetwo dimensional central charge c , which in two dimensions turns up as the central charge of theconformal algebra, as the coefficient of the two point function of the energy-momentum tensor,and as the anomalous contribution to the trace of the energy-momentum tensor.In the four dimensional trace anomaly, the following constants appear (cid:10) T mm (cid:11) = c C + a ˜ R + b R + f (cid:3) R , (8.23)where R is the scalar curvature (Ricci scalar), C is the square of the Weyl tensor, and ˜ R is theEuler density, C := C abcd C abcd = R abcd R abcd − R ab R ab + R , (8.24)˜ R := R abcd R abcd − R ab R ab + R . (8.25)There are known counter examples for a “ c ”-theorem in four space-time dimensions but thatstill leaves open the possibility of an a -theorem [50], which holds in all examples that permitexplicit checking. Since these are supersymmetric theories, it may well be that supersymmetryis a necessary ingredient for the irreversibility of RG flows. (As an aside in all known examplesof holographic renormalisation group flows that permit determination of the anomaly coefficientson both ends of the flow it holds c = a . On the supergravity side monotonicity of the flow isrelated to energy conditions as they have to be employed in causality considerations in Einsteingravity [7].) Often by an abuse of language the a -theorem is also called c -theorem, even thoughthe prefactor of Euler density is conventionally denoted “ a ”.8.4 Local RG Equation and the c -TheoremThe analysis of this Section will be confined to idealised renormalisable field theories that are clas-sically conformally invariant and involve a set of coupling constants λ i corresponding to local scalaroperators O i . Due to conformal invariance the coupling constants should have mass dimensionzero such that the operator’s mass dimension should be equal to the space-time dimension.When the theory is not conformally invariant on the quantum level the trace of the energy-mo-mentum tensor is non-vanishing and can be expressed in terms of some operator basis formed by O i (cid:10) T mm (cid:11) = β i (cid:10) [ O i ] (cid:11) , (8.26)where [ O i ] denotes a (by some renormalisation scheme) well-defined operator insertion and β i arethe beta functions associated to the corresponding couplings λ i .When Weyl symmetry is preserved during quantisation, the beta functions and therefore thetrace of the energy-momentum tensor vanish.Promoting the coupling constants λ i to fields as well as the metric, λ i λ i ( x ) , (8.27) η mn g mn ( x ) , (8.28)allows to give well-defined expressions for the operators O i (the bracket indicating that the oper-ator is well-defined will be silently dropped, henceforth) and the energy-momentum tensor, O i ( x ) := δδλ i ( x ) W, T mn ( x ) := 2 δδg mn ( x ) W. (8.29) This requires the theory to be defined for a general curved background metric g mn . In addition tothe counterterms present in the QFT on flat space with constant couplings, which give rise to theusual running of couplings, generically there should be now also counterterms A depending on thecurvature and on ∂ m λ i , which vanish in the limit of constant couplings and metric. In particular(8.26) acquires additional contributions according to (cid:10) T mm (cid:11) = β i (cid:10) O i (cid:11) + ∇ m (cid:10) J m (cid:11) + A , (8.30)with J m a local current. In general the trace above is not a local expression, which is why itwas important to introduce space-time dependent couplings to give a meaning to any products offinite operators by functional derivatives with respect to couplings or the metric. The essentialassumption is that the anomaly A stays a local expression to all orders, or in other words thatthe non-local contribution to the vacuum expectation value of the trace is contained in (cid:10) O i (cid:11) .The statement (8.30) can be recast in the form∆ Wσ W = ∆ βσ W − Z d D x √ g A ( σ, R abcd , ∂ m λ i ) , (8.31)where W = ln R [ dφ ] exp( − S/ ~ ) is the generating functional of the connected Green’s functions, σ is the parameter of Weyl transformation generated by ∆ Wσ and∆ Wσ := 2 Z dV g mn δδg mn , dV = d D x √ g, (8.32)∆ βσ := Z dV σβ i δδλ i , (8.33)with D the number of space-time dimensions.Equation (8.31) is in effect a local version of the (anomalous) Callan–Symanzik equation (cid:20) µ ∂∂µ + β i ∂∂λ i (cid:21) W = A . (8.34)The shape of A ( σ, R abcd , ∂ m λ i ) is restricted by power counting and the requirement to vanish inthe flat space/constant coupling limit, such that in this limit the local RG equation (8.31) reducesto the homogeneous Callan–Symanzik equation when imposing the condition (cid:20) µ ∂∂µ + 2 g mn δδg mn (cid:21) W = 0 , (8.35)which is a consequence of na¨ıve dimensional analysis.As a simple example a possible parametrisation of the ambiguous anomaly in two dimensionsis (∆ Wσ − ∆ βσ ) W = Z dV (cid:20) σ (cid:0) c R + χ ij ∂ m λ i ∂ m λ i (cid:1) + ( ∂ m σ ) w i ∂ m λ i (cid:21) , (8.36)with c , χ ij and w i arbitrary function of the couplings, which may be determined in a perturba-tive expansion with the assumption that the above shape is preserved to all orders, and partialderivatives ∂ i := ∂ λ i .A further constraint on the anomaly with far less trivial consequences arises from Weyl trans-formations being Abelian, which implies (cid:2) ∆ Wσ − ∆ βσ , ∆ Wσ ′ − ∆ βσ ′ (cid:3) = 0 . (8.37) In this formulation the anomaly is of course only determined up to partial integrations. Furthermore it isonly defined up to adding local counterterms to the vacuum energy functional W . This Wess–Zumino consistency condition renders the determination of the trace anomaly an al-gebraic (cohomological) problem.In the case of two dimensions (8.36) the consistency condition yields (cid:2) ∆ Wσ − ∆ βσ , ∆ Wσ ′ − ∆ βσ ′ (cid:3) = Z dV ( σ ′ ∂ m σ − σ∂ m σ ′ ) V m , (8.38) V m = ( ∂ m λ i )( ∂ i ( c + w j β j ) − χ ij β j + ( ∂ i w j − ∂ j w i ) β j ) (8.39)and therefore the following coefficient consistency condition holds β i ∂ i ( c + w j β j ) = χ ij β i β j . (8.40)The arbitrariness of W with respect to local functionals of the fields δW = Z dV ( b R − c ij ∂ m λ i ∂ m λ j ) (8.41)implies for the coefficients δc = β i ∂ i b, δχ ij = L β χ ij = β k ∂ k c ij + 2 β i β k c kj , (8.42) δw i = − ∂ i b + c ij β j , δ ( c + w j β j ) = c ij β i β j . (8.43)The Zamolodchikov metric G ij , G ij ( t ) = 18 ( x ) (cid:10) O i ( x ) O j (0) (cid:11) , t = ln µ x , (8.44)is positive by unitarity (or reflection positivity in Euclidean space). It can be shown that G ij = χ ij + L β c ij .Then the function C := 3( c + w i β i + c ij β i β j ) (8.45)is monotonic by (8.40) and positive definiteness of G ij , C ′ = − β i ∂ i C = − G ij β i β j < . (8.46)This is Zamolodchikov’s famous c -theorem.Of course there is more to be said about renormalisation scheme dependence, for details see[52]. Here it shall suffice to mention that equation (8.40) is invariant under (8.41).8.4.1 a -TheoremThe same calculation can be repeated in four space-time dimensions, giving rise to a system ofcoefficient consistency equations much more involved than the two dimensional example. Thecomplete set of anomaly terms and consistency equations shall not be reproduced here, the inter-ested reader is referred to [52] instead.Omitting a number of less interesting terms, a sketch of the four dimensional trace anomaly isgiven by [∆ Wσ − ∆ βσ ] W = Z dV σ (cid:2) a ˜ R + c C + b R + χ gij G mn ∂ m λ i ∂ n λ j + χ aij ∇ λ i ∇ λ j + χ bijk ∂ m λ i ∂ m λ j ∇ λ k + . . . (cid:3) (8.47)+ Z dV ∂ m σ (cid:2) S ij ∂ m λ i ∇ λ j + . . . (cid:3) , with ˜ R , C , R , G mn the Euler density, square of the Weyl tensor and Ricci scalar and theEinstein tensor, respectively.The coefficient consistency equation analogue to (8.40) reads β i ∂ i ( a + w j β j ) = χ gij β i β j . (8.48)By virtue of a further consistency equation, χ gij + 2 χ aij + 2 ∂ i β k χ akj + β k χ bkij = L β S ij , (8.49)where − χ aij can be shown to be positive definite, there might be hope to find a four-dimensional“a-theorem”, when getting under control the other coefficients χ bkij and S ij . In the bosonic sectordiscussed by Osborn, this seems not feasible. However there might be additional constraints insupersymmetric theories. This is the topic of the next Chapter. This Chapter generalises the local renormalisation group equation reviewed in the previous Chap-ter to a minimal supergravity framework. A basis for the trace anomaly is found and the conse-quences of the Wess–Zumino consistency conditions for super-Weyl transformations are evaluated.9.1
SUSY
Local RG EquationThe (integrated) local Callan–Symanzik ( CS ) equation of the previous Chapter reads[ Z d x √− g σ ( x ) 2 g mn δδg mn + Z d x √− g σ ( x ) β i δδλ i ( x ) ] W = Z d x √− gA ( σ, λ i ) . (9.1)Generically the action for a supersymmetric Yang–Mills theory reads S = 18 π λ Z d z Tr W α W α + c.c. , (9.2) W α = −
18 ¯ D (e − V D α e V ) , (9.3)with λ the coupling constant, which may be complex, λ = 4 πg − iθ π . (9.4)Because the action is chiral it is natural to promote the complex couplings to chiral fields as well.Coupling to minimal supergravity, which is both the simplest and best explored choice, impliesthat the Weyl parameter σ ( x ) becomes a chiral field too. Furthermore the supersymmetric gen-eralisation of the trace of the energy-momentum tensor (“supertrace”) is also chiral and definedby T = ϕ δSδϕ . (9.5)The supertrace is related to the supercurrent by¯ D ˙ α T α ˙ α = − D α T , (9.6) where the supercurrent is defined by T α ˙ α = δSδ H α ˙ α , (9.7)with H α ˙ α corresponding to the gravitational superfield. Accordingly a
SUSY version of (9.1) should be given by [106] (cid:20)Z d z σ ϕ δδϕ − Z d z σ β i δδλ i + c.c. (cid:21) W = A + c.c. , (9.8)where A denotes the anomaly which consists entirely of terms that contain supergravity fields ordepend on a derivative of λ or ¯ λ , A = Z d zφ σ A . (9.9)Using the differential operators ∆ Wσ, ¯ σ := ∆ W + ¯∆ W , (9.10)∆ βσ, ¯ σ := ∆ β + ¯∆ β , (9.11)∆ W := Z d z σ φ δδφ , (9.12)∆ β := Z d z σβ δδλ , (9.13)the SUSY local RG equation can be recast into the form(∆ W − ∆ β ) W = A + ¯ A. (9.14)It is convenient to additionally split this local CS equation into a chiral and anti-chiral equation,(∆ W − ∆ β ) W = A, (9.15)( ¯∆ W − ¯∆ β ) W = ¯ A, (9.16)which gives rise to the following two Wess–Zumino consistency conditions, (cid:2) ∆ Wσ − ∆ βσ , ∆ Wσ ′ − ∆ βσ ′ (cid:3) W = 0 , (9.17) (cid:2) ∆ W ¯ σ − ∆ β ¯ σ , ∆ Wσ − ∆ βσ (cid:3) W = 0 . (9.18)It remains to find a suitable expression for the anomaly A .9.2 Basis for the Trace AnomalyIn this Section a basis of dimension two operators is constructed that consists strictly of super-gravity superfields (supertorsions) and covariant chiral derivatives and furthermore contains nofields with negative powers. By assumption (see Section 9.1) the Weyl parameter σ and the couplings λ i are chiral scalarfields.The strategy for finding a basis of dimension two operators is as follows. To be precise, it is the quantum superfield associated to the gravitational superfield H α ˙ α in quantum-background splitting. In Wess–Zumino gauge the lowest component of the gravitational superfield H α ˙ α containsthe vierbein. Due to the peculiarities of curved superspace there is actually a seemingly non-local term namely R − W αβγ W αβγ , which is Weyl covariant by itself and could be trivially included in the discussion. The expressionis related to the Pontryagin invariant. Supergravity Fieldsquantity dimension undotted dotted R R D D D α ˙ α G W W Table 10
Dimensional Analysis for Supergravity Fields: The total dimension of any basis term has tobe two, the number of respective dotted and undotted indices even.
1. Use the freedom to partially integrate to remove any derivatives on the Weyl parameter σ .The anomaly then has the shape∆ W Γ = Z d z E − σ B ( λ, ¯ λ ) · A , (9.19)with A = A ( R, ¯ R, G α ˙ α , W αβγ , ¯ W ˙ α ˙ β ˙ γ , D , ¯ D , D λ, ¯ D ¯ λ ).2. Expand in derivatives on couplings. Since the overall scaling dimension is supposed to betwo, there are at most four derivatives and consequently at most four couplings in A .Furthermore since all basis terms for A should be scalars, the total number of indices shouldbe even (dotted and undotted indices respectively). The properties relevant to these simplecounting arguments are summarised in Table 10.The following combinations (bars not yet included) have a chance to yield the right dimensionand index structure:2 × R, × G, (1 × R, × D ) , (1 × G, × D ) , × D . Taking into account the algebra and Bianchi identities, several derivatives acting on the samecoupling λ can be brought to a standard order. I chose D α λ, D λ, D α ˙ α λ, D α ˙ α D β λ, D α ˙ α D λ, D α ˙ α D α ˙ α λ, (9.20)and accordingly for ¯ λ .In total there arise 38 terms, such that the basis ansatz for the anomaly reads B · A = b ( A ) G α ˙ α G α ˙ α + b ( B ) R ¯ R + b ( C ) R + b ( ¯ C ) ¯ R + b ( D ) ( D R ) + b ( ¯ D ) ( ¯ D ¯ R )+ b ( E ) i R D λ i + b ( ¯ E )¯ ı ¯ R ¯ D ¯ λ ¯ ı + b ( F )¯ ı R ¯ D ¯ λ ¯ ı + b ( ¯ F ) i ¯ R D λ i + b ( G ) i ( D α R )( D α λ i ) + b ( ¯ G )¯ ı ( ¯ D ˙ α ¯ R )( ¯ D ˙ α ¯ λ ¯ ı )+ b ( H ) i G α ˙ α D α ˙ α λ i + b ( ¯ H )¯ ı G α ˙ α D α ˙ α ¯ λ ¯ ı + b ( I ) i D α ˙ α D α ˙ α λ i + b (¯ I )¯ ı D α ˙ α D α ˙ α ¯ λ ¯ ı + b ( J ) ij R ( D α λ i )( D α λ j ) + b ( ¯ J )¯ ı ¯ ¯ R ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ )+ b ( K ) ij ¯ R ( D α λ i )( D α λ j ) + b ( ¯ K )¯ ı ¯ R ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ )+ b ( L ) i ¯ G α ˙ α ( D α λ i )( ¯ D ˙ α ¯ λ ¯ ) + b ( M ) i ¯ ( D α ˙ α λ i )( D α ˙ α ¯ λ ¯ )+ b ( N ) ij ( D α ˙ α λ i )( D α ˙ α λ j ) + b ( ¯ N )¯ ı ¯ ( D α ˙ α ¯ λ ¯ )( D α ˙ α ¯ λ ¯ )+ b ( O ) i ¯ ( D α λ i )( D α ˙ α ¯ D ˙ α ¯ λ ¯ ) + b ( ¯ O )¯ ıj ( ¯ D ˙ α ¯ λ ¯ ı )( D α ˙ α D α λ j )+ b ( P ) i ¯ ( D λ i )( ¯ D ¯ λ ¯ )+ b ( Q ) ij ( D λ i )( D λ j ) + b ( ¯ Q )¯ ı ¯ ( ¯ D ¯ λ ¯ ı )( ¯ D ¯ λ ¯ ) (9.21)+ b ( R ) ijk ( D α λ i )( D α λ j )( D λ k ) + b ( ¯ R )¯ ı ¯ ¯ k ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ )( ¯ D ¯ λ ¯ k )+ b ( S ) ij ¯ k ( D α λ i )( D α λ j )( ¯ D ¯ λ ¯ k ) + b ( ¯ S )¯ ı ¯ k ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ )( D λ k )+ b ( T ) ij ¯ k ( D α ˙ α λ i )( D α λ j )( ¯ D ˙ α ¯ λ ¯ k ) + b ( ¯ T )¯ ı ¯ k ( D α ˙ α ¯ λ ¯ ı )( D α λ k )( ¯ D ˙ α ¯ λ ¯ )+ b ( U ) ij ¯ k ¯ l ( D α λ i )( D α λ j )( ¯ D ˙ β ¯ λ ¯ k )( ¯ D ˙ β ¯ λ ¯ l )+ b ( V ) ijkl ( D α λ i )( D α λ j )( D β λ k )( D β λ l )+ b ( ¯ V )¯ ı ¯ ¯ k ¯ l ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ )( ¯ D ˙ β ¯ λ ¯ k )( ¯ D ˙ β ¯ λ ¯ l ) . where b ( A... ¯ V ) are potentially functions of λ and ¯ λ . However, this choice is not minimal as it stillallows for partial integration with respect to ¯ D ˙ α because the chiral field σ ignores these. Singlederivatives on ¯ λ cannot be removed by partial integration in general, since a derivative acting onthe coefficient b reproduces the same term again.More precisely, due to Z d z b ¯ ( ¯ D ˙ α ¯ λ ¯ ) ¯ X ˙ α = Z d z (cid:20) ˜ b ¯ + ( ∂ ¯ ˜ b ¯ ı )¯ λ ¯ ı (cid:21) ( ¯ D ˙ α ¯ λ ¯ ) ¯ X ˙ α = − Z d z ˜ b ¯ ¯ λ ¯ ( ¯ D ˙ α ¯ X ˙ α ) ,b ¯ = ∂ ¯ (˜ b ¯ ı ¯ λ ¯ ı ) (9.22)a basis term with a single derivative on ¯ λ can only be removed from the tentative basis if a ˜ b obeying (9.22) exists; i.e. the integrability conditions ∂ ¯ ı b ¯ = ∂ ¯ b ¯ ı are fulfilled. This is certainly nottrue in general, but for only one coupling or if the theory is invariant under arbitrary exchange ofthe coupling constants ¯ λ ¯ ı ↔ ¯ λ ¯ , the basis reduces further.Apart from this complication, removable terms are those which either have an outer ¯ D derivative(as opposed to one being hidden behind a D α ) or can be brought to that form by using the Bianchiidentities and the supergravity algebra.The above “basis” not being a minimal set of operators is not really a problem (except forcreating a bit of extra work in the followings), since it will be possible to consistently set to zerothe prefactors to such superfluous terms belatedly. Note that b ( T ) and b ( ¯ T ) are the only coefficients which potentially can be asymmetric in two indices of thesame type. As we will see later, the variations are symmetric, so consistency conditions can only give results forthe respective symmetric part. (cid:2) ∆ Wσ − ∆ βσ , ∆ Wσ ′ − ∆ βσ ′ (cid:3) W = 0 , (9.23) (cid:2) ∆ W ¯ σ − ∆ β ¯ σ , ∆ Wσ − ∆ βσ (cid:3) W = 0 . (9.24)As shall be seen, all necessary expressions can be determined from(∆ Wσ − ∆ βσ )(∆ Wσ ′ − ∆ βσ ′ ) W, (9.25)which requires to calculate the Weyl variation of all basis terms as well as to determine theexpressions ∆ Wσ (∆ Wσ ′ − ∆ βσ ′ ) W, ∆ βσ (∆ Wσ ′ − ∆ βσ ′ ) W. (9.26)Since the calculation is straight-forward but tedious, the results have been banned to appendices A,B and C.The general structure of (9.25) is(∆ Wσ − ∆ βσ )(∆ Wσ ′ − ∆ βσ ′ ) W = Z d z E − σ ′ (cid:8) σ F + ( D α σ ) F α + ( D σ ) F + ( D α ˙ α σ ) F α ˙ α + ( D α ˙ α D α σ ) ¯ F ˙ α + ( D α ˙ α D α ˙ α σ ) F (cid:9) , (9.27)where the coefficients F can be determined from the intermediate results in appendix A and arelisted in appendix B.The naming scheme for the anomaly terms has been chosen such that the calculation of theWeyl consistency conditions only requires ∆ σ ∆ σ ′ W (9.28)to be computed by variation. The reader may convince himself that the other three operatorcombinations can be determined from the following simple set of rules.∆ σ ′ ∆ σ W = (∆ σ ∆ σ ′ W ) σ ↔ σ ′ ; (9.29)∆ σ ¯∆ ¯ σ W = (∆ σ ∆ σ ′ W ) N , (9.30)( b ( x ) ) N := ¯ b (¯ x ) , ( σ ′ ) N := ¯ σ, ( σ ) N := σ, ( . . . ) N := ( . . . ); (9.31)¯∆ ¯ σ ∆ σ ′ W = (∆ σ ∆ σ ′ W ) N , (9.32)where ( . . . ) denotes anything that is not covered by explicit prior rules. Note that for the few realterms, it holds b (¯ x ) = b ( x ) .So the (cid:2) ∆ , ∆ (cid:3) Wess–Zumino consistency condition (9.23) is[∆ Wσ − ∆ βσ , ∆ Wσ ′ − ∆ βσ ′ ] W = Z d z E − ( σ ′ D α σ − σ D α σ ′ ) (cid:8) F α − D α ( F − i ¯ D ˙ α ¯ F ˙ α )+ i ¯ D ˙ α ( F α ˙ α − D α ˙ α F ) + iG α ˙ α ¯ F ˙ α (cid:9) , (9.33) while the (cid:2) ∆ , ¯∆ (cid:3) Wess–Zumino consistency condition (9.24) yields (cid:2) ∆ W ¯ σ − ∆ β ¯ σ , ∆ Wσ − ∆ βσ (cid:3) W = Z d z E − (cid:20) σ ¯ σ (b) + σ ( D α ˙ α ¯ σ ) (c) + ( D α ˙ α σ )( D α ˙ α ¯ σ ) (d) (cid:21) , (9.34)with (b), (c) and (d) the respective left hand sides of F α − D α ( F − i ¯ D ˙ α ¯ F ˙ α )+ i ¯ D ˙ α ( F α ˙ α − D α ˙ α F ) + iG α ˙ α ¯ F ˙ α = 0 , (9.35a) (cid:8) F − ( D α F α ) + ( D F ) − D α ˙ α ( F α ˙ α − D α ˙ α F − D α ¯ F α ) (9.35b) − i ( ¯ D ˙ α ¯ R ) ¯ F ˙ α − iG α ˙ α ( D α ¯ F α ) (cid:9) N − c.c. = 0 , (9.35c) (cid:8) F α ˙ α − D α ˙ α F − D α ¯ F α (cid:9) N + c.c. = 0 , (9.35d) F N = ¯ F N , (9.35e)which constitute the full set of consistency conditions on the level of abbreviations F . The complexconjugate of (9.35a) is an additional part of this system.These coefficient consistency equations are the main result of this Part. Unfortunately expandedout they fill about three pages and have been put into Appendix C, therefore.9.4 Local CountertermsThe vacuum energy functional W is only determined up to the addition of local counter terms δW , a convenient choice for which is provided by the basis used for the anomaly, since it allowsto reuse the results from the Wess–Zumino consistency condition: W ≡ W + δW, (9.36) δW = Z d z E − δ B · A , (9.37)with δ B · A analogous to (9.21). To fulfil the reality requirement δW = δW , it is necessary (andsufficient) to choose the coefficients δb from δ B according to δ ¯ b ( x ) = δb (¯ x ) for any x . Realising that ∆ σ W = Z d z E − σ B · A , (9.38)= ⇒ δW = ∆ σ ′ W (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ ′ b ( x ) δb ( x ) b (¯ x ) δ ¯ b ( x ) =: ∆ σ ′ W (cid:12)(cid:12) δ , (9.39) In particular for coefficients of the single, real terms ( A ), ( B ), ( L ), ( M ), ( P ), ( U ), this amounts to taking b ( x ) = ¯ b ( x ) . the effect of adding the local counter terms δW to the generating functional W is seen to be∆ σ ( W + δW ) = ∆ σ ( W + ∆ σ ′ W (cid:12)(cid:12) δ ) (9.40)= Z d z E − σ B · A + Z d z E − σ (cid:8) F − D α F α + D F − D α ˙ α F α ˙ α + D α D α ˙ α ¯ F ˙ α + D α ˙ α D α ˙ α F (cid:9)(cid:12)(cid:12) δ , (9.41)where in the last line equation (9.27) has been used.In other words, the addition of local counter terms corresponds to the mapping B · A 7→ B · A + (cid:8) F − D α F α + D F − D α ˙ α F α ˙ α + D α D α ˙ α ¯ F ˙ α + D α ˙ α D α ˙ α F (cid:9)(cid:12)(cid:12) δ . (9.42)9.5 S-duality N = 4 SYM is invariant under an SL(2 , R ) symmetry that is preserved on the quantum level.Explicit calculations indicate the symmetry is also maintained to one loop during coupling togravity. Assuming that this is true to all orders, one might restrict the discussion of anomaly termsto superfield expressions that are manifestly invariant under that symmetry for the discussion ofan N = 4 fixed point.The theory of modular forms easily fills an entire book [107], but the consideration here shallbe restricted to SL(2 , R ) invariant terms that can be build from the basis of anomaly terms (9.21).In terms of the complex coupling λ := πg − iθ π , the SL(2 , R ) symmetry is generated by the twotransformations λ λ , λ λ + i, (9.43)which have this unusual form due to employing the convention of taking the coupling constant g − as the real part of λ .It follows immediately that for coefficient functions b ( λ, ¯ λ ) in the anomaly it holds b = b ( λ + ¯ λ ).In addition one observes 1 λ + ¯ λ λ ¯ λ λ + ¯ λ , (9.44) D α λ
7→ − λ D α λ, (9.45)¯ D ˙ α ¯ λ
7→ − λ ¯ D ˙ α ¯ λ, (9.46) D λ
7→ − λ D λ, (9.47)¯ D ¯ λ
7→ − λ ¯ D ¯ λ, (9.48) D α ˙ α D α λ
7→ − λ D α ˙ α D α λ, (9.49)where D λ := D λ − λ + ¯ λ ( D α λ )( D α λ ) , (9.50)¯ D ¯ λ := D λ = ¯ D ¯ λ − λ + ¯ λ ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ ) , (9.51) Therefore S-invariant expressions are given by1( λ + ¯ λ ) ( D λ )( ¯ D ¯ λ ) , ∼ ( P ) , ( S ) , ( ¯ S ) , ( U ) (9.52)1( λ + ¯ λ ) ( D α λ )( D α ¯ D ¯ λ ) , ∼ ( L ) , ( O ) , ( U ) , ( ¯ T ) (9.53)1( λ + ¯ λ ) ( ¯ D ˙ α D λ )( ¯ D ˙ α ¯ λ ) , (9.54)1( λ + ¯ λ ) ( D α ˙ α λ )( D α ˙ α ¯ λ ) , ∼ ( M ) (9.55)1( λ + ¯ λ ) G α ˙ α ( D α λ )( ¯ D ˙ α ¯ λ ) , ∼ ( L ) (9.56)1( λ + ¯ λ ) ( D α λ )( D α λ )( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ ) ∼ ( U ) (9.57)and moreover the λ, ¯ λ independent terms ( A ) to ( ¯ D ).9.6 Towards a ProofFor the proof of Zamolodchikov’s theorem in two dimensions, the crucial ingredient is the connec-tion of the anomaly coefficients to correlation functions from which the positive definite Zamolod-chikov metric was defined, see Sections 8.4 and 8.4.1 in particular.As an example of how this procedure works the consistency condition (C.3f) from the appendixshall be discussed, − i b ( M ) j ¯ k + β i b ( T ) ji ¯ k + ib ( L ) j ¯ k + i b ( N ) ij ( ∂ ¯ k β i ) + i β i ( ∂ ¯ k b ( N ) ij ) − b ( T ) ij ¯ k β i = 0 .b ( T ) ij ¯ k is the only coefficient function that is not (anti-)symmetric in indices of the same kind. Fromthe expression above it can however be projected out by multiplying with β j , which leaves β j (cid:2) b ( M ) j ¯ k − b ( L ) j ¯ k − ∂ ¯ k ( β i b ( N ) ij ) (cid:3) = 0 , (9.58)In fact b ( N ) ij vanishes identically as a consequence of the RG equation, which for the anomalyrestricted to that coefficient reads µ ∂∂µ W + β i ∂ i W = b ( N ) ij ( D λ i )( D λ j ) . (9.59)Acting on it with δδλ k δδλ l , gives µ ∂∂µ (cid:10) O k O l (cid:11) + β i ∂ i (cid:10) O k O l (cid:11) = b ( N ) kl ( D δ ( z ))( D δ ( z ′ )) , (9.60)where the left-hand side vanishes by non-renormalisation of chiral correlation functions. It imme-diately follows that b ( N ) ij ≡
0, which means that equation (9.58) implies β j ¯ β ¯ k (cid:2) b ( M ) j ¯ k − b ( L ) j ¯ k (cid:3) = 0 . (9.61)This is the supersymmetric version of equation (8.49), which reads χ gij − χ aij = L β S ij − ∂ i β k χ akj − β k χ bkij , though from (9.61) the right hand side is zero when taking into account b ( M ) ∼ χ g − χ a , b ( L ) ∼ χ g , (9.62)as will be seen from the component expansions (9.68)–(9.70) of the next Section. This is just asrequired for a proof of the a -theorem, since χ ( a ) can be shown to be positive definite in a particularscheme. In that scheme, − c χ a = x S (cid:10) O i ( x ) O j (0) (cid:11) , (9.63)where the right hand side is positive definite by unitarity. The set of counterterms which areneeded to change to a scheme where χ a = c χ a were determined in [108].Of course the other anomaly terms might contribute further terms to the simple identificationbetween b ( M ) , b ( L ) and χ a , χ g , thus spoiling the success. Actually from the whole basis for theanomaly, there is only one term which could do so, namely ( D λ )( ¯ D ¯ λ ), which seems harmlesssince its component expansion yields only ( ∇ λ )( ∇ λ ∗ ). Moreover it is expected to conspire withthe ( M ) and ( L ) terms from the anomaly basis to form a supersymmetric version of the “Riegertoperator” as shall be explained now.9.7 Superfield Riegert OperatorFor N = 4 Yang–Mills theory [53] obtains a one-loop trace anomaly that contains the operator1( λ + λ ∗ ) (cid:0) ∇ λ ∇ λ ∗ − G mn ∇ m λ ∇ n λ ∗ − R∇ m λ ∇ m λ ∗ (cid:1) , (9.64)which basically is the Riegert operator (8.17). Note that the bosonic Riegert operator is a directconsequence of the (bosonic) consistency conditions for the N = 4 case. It is therefore important toreproduce the Riegert operator in the component expansion of the superfield formulation employedhere.This result indicates an inconsistency with our result because there does not seems to exist asuperfield expression that generates this Riegert operator in a component expansion. Thereforeit cannot be generated as part of the derived superfield trace anomaly.Strange enough in components a super-Weyl covariant version of this operator is known suchthat the following expression [55] is invariant under super-Weyl transformations, L = e − ∇ φ ∗ ∇ φ − R mn − g mn R ) ∇ m φ ∗ ∇ n φ − ¯ χ [ D + ( R mn − g mn R ) γ m D n ] χ − ¯ χγ m D n χF mn + F ∗ [ D −
16 ( R − ¯ ψ m R m )] F +(gravitino terms) , (9.65)with D m χ = ∇ m χ + i γ A m χ, D m = ( ∂ m + i A m ) F, (9.66)and φ, ψ, F the components of a chiral field of Weyl weight 0.Therefore one should expect a superfield version ∆ R of this operator to exist such that δ Weyl (cid:2)Z d zE − λ ∆ R ¯ λ (cid:3) = 0 , (9.67) The factor in front plus some further terms are required to make the operator SL(2 , R ) invariant in addition.
00 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings with δ Weyl indicating a super-Weyl transformation.On the other hand one might simply use a component expansion of all basis terms and determinethe linear combination that yields the bosonic Riegert operator (8.17) as its lowest component.Such a component expansion can be quite involved, but fortunately there is only a limit numberof terms that can contribute. Here the discussion shall be restricted to a few natural candidateterms which already produces some interesting results. D ( ¯ D − R )( D λ )( ¯ D ¯ λ ) (cid:12)(cid:12) b = 256( ∇ λ )( ∇ λ ∗ ) , (9.68) D ( ¯ D − R ) G α ˙ α ( D α λ )( ¯ D ˙ α ¯ λ ) (cid:12)(cid:12) b (9.69)= 64( G ( µν ) + g µν R )( ∇ µ λ )( ∇ ν λ ∗ ) − R ( ∇ µ λ )( ∇ µ λ ∗ ) + (imag.) , D ( ¯ D − R )( D α ˙ α λ )( D α ˙ α ¯ λ ) (cid:12)(cid:12) b = R g µν ( ∇ µ λ )( ∇ ν λ ∗ ) − ∇ µ λ )( ∇ ∇ µ λ ∗ ) + (imag.) (9.70)= R g µν ( ∇ µ λ )( ∇ ν λ ∗ ) − R µν ( ∇ µ λ )( ∇ ν λ ∗ )+32( ∇ λ )( ∇ λ ∗ ) + (total deriv.) , where the following relations have been used, (cid:2) ∇ µ , ∇ ν (cid:3) V ρ = R ρσµν V σ , (9.71) ∇ ∇ µ V = ∇ µ ∇ V + R νµ ∇ ν V. (9.72)First of all one should note that (9.68) can be expressed by a linear combination of (9.69) and(9.70) and a total derivative, which is just the component version of (7.94),( D λ )( ¯ D ¯ λ ) = 4 G α ˙ α ( D α λ )( ¯ D ˙ α ¯ λ ) + 8( D α ˙ α λ )( D α ˙ α ¯ λ )+(total derivative) . (9.73)This relation being preserved in the component expansion is a strong indication for equations(9.68)–(9.70) to be correct.Up to this identity the only combination of the candidate terms (9.68)–(9.70) that yields thebosonic Riegert operator as its lowest component is( D λ )( ¯ D ¯ λ ) − G α ˙ α ( D α λ )( ¯ D ˙ α ¯ λ ) . (9.74)This combination is not super-Weyl covariant however and it turns out that for the anomalybasis (9.21), there is no non-trivial super-Weyl invariant expression that includes ( D λ )( ¯ D ¯ λ )—or( D α ˙ α λ )( D α ˙ α ¯ λ ) by (9.73). In other words, there is no superfield version of the Riegert operatorfor chiral fields of Weyl weight
0. This is rather puzzling since the component version does exist.What may have gone wrong?9.8 DiscussionEquation (9.73) provides a rather non-trivial consistency check for the component expansion andthe Weyl variations are simple to check. One should therefore be confident that the result of theprevious Section is correct.Since the Weyl parameter in minimal supergravity is a chiral field, it naturally also encodessuperlocal U(1) R transformations. So perhaps one is simply requiring too much symmetry. Sincethe expressions are global U(1) R invariant anyway, neglecting the local symmetry corresponds toallowing terms that contain derivatives acting on σ − ¯ σ . Due to D α ( σ − ¯ σ ) = D α ( σ + ¯ σ ) thiscannot be distinguished from super-Weyl transformations. In non-minimal supergravity it is possible to not require invariance under local U(1) R , and apossible super-Weyl covariant operator (in the conventions of [103]) is given by( D α ˙ α λ )( D α ˙ α ¯ λ ) (9.75)with the Weyl covariant vector derivative for scalar chiral yields of U(1) R charge y given by D α ˙ α := i ( ¯ ∇ ˙ α − i ( + y )¯Γ ˙ α )( ∇ α + iy Γ α ) , (9.76) δ [ D α ˙ α λ ] = L D α ˙ α λ. (9.77)In new-minimal supergravity the U(1) R drops from the formulation and it is possible to give asuperfield Riegert operator for linear superfields of Weyl-weight 0 that is covariant under the fullinvariance group of the supergravity algebra [56] D α ˙ α D α ˙ α + i D α ¯ T ˙ α + ¯ D ˙ α T α ) D α ˙ α , (9.78)where D α ˙ α = D α ˙ α − i ( T α ¯ D ˙ α + ¯ T ˙ α D α ) is a super-Weyl covariant derivative.The difficulties to formulate fields of arbitrary Weyl and U(1) R weight in a superconformalframework are long known (see for example [109]) and led to the introduction of a chiral compen-sating field. This can be most easily illustrated taking a chiral field λ as an example. It clearlyshould transform under generalised super-Weyl transformations according to λ e n + σ + n − ¯ σ λ, (9.79)with n + a real number and n − = 0 in order to stay a chiral field. In other words the type of thefield dictates a fixed relation between its U(1) R charge and its Weyl weight. Therefore a singlefield transforming as Φ e σ Φ can be used to bring all other fields to a fixed Weyl and U(1)weight, by redefinitions of the type ˜ λ = Φ − n + λ for example.A suitable set of invariant supergravity fields is given by D α = U D α − D β U ) M αβ , U = [Ψ n +1 ¯Ψ n − ] − n +18 n , ¯ D ˙ α = ¯ U ¯ D ˙ α −
2( ¯ D ˙ β ¯ U ) ¯ M ˙ α ˙ β , D α ˙ α = i (cid:8) D α , ¯ D ˙ α (cid:9) , T α = D α T , T = ln U ¯ U , (9.80) R = −
14 ( ¯ D − R ) ¯ U , W αβγ = ¯ U U W αβγ , G α ˙ α = ¯ UU G α ˙ α + ( ¯ D ˙ α ln U )( D α ln U )+ ¯ D ˙ α D α ln( U ¯ U − ) − D α ¯ D ˙ α ln( ¯ U U − ) , where Ψ is a linear conformal compensator which transforms under Weyl transformation ϕ e σ ϕ according to Ψ Ψ ′ = exp (cid:20) n − n + 1 σ − ¯ σ (cid:21) Ψ , ¯ D ˙ α σ = 0 . (9.81)The case n = corresponds to minimal supergravity and the compensator Φ := ¯Ψ is a chiral field.It should be remarked that the expressions (9.80) can be easily obtained by replacing σ and ¯ σ in the Weyl transformed objects by − ln Φ and − ln ¯Φ in a similar way as in the bosonic case inSection 8.1.4.
02 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings
One might think of taking the already known chiral compensator ϕ − as the compensator Φin (9.80). However this use of the chiral compensator ϕ , which is also a prepotential that trans-forms under the Λ supergroup, would break invariance under that symmetry. Another interestingpossibility is the use of Ω = 1 + Z d z ′ E − ( z ′ ) G + − ( z, z ′ ) , (9.82)where G + − is the Feynman superpropagator defined by ( D − R ) z G + − ( z, z ′ ) = δ ( z, z ′ ) (9.83)and δ ( z, z ′ ) is the chiral delta distribution.A simple consequence of the defining relation is¯ D ˙ α Ω = 0 , ( D − R )Ω = 0 , (9.84)which implies Ω e − σ Ω under super-Weyl transformation and Ω is a suitable (though non-local)compensator. For superconformal backgrounds Ω actually becomes local and take the formΩ = ϕ − + O ( H ) . (9.85)With such a compensator the expression( D λ )( ¯ D ¯ λ ) − G α ˙ α ( D α λ )( ¯ D ˙ α ¯ λ ) (9.86)yields the bosonic Riegert operator and is super-Weyl invariant. Unfortunately the latter is alsotrue for any other expression, so not much has been gained. In particular in the presence of acompensator the criterion for Weyl invariance of a term is the absence of any functional dependenceon that compensating field, which is certainly not true for (9.86).Another approach may be to ask what is a natural Weyl invariant operator for an arbitraryfield, such that the operator does not coincide with the Riegert operator. For example E − [( D − R ) ψ ][( ¯ D − R ) ¯ ψ ] (9.87)is invariant when ψ e ¯ σ − σ ψ . This transformational behaviour is incompatible with ψ beinga chiral field. It is possible for ψ being linear, but that assumption annihilates the operator ofcourse.For a real field V , a Weyl invariant operator is given by E − V D α ( ¯ D − R ) D α V ≡ E − V ¯ D ˙ α ( D − R ) ¯ D ˙ α V, (9.88)with additional gauge invariance V V + λ + ¯ λ , where λ and ¯ λ are chiral and anti-chiral fieldsrespectively.Since the N = 4 case should also incorporate SL(2 , R ) symmetry with invariance of the anomalyunder λ λ + i, λ λ , (9.89)one might be tempted to use the SL(2 , R ) K¨ahler form V = ln λ + ¯ λ to also include that symmetry. Of course the operator will then contain additional pieces actingon more than two fields. However those pieces which do act on only two fields form exactly thecombination (9.73), such that the Riegert operator is missing again.It seems that there is something in the minimal supergravity formalism that does not allow forsuperfield formulation of the Riegert operator. I strongly suspect that it is the U(1) R symmetrythat spoils the formulation of the operator by being inevitable tied to the super-Weyl transforma-tions.
10 Conclusions
For the understanding of quantum field theories, its coupling to gravity backgrounds has proved avaluable tool. The discovery of
AdS / CFT correspondence, which realises such a coupling holograph-ically, has revived the interest in this idea and been a major break-through in the understandingof strongly coupled Yang–Mills theories. While the original
AdS / CFT duality involves N = 4supersymmetric Yang–Mills theory, it has soon been extended to less symmetric, more realistictheories.In this work, such an extension is explored in more detail, taking as a starting point the N = 2supersymmetric D3/probe D7-brane framework of [20], which is dual to N = 4 supersymmetric,large N c SU( N c ) Yang–Mills theory augmented by a small number N f of N = 2 hypermultipletsin the fundamental representation. By holographic methods, this theory’s meson spectrum canbe calculated analytically at quadratic order [24].I considered first a geometry more general than the conventional AdS × S and second aninstanton gauge configuration on the D7-branes. The general strategy was to introduce back-ground configurations that reproduce the conventional setting in certain limits. This allowed tomake contact with the ordinary AdS / CFT dictionary and is an important feature of this approachcompared to others in the area that is sometimes referred to as
AdS / QCD .The following results were obtained: ˆ A holographic dual of spontaneous chiral symmetry breaking by a bilinear quark condensate (cid:10) ¯ ψψ (cid:11) was found. Since such a condensate is prohibited by supersymmetry, this required touse a background that completely breaks supersymmetry and approximates AdS × S onlytowards the boundary. By standard AdS / CFT , the boundary of the space-time is associatedto the ultraviolet of the dual field theory, such that the configuration describes an N = 2theory that is relevantly deformed and flows to a non-supersymmetric infrared.I calculated the quark condensate (cid:10) ¯ ψψ (cid:11) as a function of the quark mass m q , which gavea non-vanishing quark condensate in the limit m q →
0; i.e. sponetaneous chiral symmetrybreaking. Moreover I determined the meson spectrum and demonstrated that the mesonmode associated to the U(1) A axial symmetry, which is geometrically realised as rotations,becomes massless in the m q → m q = 0this mode becomes a pseudo-Goldstone mode, which obeys the Gell-Mann–Oakes–Rennerrelation M π ∼ m q . In the large quark mass limit, the mesons lie in the supersymmetricregime such that their mass is degenerate and approximates the analytic results of the N = 2theory.In addition I determined the mass of highly excited scalar and pseudoscalar mesons, whichhave the interesting feature of not being degenerate in this setup. ˆ The dual description of the mixed Coulomb–Higgs branch of the N = 2 theory was found.The Higgs VEV corresponds to the size of an instanton configuration on the supergravity side,establishing a link between supersymmetry and the
ADHM construction that was known toexist. Such an instanton configuration can only exist when there are at least two flavours,such that a non-Abelian
Dirac–Born–Infeld action had to be used. Ordering ambiguities canbe avoided since a calculation to quadratic order is sufficient, but a crucial insight was theuse of a singular gauge transformation to obtain the correct boundary behaviour consistentwith the
AdS / CFT dictionary.Having overcome this major obstacle, I numerically determined the meson spectrum andfound it to approach the analytic N = 2 spectrum in the limit of vanishing and infinite Higgs VEV , though in the latter case a non-trivial rearrangement was observed, which could beexplained to arise from above singular gauge transformation. Here a background by Gubser and Kehagias–Sfetsos [40, 39] was chosen.
04 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings ˆ A geometric realisation of heavy-light mesons was developed; i.e. mesons build up from a lightand heavy quark providing a framework for the description of B mesons not available before.Since a realisation in terms of a non-Abelian D7-brane action only makes sense for small massdifferences, a different approach has to be chosen. The configuration under consideration isthat of a long string stretched between two D7-branes with a large separation, where theD7-branes are arranged to correspond to a massless and a heavy quark respectively.I describe an effective point-particle action derived from the Polyakov action for a straightstring in a semi-classical approximation. After quantisation the equation of motion givesrise to the spectrum of mesons consisting of a massless and a heavy quark. I evaluated thespectrum in the standard AdS × S background, where I could find an analytic formulafor the numerically determined heavy-light meson masses, and for the non-supersymmetricbackgrounds by Constable–Myers [46] and by Gubser, Kehagias–Sfetsos discussed earlier. Inthe former case a comparison with the experimental values of the B meson mass yields adeviation of about 20%.The models considered in this thesis are not meant to be realistic duals of QCD , but insteadfocus on a particular aspect like chiral symmetry breaking by a chiral quark condensate, themeson spectrum for D3/D7
AdS / CFT either non-supersymmetric deformed or with a Higgs
VEV switched on, and the spectrum of heavy-light mesons in several backgrounds, giving a descriptionof B mesons.It would be certainly interesting to extend the techniques developed in this thesis to a morerealistic example of
AdS / QCD . Over the last years there has been steady progress towards sucha description, including string theory duals of theories that exhibit chiral symmetry breaking[86, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128].There are however three major points that need to be addressed in future refinements of
AdS / QCD .The models considered here have a UV fixed point, but they are not asymptotically free. Theweak-strong nature of the duality, which makes AdS / CFT so interesting, unfortunately means thatweak coupling in the field theory’s UV implies strong curvature towards the boundary of the AdS space, thus requiring a full string theoretical treatment, which currently is not feasible. Lackingthat, there are recent attempts to circumvent the situation by introducing a UV cut-off in thegeometry to produce phenomenological models of QCD dynamics [99, 100, 129, 130, 131, 132, 133,101, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143].A second problematic property is the probe limit N f ≪ N c , which corresponds to the quenchedapproximation of lattice QFT . Additional contributions are roughly of the order N f N c . Includingthe backreaction of the D7-branes on the geometry would allow the number of flavours to be ofthe same order of magnitude as the number of colours. Such backgrounds have been consideredin [85].The last important aspect is the separation of the SUSY and confinement scales. In the B physicsexample discussed in Section 6.3, the B meson is far in the supersymmetric regime. To changethis situation one needs a background configuration that incorporates at least two different scales.From the recent works cited above one can read off a tendency to focus on particular aspects ofthe larger problem of finding a holographic dual of
QCD and YM theories, an approach also to befound in this thesis. A challenge for the future will be to incorporate into one model as many aspossible of the insights gained here and elsewhere since the discovery of AdS / CFT duality almostten years ago.In the second Part of this thesis the coupling of supersymmetric quantum field theories tominimal supergravity was investigated. Coupling a gravity background to a conformal quantum In particular the heavy-light meson construction could be easily extended to other, more realistic models. field theory gives rise to a conformal anomaly (cid:10) T mm (cid:11) = c C − a ˜ R + b R + f (cid:3) R. ( ⋆ )In [52] a space-time dependent coupling approach was used to calculate consistency conditions forthe coefficients in the two-dimensional anomaly providing an alternative proof for Zamolodchikov’s c -theorem. However [52] did not obtain consistency conditions sufficiently restrictive to extendthe theorem to four dimensions.The specific project pursued here was to extend this technique to superfields and determinethe conformal anomaly for those supersymmetric field theories whose coupling constants can bepromoted to chiral fields λ . A prominent example for such is given by super-Yang–Mills theories.The steps performed in detail were: ˆ I determined a complete ansatz for the conformal anomaly by finding a basis of 38 localsuperfield expressions of dimension 2 and composing a linear combination with arbitrarycoefficient functions b ( λ, ¯ λ ). In the constant coupling limit, these coefficient functions becomethe superspace analogue of the coefficients c , a , b and f that appear in the bosonic conformalanomaly ( ⋆ ). ˆ Then I calculated the Wess–Zumino consistency conditions for the coefficient functions, whicharise from the fact that Weyl transformations are Abelian. ˆ Furthermore I discussed the dependence on local counterterms and possible consequences ofS-duality in the N = 4 case. ˆ It is noted that a superfield version of the Riegert operator is needed to make contactwith an existing one-loop calculation [53]. Various approaches to the problem of finding asuperfield Riegert operator (which is independent of the anomaly calculation presented) havebeen discussed. The conclusion is that the problem is rooted in the U(1) R symmetry beingbuilt into the formalism of minimal supergravity in superfield formulation in a local way ,while on the component level the U(1) R is only realised as a global symmetry.In order to check this assumption it would be desirable to repeat the full calculation in acomponent approach. The sheer size of this task is daunting however: The basis for the anomalyI found contains about 40 terms in superfield formulation plus their complex conjugates. Asa consequence the calculation of the Wess–Zumino consistency conditions is very involved andpotentially error prone. A component based approach will probably incorporate even more termsand should therefore be implemented with the help of a computer. Unfortunately a computer basedtreatment of supergravity has a number of requirements not satisfied by any existing computeralgebra system ( CAS ) today. These requirements are ˆ an efficient mechanism for the representation of tensors and contracted indices, ˆ handling of commuting, anticommuting and non-commuting objects (this should include theability to reduce a number of terms to a canonical basis of terms using the supergravityalgebra and Bianchi identities), ˆ a way to represent non-commuting tensor valued functions of other objects (e.g. for non-anticommuting spinorial derivatives), ˆ making no assumption about the symmetries of the metric, The Riegert operator is the unique conformally covariant differential operator of fourth order acting on ascalar field of Weyl weight 0.
06 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings ˆ allowing torsion, and ˆ no automatic expansion of compact parenthesised expressions into a lengthy sum of terms.Of the existing systems, FORM [144] seems to be coming the closest to these requirements since itprovides a rather low-level tensor support without restrictive internal assumptions. Its summaris-ing capabilities are unsatisfactory however and may be a major obstacle in the implementation ofa computer based analysis of the trace anomaly.Another promising program is
Cadabra [145, 146], which meets all of the above requirementsbut is still in a development stage.Nevertheless the next steps in a future analysis of the trace anomaly are the implementationof a supergravity computer algebra package and a component based analysis. As outlined abovethis is a difficult task, but the results presented in this thesis can serve as a highly non-trivial unittest to confirm the correctness of such a package. Then one may carry out a complete componentexpansion of all basis terms and reexamine the question of whether a superfield version of theRiegert operator does exist in minimal superfield supergravity. This analysis can then be easilyextended to non-minimal
SUGRA and as a check one may reproduce the Riegert operator in new-minimal
SUGRA as well.A reimplementation of the whole calculation in a component based approach would providean independent source of confirmation for the results of this thesis. If a superfield based treat-ment of minimal supergravity is consistent on the quantum level, the two calculations shouldactually yield the same result, strengthening confidence in the results presented here. Of courseinconsistency would be an interesting result in its own right.In any case I hope to have provided a basis for understanding the structure of the conformalanomaly in supersymmetric field theories coupled to supergravity. Acknowledgements
I would like to thank my wife Olga, Johanna Erdmenger and Dieter L¨ust. Without their supportthis work would not have been possible. Many thanks also to my friend Robert Eisenreich for ordespite his being esopucative all the time.I am thankful to Hugh Osborn, Peter Breitenlohner, Sergei Kuzenko and Gabriel Lopez Car-doso for helpful discussions on the subjects of supergravity, conformal covariance and space-timedependent couplings.Part of this thesis was supported by the
DFG
Graduiertenkolleg “The Standard Model of Par-ticle Physics—structure, precision tests and extensions” and the Max-Planck-Institut f¨ur Physik,M¨unchen.
A Weyl Variation of the Basis
This Chapter provides the Weyl variations of all basis terms. The terms for ∆ Wσ can be extractedfrom those proportional to σ , and correspondingly for the complex conjugated fields. Eδ (cid:2) E − G α ˙ α G α ˙ α (cid:3) = 2 iG α ˙ α D α ˙ α (¯ σ − σ ) (A.1 A ) Eδ (cid:2) E − R ¯ R (cid:3) = − ( ¯ D ¯ σ ) ¯ R − ( D σ ) R (A.1 B ) Eδ (cid:2) E − R (cid:3) = 3(¯ σ − σ ) R − ( ¯ D ¯ σ ) R (A.1 C ) Eδ (cid:2) E − ¯ R (cid:3) = 3( σ − ¯ σ ) ¯ R − ( D σ ) ¯ R (A.1 ¯ C ) See [57] on why a superfield treatment of minimal supergravity should be consistent and [58] on the questionof consistence of anomaly calculations in the presence of compensating fields. Eδ (cid:2) E − D R (cid:3) = − D α σ )( D α R ) − D σ ) R − D ¯ D ¯ σ = − D α σ )( D α R ) − D σ ) R + 2( D α ˙ α D α ˙ α ¯ σ ) − iG α ˙ α ( D α ˙ α ¯ σ ) (A.1 D ) Eδ (cid:2) E − ¯ D ¯ R (cid:3) = − D ˙ α ¯ σ )( ¯ D ˙ α ¯ R ) −
2( ¯ D ¯ σ ) ¯ R − ¯ D D σ = −
2( ¯ D ˙ α ¯ σ )( ¯ D ˙ α ¯ R ) −
2( ¯ D ¯ σ ) ¯ R + 2( D α ˙ α D α ˙ α σ ) + 2 iG α ˙ α ( D α ˙ α σ ) (A.1 ¯ D ) Eδ (cid:2) E − R D λ (cid:3) = − ( ¯ D ¯ σ )( D λ ) + 2 R ( D α σ )( D α λ ) (A.1 E ) Eδ (cid:2) E − ¯ R ¯ D ¯ λ (cid:3) = − ( D σ )( ¯ D ¯ λ ) + 2 ¯ R ( ¯ D ˙ α ¯ σ )( ¯ D ˙ α ¯ λ ) (A.1 ¯ E ) Eδ (cid:2) E − R ¯ D ¯ λ (cid:3) = 3(¯ σ − σ ) R ( ¯ D ¯ λ ) − ( ¯ D ¯ σ )( ¯ D ¯ λ )+ 2 R ( ¯ D ˙ α ¯ σ )( ¯ D ˙ α ¯ λ ) (A.1 F ) Eδ (cid:2) E − ¯ R D λ (cid:3) = 3( σ − ¯ σ ) ¯ R ( D λ ) − ( D σ )( D λ )+ 2 ¯ R ( D α σ )( D α λ ) (A.1 ¯ F ) Eδ (cid:2) E − ( D α R )( D α λ ) (cid:3) = − D α σ )( D α λ ) R − ( D α ¯ D ¯ σ )( D α λ )= − D α σ )( D α λ ) R + [( G α ˙ α − i D α ˙ α )( ¯ D ˙ α ¯ σ )]( D α λ ) (A.1 G ) Eδ (cid:2) E − ( ¯ D ˙ α ¯ R )( ¯ D ˙ α ¯ λ ) (cid:3) = −
2( ¯ D ˙ α ¯ σ )( ¯ D ˙ α ¯ λ ) ¯ R − ( ¯ D ˙ α D σ )( D ˙ α ¯ λ )= −
2( ¯ D ˙ α ¯ σ )( ¯ D ˙ α ¯ λ ) ¯ R − [( G α ˙ α + i D α ˙ α )( D α σ )]( ¯ D ˙ α ¯ λ ) (A.1 ¯ G ) Eδ (cid:2) E − G α ˙ α D α ˙ α λ (cid:3) = i [ D α ˙ α (¯ σ − σ )]( D α ˙ α λ ) − i G α ˙ α ( ¯ D ˙ α ¯ σ )( D α λ ) (A.1 H ) Eδ (cid:2) E − G α ˙ α D α ˙ α ¯ λ (cid:3) = i [ D α ˙ α (¯ σ − σ )]( D α ˙ α ¯ λ ) − i G α ˙ α ( D α σ )( ¯ D ˙ α ¯ λ ) (A.1 ¯ H ) Eδ (cid:2) E − ( D α ˙ α D α ˙ α λ ) (cid:3) = ( D α ˙ α ( σ + ¯ σ )) D α ˙ α λ − R ( D α σ )( D α λ ) − i ( ¯ D ˙ α ¯ σ )( D α ˙ α D α λ ) + G α ˙ α ( ¯ D ˙ α ¯ σ )( D α λ ) (A.1 I ) Eδ (cid:2) E − ( D α ˙ α D α ˙ α ¯ λ ) (cid:3) = ( D α ˙ α ( σ + ¯ σ )) D α ˙ α ¯ λ − ¯ R ( ¯ D ˙ α ¯ σ )( ¯ D ˙ α ¯ λ ) − i ( D α σ )( D α ˙ α ¯ D ˙ α ¯ λ ) − G α ˙ α ( D α σ )( ¯ D ˙ α ¯ λ ) (A.1 ¯ I ) Eδ (cid:2) E − R ( D α λ )( D α λ ) (cid:3) = − ( ¯ D ¯ σ )( D α λ )( D α λ ) (A.1 J ) Eδ (cid:2) E − ¯ R ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ ) (cid:3) = − ( D σ )( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ ) (A.1 ¯ J ) Eδ (cid:2) E − ¯ R ( D α λ )( D α λ ) (cid:3) = 3( σ − ¯ σ ) ¯ R ( D α λ )( D α λ ) − ( D σ )( D α λ )( D α λ ) (A.1 K ) Eδ (cid:2) E − R ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ ) (cid:3) = 3(¯ σ − σ ) R ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ ) − ( ¯ D ¯ σ )( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ ) (A.1 ¯ K ) Eδ (cid:2) E − G α ˙ α ( D α λ )( ¯ D ˙ α ¯ λ ) (cid:3) = i ( D α ˙ α (¯ σ − σ ))( D α λ )( ¯ D ˙ α ¯ λ ) (A.1 L ) Eδ (cid:2) E − ( D α ˙ α λ )( D α ˙ α ¯ λ ) (cid:3) = − i ( ¯ D ˙ α ¯ σ )( D α λ )( D α ˙ α ¯ λ ) − i ( D α σ )( ¯ D ˙ α ¯ λ )( D α ˙ α λ ) (A.1 M ) Eδ (cid:2) E − ( D α ˙ α λ )( D α ˙ α λ ) (cid:3) = − i ( ¯ D ˙ α ¯ σ )( D α λ )( D α ˙ α λ ) (A.1 N ) Eδ (cid:2) E − ( D α ˙ α ¯ λ )( D α ˙ α ¯ λ ) (cid:3) = − i ( D α σ )( ¯ D ˙ α ¯ λ )( D α ˙ α ¯ λ ) (A.1 ¯ N )
08 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings Eδ (cid:2) E − ( D α λ )( D α ˙ α ¯ D ˙ α ¯ λ ) (cid:3) = ( D α λ )[( D α ˙ α (¯ σ + σ ))( ¯ D ˙ α ¯ λ ) (A.1 O ) − i ( D α σ )( ¯ D ¯ λ ) + 2( ¯ D ˙ α ¯ σ )( D α ˙ α ¯ λ ) (cid:3) Eδ (cid:2) E − ( − D ˙ α ¯ λ )( D α ˙ α D α λ ) (cid:3) = − ( ¯ D ˙ α ¯ λ )[( D α ˙ α (¯ σ + σ ))( D α λ )+ i ( ¯ D ˙ α ¯ σ )( D λ ) + 2( D α σ )( D α ˙ α λ ) (cid:3) (A.1 ¯ O ) Eδ (cid:2) E − ( D λ )( ¯ D ¯ λ ) (cid:3) = 2( D α σ )( D α λ )( ¯ D ¯ λ )+ 2( ¯ D ˙ α ¯ σ )( D λ )( ¯ D ˙ α ¯ λ ) (A.1 P ) Eδ (cid:2) E − ( D λ ) (cid:3) = 3( σ − ¯ σ )( D λ ) + 4( D λ )( D α σ )( D α λ ) (A.1 Q ) Eδ (cid:2) E − ( ¯ D ¯ λ ) (cid:3) = 3(¯ σ − σ )( ¯ D ¯ λ ) + 4( ¯ D ¯ λ )( ¯ D ˙ α ¯ σ )( ¯ D ˙ α ¯ λ ) (A.1 ¯ Q ) Eδ (cid:2) E − ( D α λ )( D α λ )( D λ ) (cid:3) = 3( σ − ¯ σ )( D α λ )( D α λ )( D λ )+ 2( D α σ )( D α λ )( D β λ )( D β λ ) (A.1 R ) Eδ (cid:2) E − ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ )( ¯ D ¯ λ ) (cid:3) = 3(¯ σ − σ )( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ )( ¯ D ¯ λ )+ 2( ¯ D ˙ α ¯ σ )( ¯ D ˙ α ¯ λ )( ¯ D ˙ β ¯ λ )( ¯ D ˙ β ¯ λ ) (A.1 ¯ R ) Eδ (cid:2) E − ( D α λ )( D α λ )( ¯ D ¯ λ ) (cid:3) = 2( D α λ )( D α λ )( ¯ D ˙ α ¯ σ )( ¯ D ˙ α ¯ λ ) (A.1 S ) Eδ (cid:2) E − ( ¯ D ˙ α ¯ λ )( D ˙ α ¯ λ )( D λ ) (cid:3) = 2( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ )( D α σ )( D α λ ) (A.1 ¯ S ) Eδ (cid:2) E − ( D α ˙ α λ i )( D α λ j )( ¯ D ˙ α ¯ λ ¯ k ) (cid:3) = i ( D α λ j )( D α λ i )( ¯ D ˙ α ¯ σ )( ¯ D ˙ α ¯ λ ¯ k ) (A.1 T ) Eδ (cid:2) E − ( D α ˙ α ¯ λ ¯ ı )( D α λ k )( D ˙ α ¯ λ ¯ ) (cid:3) = − i ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ )( D α σ )( D α λ k ) (A.1 ¯ T ) Eδ (cid:2) E − ( D α λ )( D α λ )( ¯ D ˙ β ¯ λ )( ¯ D ˙ β ¯ λ ) (cid:3) = 0 (A.1 U ) Eδ (cid:2) E − ( D α λ )( D α λ )( D β λ )( D β λ ) (cid:3) = 3( σ − ¯ σ )( D α λ )( D α λ )( D β λ )( D β λ ) (A.1 V ) Eδ (cid:2) E − ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ )( ¯ D ˙ β ¯ λ )( ¯ D ˙ β ¯ λ ) (cid:3) = 3(¯ σ − σ )( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ )( ¯ D ˙ β ¯ λ )( ¯ D ˙ β ¯ λ ) (A.1 ¯ V ) B Wess–Zumino Consistency Condition
Weyl ContributionFor the coefficients defined in(∆ Wσ )(∆ Wσ ′ − ∆ βσ ′ ) W = Z d z E − σ ′ (cid:8) σ C + ( D α σ ) C α +( D σ ) C + ( D α ˙ α σ ) C α ˙ α + ( D α ˙ α D α σ ) C ˙ α + ( D α ˙ α D α ˙ α σ ) C (cid:9) , (B.1) one obtains C = − b ( C ) R + 3 b ( ¯ C ) ¯ R − (cid:2) R ( ¯ D ¯ λ ) b ( F ) (cid:3) + 3 ¯ R ( D λ ) b ( ¯ F ) + 3 b ( K ) ¯ R ( D α λ )( D α λ ) − (cid:2) b ( ¯ K ) R ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ ) (cid:3) + 3 b ( Q ) ( D λ ) − b ( ¯ Q ) ( ¯ D ¯ λ ) + 3( D α λ )( D α λ )( D λ ) b ( R ) − b ( ¯ R ) ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ )( ¯ D ¯ λ )+ 3 b ( V ) ( D α λ )( D α λ )( D β λ )( D β λ ) − b ( ¯ V ) ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ )( ¯ D ˙ β ¯ λ )( ¯ D ˙ β ¯ λ ) , (B.2a) C α = − b ( D ) ( D α R ) − R ( D α λ ) b ( E ) + 2 ¯ R ( D α λ ) b ( ¯ F ) − (cid:2) D α λ ) Rb ( G ) (cid:3) − (cid:2) G α ˙ α ( ¯ D ˙ α ¯ λ ) b ( ¯ G ) (cid:3) − i G α ˙ α ( ¯ D ˙ α λ ) b ( ¯ H ) − i b ( M ) ( ¯ D ˙ α ¯ λ )( D α ˙ α λ ) − i ( ¯ D ˙ α ¯ λ )( D α ˙ α ¯ λ ) b ( ¯ N ) − (cid:2) i ( D α λ )( ¯ D ¯ λ ) b ( O ) (cid:3) − (cid:2) b ( ¯ O ) ( D α ˙ α λ )( ¯ D ˙ α ¯ λ ) (cid:3) + 2 b ( P ) ( D α λ )( ¯ D ¯ λ )+ 4 b ( Q ) ( D λ )( D α λ ) + 2( D α λ )( D β λ )( D β λ ) b ( R ) + 2 b ( ¯ S ) ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ )( D α λ ) − i b ( ¯ T ) ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ )( D α λ ) , (B.2b) C = − b ( B ) R − b ( ¯ C ) ¯ R − b ( D ) R − ( ¯ D ¯ λ ) b ( ¯ E ) − ( D λ ) b ( ¯ F ) − ( ¯ D ˙ α ¯ λ )( ¯ D ˙ α ¯ λ ) b ( ¯ J ) − b ( K ) ( D α λ )( D α λ ) , (B.2c) C α ˙ α = − ib ( A ) G α ˙ α + 2 iG α ˙ α b ( ¯ D ) − i ( D α ˙ α λ ) b ( H ) − i ( D α ˙ α ¯ λ ) b ( ¯ H ) − ib ( L ) ( D α λ )( ¯ D ˙ α ¯ λ ) + (cid:2) b ( O ) ( D α λ )( ¯ D ˙ α ¯ λ ) (cid:3) − (cid:2) b ( ¯ O ) ( D α λ )( ¯ D ˙ α ¯ λ ) (cid:3) , (B.2d)¯ C ˙ α = (cid:2) ib ( ¯ G ) ( ¯ D ˙ α ¯ λ ) (cid:3) , (B.2e) C = (cid:2) b ( ¯ D ) (cid:3) . (B.2f)For further discussion, it proves useful to sort its contents with respect to derivatives on λ or ¯ λ . C = − b ( C ) R + 3 b ( ¯ C ) ¯ R − (cid:2) R ( ¯ D ¯ λ ¯ ı ) b ( F )¯ ı (cid:3) + 3 ¯ R ( D λ i ) b ( ¯ F ) i + 3 b ( K ) ij ¯ R ( D α λ i )( D α λ j ) − (cid:2) b ( ¯ K )¯ ı ¯ R ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ ) (cid:3) + 3 b ( Q ) ij ( D λ i )( D λ j ) − b ( ¯ Q )¯ ı ¯ ( ¯ D ¯ λ ¯ ı )( ¯ D ¯ λ ¯ )+ 3( D α λ i )( D α λ j )( D λ k ) b ( R ) ijk − b ( ¯ R )¯ ı ¯ ¯ k ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ )( ¯ D ¯ λ ¯ k )+ 3 b ( V ) ijkl ( D α λ i )( D α λ j )( D β λ k )( D β λ l ) − b ( ¯ V )¯ ı ¯ ¯ k ¯ l ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ )( ¯ D ˙ β ¯ λ ¯ k )( ¯ D ˙ β ¯ λ ¯ l ) , (B.3a)
10 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings C α = − b ( D ) ( D α R ) + ( D α λ i ) (cid:18) − R ( b ( E ) i + (cid:2) b ( G ) i (cid:3) + (cid:2) b ( I ) i (cid:3) ) + 2 ¯ Rb ( ¯ F ) i (cid:19) + ( ¯ D ˙ α ¯ λ ¯ ı ) (cid:18) − i G α ˙ α b ( ¯ H )¯ ı − G α ˙ α b (¯ I )¯ ı − (cid:2) G α ˙ α b ( ¯ G )¯ ı (cid:3)(cid:19) + ( D α ˙ α ¯ D ˙ α ¯ λ ¯ ı ) (cid:18) − i b (¯ I )¯ ı (cid:19) + ( D α ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ ) (cid:18) − ib ( ¯ N )¯ ı ¯ (cid:19) + ( D α ˙ α λ i )( ¯ D ˙ α ¯ λ ¯ ) (cid:18) − i b ( M ) i ¯ − (cid:2) b ( ¯ O )¯ i (cid:3)(cid:19) + ( D α λ i )( ¯ D ¯ λ ¯ ) (cid:18) − (cid:2) i b ( O ) i ¯ (cid:3) + 2 b ( P ) i ¯ (cid:19) + ( D λ i )( D α λ j ) (cid:18) b ( Q ) ij (cid:19) + ( D α λ i )( D β λ j )( D β λ k ) (cid:18) b ( R ) ijk (cid:19) + ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ )( D α λ k ) (cid:18) b ( ¯ S )¯ ı ¯ k − i b ( ¯ T ) { ¯ ı ¯ } k (cid:19) , (B.3b) C = − ( b ( B ) + 2 b ( D ) ) R − b ( ¯ C ) ¯ R − ( ¯ D ¯ λ ¯ ı ) b ( ¯ E )¯ ı − ( D λ i ) b ( ¯ F ) i − ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ ) b ( ¯ J )¯ ı ¯ − b ( K ) ij ( D α λ i )( D α λ j ) , (B.3c) C α ˙ α = 2 iG α ˙ α ( (cid:2) b ( ¯ D ) (cid:3) − b ( A ) ) + ( D α ˙ α λ i )( (cid:2) b ( I ) i (cid:3) − (cid:2) ib ( H ) i (cid:3) )+ ( D α ˙ α ¯ λ ¯ ı )( b (¯ I )¯ ı − ib ( ¯ H )¯ ı ) + ( (cid:2) b ( O ) i ¯ (cid:3) − (cid:2) b ( ¯ O ) i ¯ (cid:3) − ib ( L ) i ¯ )( D α λ i )( ¯ D ˙ α ¯ λ ¯ ) , (B.3d)¯ C ˙ α = (cid:2) ib ( ¯ G )¯ ( ¯ D ˙ α ¯ λ ¯ ) (cid:3) , (B.3e) C = (cid:2) b ( ¯ D ) (cid:3) . (B.3f)Beta ContributionActing with the operator ∆ β on the conformal anomaly, one first notices, that δδλ i should onlyact on derivatives of λ , since otherwise a σσ ′ contribution, which vanishes from the commutator,is created. ∆ βσ (∆ Wσ ′ − ∆ βσ ′ ) W = Z d z E − σ ′ (cid:8) σβ i ( ∂ i B (0) ) A (0) + ( D α σβ i ) E iα (B.4)+ ( D σβ i ) E i (2) + ( D α ˙ α σβ i ) E iα ˙ α + ( D α ˙ α D α ˙ α σβ i ) E i + ( D α ˙ α D ˙ α σβ i ) ¯ E i ˙ α (cid:9) , E α = (cid:2) b ( G ) i ( D α R ) (cid:3) + (cid:2) b ( J ) ij R ( D α λ j ) (cid:3) + 2 b ( K ) ij ¯ R ( D α λ j )+ b ( L ) i ¯ G α ˙ α ( ¯ D ˙ α ¯ λ ¯ ) + (cid:2) b ( O ) i ¯ ( D α ˙ α ¯ D ˙ α ¯ λ ¯ ) (cid:3) + 2 b ( R ) ijk ( D α λ j )( D λ k ) + (cid:2) b ( S ) ij ¯ k ( D α λ j )( ¯ D ¯ λ ¯ k ) (cid:3) + b ( T ) ji ¯ k ( D α ˙ α λ j )( ¯ D ˙ α ¯ λ ¯ k ) + b ( ¯ T )¯ i ¯ k ( D α ˙ α ¯ λ ¯ )( ¯ D ˙ α ¯ λ ¯ k )+ 2 b ( U ) ij ¯ k ¯ l ( D α λ j )( ¯ D ˙ β ¯ λ ¯ k )( ¯ D ˙ β ¯ λ ¯ l ) + 4 b ( V ) ijkl ( D α λ j )( D β λ k )( D β λ l ) , (B.5a) E = b ( E ) i R + b ( ¯ F ) i ¯ R + b ( P ) i ¯ ( ¯ D ¯ λ ¯ )+ 2 b ( Q ) ij ( D λ j ) + b ( R ) lji ( D α λ l )( D α λ j ) + b ( ¯ S )¯ ı ¯ i ( ¯ D ˙ α ¯ λ ¯ ı )( ¯ D ˙ α ¯ λ ¯ ) , (B.5b) E α ˙ α = (cid:2) b ( H ) i G α ˙ α (cid:3) + b ( M ) i ¯ ( D α ˙ α ¯ λ ¯ ) + b ( N ) ij ( D α ˙ α λ j ) + b ( T ) ij ¯ k ( D α λ j )( ¯ D ˙ α ¯ λ ¯ k ) , (B.5c)¯ E ˙ α = (cid:2) b ( ¯ O )¯ i ( ¯ D ˙ α ¯ λ ¯ ) (cid:3) , (B.5d) E = (cid:2) b ( I ) i (cid:3) . (B.5e)SummaryThe results of the previous two Sections can be used to determine the F coefficients defined by(∆ Wσ − ∆ βσ )(∆ Wσ ′ − ∆ βσ ′ ) W = Z d z E − σ ′ (cid:8) σ F + ( D α σ ) F α + ( D σ ) F + ( D α ˙ α σ ) F α ˙ α + ( D α ˙ α D α σ ) ¯ F ˙ α + ( D α ˙ α D α ˙ α σ ) F (cid:9) , (B.6)by expanding the Weyl and beta contribution in terms of derivatives on λ and ¯ λ , keeping in mindthat the b coefficients and beta functions are functions of λ and ¯ λ in general, so it holds b = b ( λ, ¯ λ ) , (B.7a) D α b = ( D α λ k )( ∂ k b ) , (B.7b)¯ D ˙ α b = ( ¯ D ˙ α ¯ λ ¯ k )( ∂ ¯ k b ) , (B.7c) D α ˙ α b = ( D α ˙ α λ k )( ∂ k b ) + ( D α ˙ α ¯ λ ¯ k )( ∂ ¯ k b ) , (B.7d)¯ D ˙ α D α ˙ α b = ( ¯ D ˙ α D α ˙ α λ k )( ∂ k b ) + ( D α ˙ α λ k )( ¯ D ˙ α ¯ λ ¯ )( ∂ ¯ ∂ k b )+ ( ¯ D ˙ α D α ˙ α ¯ λ ¯ k )( ∂ ¯ k b ) + ( D α ˙ α ¯ λ ¯ k )( ¯ D ˙ α ¯ λ ¯ )( ∂ ¯ ∂ ¯ k b ) , (B.7e) D α ˙ α D α ˙ α b = ( D α ˙ α D α ˙ α λ k )( ∂ k b ) + ( D α ˙ α D α ˙ α ¯ λ ¯ k )( ∂ ¯ k b )+( D α ˙ α λ k )( D α ˙ α λ l )( ∂ l ∂ k b )+2( D α ˙ α λ k )( D α ˙ α ¯ λ ¯ l )( ∂ k ∂ ¯ l b )+( D α ˙ α ¯ λ ¯ k )( D α ˙ α ¯ λ ¯ l )( ∂ ¯ k ∂ ¯ l b ) (B.7f)and similarly for β i . This yields F = C + β i ( ∂ i B ) · A + ( D α λ j )( ∂ j β i ) E iα + [( D λ j )( ∂ j β i ) + ( D α λ j )( D α λ k )( ∂ j ∂ k β i )] E i + [( D α ˙ α λ j )( ∂ j β i ) + ( D α ˙ α ¯ λ ¯ )( ∂ ¯ β i )] E iα ˙ α + [( D α ˙ α D α λ j )( ∂ j β i ) + ( D α λ j )( D α ˙ α λ k )( ∂ k ∂ j β i )+ ( D α λ j )( D α ˙ α ¯ λ ¯ k )( ∂ ¯ k ∂ j β i )] ¯ E ˙ αi + [( D α ˙ α D α ˙ α λ k )( ∂ k β i ) + ( D α ˙ α D α ˙ α ¯ λ ¯ k )( ∂ ¯ k β i )+ ( D α ˙ α λ k )( D α ˙ α λ j )( ∂ j ∂ k β i ) + 2( D α ˙ α λ k )( D α ˙ α ¯ λ ¯ )( ∂ ¯ ∂ k β i )+ ( D α ˙ α ¯ λ ¯ k )( D α ˙ α ¯ λ ¯ )( ∂ ¯ ∂ ¯ k β i )] E i , (B.8a) F α = C α + β i E iα + 2( D α λ j )( ∂ j β i ) E i + [( D α ˙ α λ k )( ∂ k β i ) + ( D α ˙ α ¯ λ ¯ k )( ∂ ¯ k β i )] ¯ E ˙ αi , (B.8b)
12 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings F = C + β i E i , (B.8c) F α ˙ α = C α ˙ α + β i E iα ˙ α + ( D α λ j )( ∂ j β i ) ¯ E i α + 2( D α ˙ α λ k )( ∂ k β i ) E i + 2( D α ˙ α ¯ λ ¯ k )( ∂ ¯ k β i ) E i , (B.8d)¯ F ˙ α = ¯ C ˙ α + β i ¯ E i ˙ α , (B.8e) F = C + β i E i . (B.8f) C Coefficient Consistency Equations
Consistency equation (9.35e) yields2¯ b ( D ) − b ( D ) + β i ¯ b (¯ I ) i − ¯ β ¯ ı ¯ b (¯ I )¯ ı = 0 . (C.1)From consistency equation (9.35d) one obtains − ¯ b ( A ) + b ( A ) = 0 , (C.2a)¯ b ( ¯ N ) ij β i + ( ∂ j β i )¯ b (¯ I ) i + 2 ∂ j ¯ b ( D ) + ( ∂ j ¯ b (¯ I ) i ) β i + b ( I ) j + ib ( H ) j + b ( M ) j ¯ ı ¯ β ¯ ı + ( ∂ j ¯ β ¯ ı ) b (¯ I )¯ ı + 2( ∂ j b ( D ) ) + ( ∂ j b (¯ I )¯ ı ) ¯ β ¯ ı − b ( G ) j − i ¯ β ¯ ı b ( O ) j ¯ ı = 0 , (C.2b) b ( ¯ N )¯ ı ¯ β ¯ ı + ( ∂ ¯ ¯ β ¯ ı ) b (¯ I )¯ ı + 2 ∂ ¯ b ( D ) + ( ∂ ¯ b (¯ I )¯ ı ) ¯ β ¯ ı + ¯ b ( I )¯ − i ¯ b ( H )¯ + ¯ b ( M ) i ¯ β i + ( ∂ ¯ β i )¯ b (¯ I ) i + 2( ∂ ¯ ¯ b ( D ) ) + ( ∂ ¯ ¯ b (¯ I ) i ) β i − b ( G )¯ + 2 iβ i ¯ b ( O )¯ i = 0 , (C.2c) − i ¯ b ( L ) i ¯ + ¯ b ( ¯ T ) ki ¯ β k − i ( ∂ i ¯ b ( G )¯ ) − ( ∂ i ¯ b ( O )¯ k ) β k + ib ( L )¯ i + b ( ¯ T )¯ k ¯ i ¯ β ¯ k + i ( ∂ ¯ b ( G ) i ) − ( ∂ ¯ b ( O ) i ¯ k ) ¯ β ¯ k = 0 . (C.2d)The following sets of equations have to be augmented by their complex conjugates. From consis-tency condition (9.35a) one gets − b ( D ) + b ( B ) + b ( A ) + 2 b ( D ) − β i b ( E ) i = 0 , (C.3a) − b ( E ) j + 2 β i b ( K ) ij + b ( E ) i ( ∂ j β i ) + ( ∂ j b ( B ) )+ 2( ∂ j b ( D ) ) − β i ( ∂ j b ( E ) i ) + b ( N ) ij β i = 0 , (C.3b)2 b ( ¯ F ) j + b ( ¯ F ) i ( ∂ j β i ) + ( ∂ j b ( ¯ C ) )+ b ( ¯ F ) j − β i ( ∂ j b ( ¯ F ) i ) + 8 β i b ( Q ) ij = 0 , (C.3c) − i b ( ¯ H )¯ − b (¯ I )¯ + β i b ( L ) i ¯ + b ( ¯ E )¯ − β i b ( P ) i ¯ + ( ∂ ¯ b ( A ) ) − b (¯ I )¯ + ib ( ¯ H )¯ = 0 , (C.3d) − i b (¯ I )¯ − ib ( ¯ E )¯ + 4 iβ i b ( P ) i ¯ + i b (¯ I )¯ + b ( ¯ H )¯ + i β i b ( M ) i ¯ = 0 , (C.3e) − i b ( M ) j ¯ k + β i b ( T ) ji ¯ k + ib ( L ) j ¯ k + i b ( N ) ij ( ∂ ¯ k β i ) + i β i ( ∂ ¯ k b ( N ) ij ) − b ( T ) ij ¯ k β i = 0 , (C.3f) − ib ( ¯ N )¯ ¯ k + β i b ( ¯ T )¯ i ¯ k − ib ( ¯ J )¯ ¯ k + 2 iβ i b ( ¯ S )¯ ¯ ki + i ( ∂ ¯ k b (¯ I )¯ − i∂ ¯ k b ( ¯ H )¯ ) + i ( ∂ ¯ k β i ) b ( M ) i ¯ + i ( ∂ ¯ k b ( M ) i ¯ ) β i = 0 , (C.3g) b ( Q ) kj + 2 β i b ( R ) ijk + 2 b ( Q ) ik ( ∂ j β i ) + ( ∂ j b ( ¯ F ) k ) − b ( K ) kj − β i ( ∂ j b ( Q ) ik ) − β i b ( R ) jki = 0 , (C.3h)2 b ( P ) i ¯ + b ( P ) k ¯ ( ∂ i β k ) + ( ∂ i b ( ¯ E )¯ ) − β k ( ∂ i b ( P ) k ¯ ) + b ( L ) i ¯ + i b ( T ) ki ¯ β k = 0 , (C.3i)2 b ( R ) jkl + 4 β i b ( V ) ijkl + b ( R ) kli ( ∂ j β i ) + ∂ j b ( K ) kl − β i ( ∂ j b ( R ) kli ) = 0 , (C.3j)2 b ( ¯ S )¯ ı ¯ k − i b ( ¯ T ) { ¯ ı ¯ } k + 2 β l b ( U ) lk ¯ ı ¯ + b ( ¯ S )¯ ı ¯ l ( ∂ k β l ) + ∂ k b ( ¯ J )¯ ı ¯ − β l ( ∂ k b ( ¯ S )¯ ı ¯ l ) + ( ∂ ¯ ı b ( L ) k ¯ ) + i b ( T ) lk ¯ ı ( ∂ ¯ β l ) + i ( ∂ ¯ b ( T ) lk ¯ ı ) β l = 0 , (C.3k)while consistency equation (9.35c) yields β l ( ∂ l ¯ b ( A ) ) − ¯ β ¯ l ( ∂ ¯ l b ( A ) ) = 0 , (C.4a) β l ( ∂ l ¯ b ( ¯ B ) ) − ¯ β ¯ l ( ∂ ¯ l b ( B ) ) = 0 , (C.4b) − b ( ¯ C ) + β l ( ∂ l ¯ b ( ¯ C ) ) − b ( C ) − ¯ β ¯ l ( ∂ ¯ l b ( C ) ) = 0 , (C.4c)+ 2¯ b ( ¯ D ) + (¯ b ( D ) − ¯ b ( A ) ) − (¯ b ( B ) + 2¯ b ( ¯ D ) ) + β l ¯ b ( ¯ G ) l + β l ¯ b ( ¯ E ) l − i β l ¯ b ( ¯ H ) l + β l ( ∂ l ¯ b ( ¯ D ) ) + ( b ( D ) − b ( A ) ) + i ¯ β ¯ l b ( ¯ H )¯ l − ¯ β ¯ l ( ∂ ¯ l b ( D ) ) = 0 , (C.4d)+ 2¯ b ( ¯ E ) i + 2¯ b ( ¯ G ) i + ¯ b (¯ I ) i − ∂ k ( ¯ b ( B ) + 2¯ b ( ¯ D ) ) − ¯ b ( F ) i + 2 β l ¯ b ( ¯ J ) li + β l ∂ i ¯ b ( ¯ E ) l + 8 β l ¯ b ( ¯ Q ) li + β l ( ∂ l ¯ b ( ¯ E ) i ) + b ( I ) i + 2 b ( E ) i − β ¯ l b ( ¯ P )¯ li − ¯ β ¯ l ( ∂ ¯ l b ( E ) i ) = 0 , (C.4e) − b ( ¯ F )¯ ı + 2 iβ l ¯ b ( ¯ O ) l ¯ ı + β l ( ∂ l ¯ b ( ¯ F )¯ ı ) − b ( F )¯ ı + ∂ ¯ ı b ( C ) − β ¯ l b ( ¯ K )¯ l ¯ ı − ¯ β ¯ l ∂ ¯ ı b ( F )¯ l − ¯ β ¯ l ( ∂ ¯ l b ( F )¯ ı ) = 0 , (C.4f)+ 2 ∂ i ¯ b ( ¯ D ) + 2¯ b ( ¯ E ) i + 2¯ b ( ¯ G ) i + ¯ b (¯ I ) i − ∂ i ( ¯ b ( B ) + 4¯ b ( ¯ D ) ) + β l ∂ i ¯ b ( ¯ G ) l + 2 β l ¯ b ( ¯ J ) li + 2 β l ∂ i ¯ b ( ¯ E ) l + β l ( ∂ l ¯ b ( ¯ G ) i ) + i b ( H ) i − b ( I ) i − b ( G ) i + b ( E ) i − ¯ β ¯ l b ( L )¯ li − β ¯ l b ( P )¯ li − ¯ β l ( ∂ ¯ l b ( G ) i ) = 0 , (C.4g) − i∂ i (¯ b ( D ) − ¯ b ( A ) ) + ¯ b ( ¯ H ) l ∂ i β l − β l ∂ i ¯ b ( ¯ H ) l + β l ( ∂ l ¯ b ( ¯ H ) i ) − b ( H ) i − ib ( G ) i − i∂ i ( b ( D ) − b ( A ) ) − ib ( E ) i − i ¯ β ¯ l b ( L )¯ li − β ¯ l b ( ¯ O )¯ li + 8 i ¯ β ¯ l b ( P )¯ li − b ( ¯ H )¯ l ∂ i ¯ β ¯ l + ¯ β ¯ l ∂ i b ( ¯ H )¯ l − ¯ β ¯ l ( ∂ ¯ l b ( H ) i ) = 0 , (C.4h) − (¯ b (¯ I ) i − i ¯ b ( ¯ H ) i ) + ∂ i ¯ b ( D ) − β l ¯ b ( ¯ N ) li + β l ∂ i ¯ b (¯ I ) l + ¯ b (¯ I ) l ∂ i β l + β l ( ∂ l ¯ b (¯ I ) i ) − b ( I ) i + ( b ( I ) i + ib ( H ) i ) − b ( E ) i − ∂ i b ( D ) − i ¯ β ¯ l b ( ¯ O )¯ li + 8 ¯ β ¯ l b ( P )¯ li + ¯ β ¯ l b ( M )¯ li − ¯ β ¯ l ∂ i b (¯ I )¯ l − b (¯ I )¯ l ∂ i ¯ β ¯ l − ¯ β ¯ l ( ∂ ¯ l b ( I ) i ) = 0 , (C.4i)
14 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings + 2 ∂ j (¯ b ( ¯ E ) i + ¯ b ( ¯ G ) i + ¯ b (¯ I ) i ) − ∂ i ∂ j ¯ b ( B ) − ∂ i ∂ j ¯ b ( ¯ D ) − ∂ i ¯ b ( F ) j + 2 β l ∂ i ¯ b ( ¯ J ) lj + β l ∂ i ∂ j ¯ b ( ¯ E ) l + β l ( ∂ l ¯ b ( ¯ J ) ij ) + 2 b ( N ) ij + 2 b ( J ) ij − β ¯ l b ( S ) ij ¯ l + 2 i ¯ β ¯ l b ( T ) i ¯ lj − ¯ β ¯ l ( ∂ ¯ l b ( J ) ij ) = 0 , (C.4j)+ 4¯ b ( ¯ K ) ij − ∂ j ¯ b ( F ) i − b ( ¯ Q ) ij − ∂ i ∂ j ¯ b ( C ) + 2 β l ∂ i ¯ b ( ¯ K ) lj + 8 β l ¯ b ( ¯ R ) lij + β l ∂ i ∂ j ¯ b ( F ) l + 16 β l ∂ i ¯ b ( ¯ Q ) lj − β l ¯ b ( ¯ R ) ijl + β l ( ∂ l ¯ b ( ¯ K ) ij ) + 3 b ( K ) ij − ¯ β ¯ l ( ∂ ¯ l b ( K ) ij ) = 0 , (C.4k) i ∂ i ¯ b ( H )¯ + ∂ i ¯ b ( I )¯ + ∂ i ¯ b ( G )¯ + ¯ b ( M ) i ¯ − i ¯ b ( O )¯ i + 2 i ¯ b ( ¯ O ) i ¯ − b ( P ) i ¯ − ∂ i ¯ b ( E )¯ + β l ∂ i ¯ b ( L ) l ¯ + 8 β l ¯ b ( ¯ S ) li ¯ − iβ l b ( T ) il ¯ + 8 β l ∂ i ¯ b ( P ) l ¯ + β l ( ∂ l ¯ b ( L ) i ¯ ) + i ∂ ¯ b ( H ) i − ∂ ¯ b ( I ) i − ∂ ¯ b ( G ) i − b ( M ) i ¯ − ib ( O ) i ¯ + 2 ib ( ¯ O )¯ i + 8 b ( P ) i ¯ + 2 ∂ ¯ b ( E ) i − ¯ β ¯ l ∂ ¯ b ( L )¯ li − β ¯ l b ( ¯ S )¯ l ¯ i − i ¯ β ¯ l ¯ b ( T )¯ ¯ li − β ¯ l ∂ ¯ b ( P )¯ li − ¯ β ¯ l ( ∂ ¯ l b ( L )¯ i ) = 0 , (C.4l)¯ b ( M ) i ¯ − i ¯ b ( ¯ O )¯ i − ∂ i (¯ b ( I )¯ − i ¯ b ( H )¯ ) − ∂ ¯ (¯ b (¯ I ) i − i ¯ b ( ¯ H ) i ) + 2 ∂ i ∂ ¯ ¯ b ( D ) − iβ l ¯ b ( ¯ T ) il ¯ + ¯ b ( M ) l ¯ ∂ i β l + ¯ b ( ¯ N ) li ∂ ¯ β l − β l ∂ i ¯ b ( M ) l ¯ − β l ∂ ¯ ¯ b ( ¯ N ) li + β l ∂ i ∂ ¯ ¯ b (¯ I ) l + ¯ b (¯ I ) l ∂ i ∂ ¯ β l + β l ( ∂ l ¯ b ( M ) i ¯ ) − b ( M ) i ¯ − ib ( ¯ O )¯ i + ∂ i ( b ( I )¯ + ib ( H )¯ ) + ∂ ¯ ( b (¯ I ) i + ib ( ¯ H ) i ) − ∂ i ∂ ¯ b ( D ) − iβ l b ( ¯ T ) il ¯ − b ( M ) l ¯ ∂ i β l − b ( ¯ N ) li ∂ ¯ β l + β l ∂ i b ( M ) l ¯ + β l ∂ ¯ b ( ¯ N ) li − β l ∂ i ∂ ¯ b (¯ I ) l − b (¯ I ) l ∂ i ∂ ¯ β l − β l ( ∂ l b ( M ) i ¯ ) = 0 , (C.4m) − ∂ j (¯ b (¯ I ) i − i ¯ b ( ¯ H ) i ) + ∂ i ∂ j ¯ b ( ¯ D ) + ¯ b ( ¯ N ) lj ∂ i β l − β l ∂ i ¯ b ( ¯ N ) lj + β l ∂ i ∂ j ¯ b (¯ I ) l + ¯ b (¯ I ) l ∂ i ∂ j β l + β l ( ∂ l ¯ b ( ¯ N ) ij ) − b ( N ) ij + ∂ j ( b ( I ) i + ib ( H ) i ) − b ( J ) ij − ∂ i ∂ j b ( D ) − i ¯ β ¯ l b ( T ) i ¯ lj + 4 ¯ β ¯ l b ( S ) ij ¯ l − b ( M )¯ lj ∂ i ¯ β ¯ l + ¯ β ¯ l ∂ i b ( M )¯ lj − ¯ β ¯ l ∂ i ∂ j b (¯ I )¯ l − b (¯ I )¯ l ∂ i ∂ j ¯ β ¯ l − ¯ β ¯ l ( ∂ ¯ l b ( N ) ij ) = 0 , (C.4n) i ∂ i ¯ b ( I )¯ + 2¯ b ( ¯ O ) i ¯ + 8 i ¯ b ( P ) i ¯ − ( ¯ b ( ¯ O ) i ¯ − ¯ b ( O ) i ¯ − i ¯ b ( L ) i ¯ ) + 2 i∂ i ¯ b ( E )¯ + β l ∂ i ¯ b ( ¯ O ) l ¯ − iβ l ¯ b ( ¯ S ) li ¯ − iβ l ∂ i b ( P ) l ¯ − β l ¯ b ( ¯ T ) li ¯ + β l ( ∂ l ¯ b ( ¯ O ) i ¯ ) + i b ( M )¯ i − b ( O ) i ¯ + ( b ( ¯ O )¯ i − b ( O )¯ i + ib ( L )¯ i ) − ¯ β ¯ l b ( ¯ T )¯ ¯ li + ¯ β ¯ l b ( ¯ T )¯ l ¯ i + ¯ β ¯ l ( ∂ ¯ l b ( O ) i ¯ ) = 0 , (C.4o) i ¯ b ( ¯ O ) i ¯ − b ( P ) i ¯ − ∂ i ¯ b ( E )¯ + 2 β l ¯ b ( ¯ S ) li ¯ + β l ∂ i ¯ b ( P ) l ¯ + β l ( ∂ l ¯ b ( P ) i ¯ )+ i b ( ¯ O ) i ¯ + 2 b ( P ) i ¯ + ∂ ¯ b ( E ) i − β ¯ l b ( ¯ S )¯ l ¯ i − ¯ β ¯ l ∂ ¯ b ( P )¯ li − ¯ β ¯ l ( ∂ ¯ l b ( P ) i ¯ ) = 0 , (C.4p) − ¯ b ( ¯ Q ) ij − ∂ i ¯ b ( F ) j − ¯ b ( ¯ K ) ij + 2 β l ¯ b ( ¯ R ) lij + 2 β l ∂ i ¯ b ( ¯ Q ) lj + β l ¯ b ( ¯ R ) ijl + β l ( ∂ l ¯ b ( ¯ Q ) ij ) + 3 b ( Q ) ij − ¯ β ¯ l ( ∂ ¯ l b ( Q ) ij ) = 0 , (C.4q) + 3¯ b ( ¯ R ) ijk − ∂ i ¯ b ( ¯ Q ) jk − ∂ k ¯ b ( ¯ K ) ij + ∂ i ¯ b ( ¯ K ) kj + 2 β l ∂ i ¯ b ( ¯ R ) ljk + 4 β l (¯ b ( ¯ V ) lkij + ¯ b ( ¯ V ) likj ) + 2 β l ∂ i ∂ j ¯ b ( ¯ Q ) lk + β l ∂ k ¯ b ( ¯ R ) ijl − β l ∂ i ¯ b ( ¯ R ) jkl + β l ( ∂ l ¯ b ( ¯ R ) ijk ) + 3 b ( R ) ijk − ¯ β ¯ l ( ∂ ¯ l b ( R ) ijk ) = 0 , (C.4r) ∂ i ( i ¯ b ( ¯ O ) j ¯ k − b ( P ) j ¯ k ) − ∂ i ∂ j ¯ b ( E )¯ k + 2 β l ∂ i ¯ b ( ¯ S ) lj ¯ k + β l ∂ i ∂ j ¯ b ( P ) l ¯ k + β l ( ∂ l ¯ b ( ¯ S ) ij ¯ k ) + 2 b ( S ) ij ¯ k + i b ( T ) ij ¯ k + ∂ i ∂ j b ( F )¯ k + ∂ ¯ k b ( J ) ij − β ¯ l b ( U ) ij ¯ k ¯ l − ¯ β ¯ l ( ∂ ¯ k b ( S ) ij ¯ l ) − ¯ β ¯ l ( ∂ ¯ l b ( S ) ij ¯ k ) = 0 , (C.4s) i ∂ j ¯ b ( M ) i ¯ k + ∂ j ¯ b ( O )¯ ki − ∂ i ( ¯ b ( ¯ O ) j ¯ k − ¯ b ( O ) j ¯ k − i ¯ b ( L ) j ¯ k ) + β l ∂ j ¯ b ( ¯ T ) il ¯ k + ¯ b ( ¯ T ) lj ¯ k ∂ i β l − β l ∂ i ¯ b ( ¯ T ) lj ¯ k + β l ( ∂ l ¯ b ( ¯ T ) ij ¯ k ) + i∂ ¯ k b ( N ) ij + 8 ib ( S ) ji ¯ k − b ( T ) ji ¯ k + ∂ i ( b ( ¯ O )¯ kj − b ( O )¯ kj + ib ( L )¯ kj ) + 2 i∂ ¯ k b ( J ) ij − ¯ β ¯ l ∂ ¯ k b ( T ) i ¯ lj − i ¯ β ¯ l b ( U ) ij ¯ k ¯ l − i ¯ β ¯ l ∂ ¯ k b ( S ) ij ¯ l − b ( ¯ T )¯ l ¯ kj ∂ i ¯ β ¯ l + ¯ β ¯ l ∂ i b ( ¯ T )¯ l ¯ kj − ¯ β ¯ l ( ∂ ¯ l b ( T ) ij ¯ k ) = 0 , (C.4t) ∂ i ( − b ( S )¯ k ¯ lj + i ¯ b ( T )¯ k ¯ lj ) − ∂ i ∂ j ¯ b ( J )¯ k ¯ l + 2 β m ∂ i ¯ b ( U ) mj ¯ k ¯ l + β m ∂ i ∂ j ¯ b ( S )¯ k ¯ lm + β m ( ∂ m ¯ b ( U ) ij ¯ k ¯ l ) − ∂ ¯ k ( − b ( S ) ij ¯ l − i b ( T ) ij ¯ l ) + ∂ ¯ k ∂ ¯ l b ( J ) ij − β ¯ m ∂ ¯ k b ( U )¯ m ¯ lij − ¯ β ¯ m ∂ ¯ k ∂ ¯ l b ( S ) ij ¯ m − ¯ β ¯ m ( ∂ ¯ m b ( U ) ij ¯ k ¯ l ) = 0 , (C.4u)3¯ b ( ¯ V ) ijkl − ∂ i ¯ b ( ¯ R ) jkl − ∂ i ∂ j ¯ b ( ¯ K ) kl + β m ∂ i ¯ b ( ¯ V ) mjkl + β m ∂ i ∂ j ¯ b ( ¯ R ) klm + β m ( ∂ m ¯ b ( ¯ V ) ijkl ) + 3 b ( V ) ijkl − ¯ β ¯ m ( ∂ ¯ m b ( V ) ijkl ) = 0 . (C.4v) D Minimal Algebra on Chiral Fields
The following follows from the superalgebra for a chiral ( λ ) or antichiral (¯ λ ) scalar superfield.Although trivial, these special cases occur sufficiently frequent to earn explicit treatment, D ¯ D ¯ λ = (8 iG α ˙ α D α ˙ α − D α ˙ α D α ˙ α + 4( ¯ D ˙ α ¯ R ) ¯ D ˙ α + 8 ¯ R ¯ D )¯ λ, (D.1a) D α D α ˙ α λ = D α ˙ α D α λ − iG α ˙ α D α λ, (D.1b) D α D α ˙ α ¯ λ = 2 i ¯ R ¯ D ˙ α ¯ λ, (D.1c)( D α ¯ D ¯ λ ) = 4( G α ˙ α − i D α ˙ α )( ¯ D ˙ α ¯ λ ) , (D.1d)( D α D λ ) = 4 ¯ R ( D α λ ) , (D.1e)( D D α λ ) = − R ( D α λ ) , (D.1f)( D ¯ D ˙ α ¯ λ ) = 4 ¯ R ( ¯ D ˙ α ¯ λ ) , (D.1g) D α ( D β λ )( D β λ ) = − ( D α λ )( D λ ) , (D.1h)( ¯ D ˙ α D α ˙ α ¯ λ ) = ( D α ˙ α ¯ D ˙ α ¯ λ ) − iG α ˙ α ( ¯ D ˙ α ¯ λ ) , (D.1i)( ¯ D ˙ α D α ˙ α λ ) = − iR ( D α λ ) , (D.1j)( ¯ D ˙ α D λ ) = 4( G α ˙ α + i D α ˙ α )( D α λ ) , (D.1k)( D α D β λ ) = ε αβ ( D λ ) , (D.1l)
16 J. Große: QFTs Coupled to SUGRA: AdS/CFT and Local Couplings ( D α D α ˙ α ¯ D ˙ α ¯ λ ) = − i D α ˙ α D α ˙ α ¯ λ + 2 i ¯ R ¯ D ¯ λ + 4 G α ˙ α D α ˙ α ¯ λ. (D.1m)Weyl variations for derivatives acting on chiral fields of Weyl weight 0 are given by δ ′ (cid:2) λ (cid:3) = 0 , (D.2a) δ ′ (cid:2) D α λ (cid:3) = ( σ ′ − ¯ σ ′ ) D α λ, (D.2b) δ ′ (cid:2) D α ˙ α λ (cid:3) = − ( σ ′ + ¯ σ ′ ) D α ˙ α λ − i ( ¯ D ˙ α ¯ σ ′ ) D α λ, (D.2c) δ ′ (cid:2) D λ (cid:3) = ( σ ′ − σ ′ ) D λ + 2( D α σ ′ ) D α λ, (D.2d) δ ′ (cid:2) ¯ D ¯ λ (cid:3) = (¯ σ ′ − σ ′ ) ¯ D ¯ λ + 2( ¯ D ˙ α ¯ σ ′ ) ¯ D ˙ α ¯ λ, (D.2e) δ ′ (cid:2) D α ˙ α ¯ λ (cid:3) = − ( σ ′ + ¯ σ ′ ) D α ˙ α ¯ λ − i ( D α σ ′ ) ¯ D ˙ α ¯ λ, (D.2f) δ ′ (cid:2) D α ˙ α ¯ D ˙ α ¯ λ (cid:3) = − σ ′ D α ˙ α ¯ D ˙ α ¯ λ + ( D α ˙ α σ ′ ) ¯ D ˙ α ¯ λ + i ( D α σ ′ ) ¯ D ¯ λ + ( ¯ D ˙ α ¯ σ ′ ) D α ˙ α ¯ λ + ( D α ˙ α ¯ σ ′ ) ¯ D ˙ α ¯ λ, (D.2g) δ ′ (cid:2) D α ˙ α D α λ (cid:3) = − ¯ σ ′ D α ˙ α D α λ + ( D α ˙ α ¯ σ ′ ) D α λ − i ( ¯ D ˙ α ¯ σ ′ ) D λ + ( D α σ ′ ) D α ˙ α λ + ( D α ˙ α σ ′ ) D α λ. (D.2h) References [1]
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