UV finiteness of Pohlmeyer-reduced form of the AdS_5xS^5 superstring theory
aa r X i v : . [ h e p - t h ] M a r Imperial-TP-AT-2009-1
UV finiteness of Pohlmeyer-reduced formof the
AdS × S superstring theory R. Roiban a, and A.A. Tseytlin b, a Department of Physics, The Pennsylvania State University,University Park, PA 16802 , USA b Blackett Laboratory, Imperial College, London SW7 2AZ, U.K.
Abstract
We consider the Pohlmeyer-type reduced theory found by explicitly solving the Vi-rasoro constraints in the formulation of
AdS × S superstring in terms of supercosetcurrents. The resulting set of classically equivalent, integrable Lagrangian equations ofmotion has the advantage of involving only a physical number of degrees of freedom andyet being 2d Lorentz invariant. The corresponding reduced theory action may be writtenas a gauged WZW model coupled to fermions with further bosonic and fermionic poten-tial terms. Since the AdS × S superstring sigma model is conformally invariant, itsclassical relation to the reduced theory may extend to the quantum level only if the latteris, in fact, UV finite. This theory is power counting renormalizable with the only possibledivergences being of potential type. We explicitly verify its 1-loop finiteness and showthat the 2-loop divergences are, in general, scheme dependent and vanish in dimensionalreduction scheme. We expect that the reduced theory is finite to all orders in the loopexpansion. [email protected] Also at Lebedev Institute. [email protected]
Introduction
Recent remarkable progress in understanding the spectrum of states with large quantum num-bers in
AdS × S string theory or dual N = 4 SYM theory was achieved via interplay of variousperturbative data from gauge theory and string theory linked together by the assumption ofexact integrability. It remains an outstanding problem to derive the corresponding asymptoticBethe ansatz equations directly from first principles – from quantum superstring theory. Thatwould be facilitated if the corresponding integrable AdS × S sigma model admitted a for-mulation in terms of elementary excitations with two-dimensional Lorentz covariant S-matrix.Such a formulation may also make more straightforward the generalization of the asymptoticBethe Ansatz to the case when both strings and dual operators have finite length, i.e. to thecase of closed strings on the cylinder R t × S .With this motivation in mind here we shall continue the study of the Pohlmeyer-reduced [1]formulation of gauge-fixed AdS × S superstring [2, 3, 4]. This theory (which we shall refer to asthe “reduced theory”) is a generalized sine-Gordon or non-abelian Toda type two-dimensionalLorentz-invariant sigma model which is closely related to the original Green-Schwarz (GS)superstring sigma model [5]. It is constructed by writing the GS superstring equations ofmotion in terms of the components of the P SU (2 , | SO (1 , × SO (5) supercoset current, fixing the conformaland κ -symmetry gauges and then reconstructing the action that reproduces the equation ofmotion for the remaining physical number of degrees of freedom.While the resulting reduced theory is classically equivalent to the original AdS × S GSsuperstring (and, in particular, it is also classically integrable) it is a priori unclear if the cor-responding quantum theories should be closely related. In general, the classical Pohlmeyer re-duction assumes two-dimensional conformal invariance but for sigma models with target spacesinvolving S n or AdS n factors (and no bosonic WZ couplings) that symmetry may hold also atthe quantum level only in very exceptional cases like the AdS × S GS superstring. The min-imal consistency requirement for the conjecture that the classical equivalence between the GSsuperstring and the reduced theory may extend to the quantum level is then the finiteness ofthe reduced theory – the cancellation of the UV divergences in world-sheet perturbation theory.This means the absence of any new dynamically generated scale in addition to the classical massparameter in the potential introduced in the process of fixing the classical conformal diffeomor-phism symmetry (this procedure spontaneously breaks the underlying conformal symmetry ofthe GS superstring in conformal gauge while preserving two-dimensional Lorentz invariance).Our aim below will be to demonstrate the cancellation of the 1-loop and 2-loop divergencesin the reduced theory which also gives a strong indication of all-loop finiteness.Let us first briefly discuss what is known about the
AdS × S superstring theory. Theclassical theory [5] generalizes the AdS × S bosonic sigma model to the presence of GSfermions incorporating self-dual 5-form coupling. The potential importance of integrability ofthis model (motivated by the known integrability of its bosonic part) was recognized early on[6, 7]; the classical intergrability was proved in the full theory including fermions in [8] (seealso [9, 10]; for a review see [11]). Given the global symmetry, uniqueness of the (2-derivative)action and analogy with WZW theory the action is expected to be UV finite to all orders [5] and1hat was directly verified at 1-loop [12, 13] and 2-loop [14] orders. The classical integrabilityappears to extend to the quantum level as is effectively verified by the matching of the 1-loop[13] and 2-loop [14] corrections to spinning string energies to the strong-coupling predictions ofthe asymptotic Bethe Ansatz (see, e.g., [15] and [16]). The GS action has a well-known peculiarity in that to carry out its perturbative expansionit is necessary to choose a non-trivial background for the closed string coordinates and expandaround it. The background introduces a fiducial mass scale (spontaneously breaking two-dimensional conformal invariance) and also spontaneously breaks the two-dimensional Lorentzinvariance at the level of interaction terms in the action. That happens, for example, when oneexpands near a null geodesic or uses a version of light-cone (l.c.) gauge in
AdS × S [6, 20, 21].While this step is a natural one when computing quantum superstring corrections to specificstring states, it is a complication in general considerations (e.g., in computing the underlyingfactorized S-matrix). In particular, the l.c. gauge fixed AdS × S GS superstring action has acomplicated interaction structure making the direct computation of the corresponding magnon-type or BMN excitation S-matrix problematic beyond the tree level [10]. Another complicationis that when formally expanded near a particular background the GS action is not power-counting renormalizable [22, 14] and one is to rely on a judicious choice of regularization (andmeasure) to verify the cancellation of the UV divergences.Remarkably, these problems are absent in the quantum theory as defined in terms of thesupercoset current variables, i.e. defined by the reduced theory action [2]. The correspondingfermionic kinetic terms have standard two-dimensional Dirac form and thus the two-dimensionalLorentz covariant fermionic propagators are defined without independently of a bosonic stringcoordinate background. Moreover, the reduced theory action is power counting renormal-izable and relatively straightforward to quantize, as its structure is similar to that of two-dimensional supersymmetric gauged
G/H
WZW model supplemented with a bosonic potentialand a “Yukawa” interaction term. The quadratic part of the reduced theory action has the same form as that of the GS super-string expanded near the BMN vacuum, i.e. as the GS action in maximally-supersymmetricplane wave background in the l.c. gauge [23, 24]: eight two-dimensional scalars together witheight two-dimensional Majorana fermions, all with equal mass µ . The interaction terms dif-fer, but one may hope that there exists a certain transformation relating the correspondingS-matrices. Since both the
AdS × S superstring and the corresponding reduced theory areexpected to be conformal theories, the parameter µ should be the only scale on which thequantum S-matrices should depend. While the S-matrix corresponding the BMN vacuum isnot two-dimensional Lorentz invariant, the one appearing in the reduced theory should beLorentz invariant (i.e. the 4-point scattering matrix should depend only on the difference ofthe two rapidities). This puts the reduced theory into the same class of integrable theories as Quantum integrability was also argued for in the closely related pure spinor formulation of
AdS × S superstring [17, 18, 19]. As we shall see, the “Yukawa” interaction is effectively responsible for the UV finiteness of the“gWZW+potential” model. These are expected to have closely related symmetries:
P SU (2 | × P SU (2 |
2) in the GS superstring case[26, 21] and SU (2) × SU (2) × SU (2) × SU (2) in the reduced theory case [2] – the latter is formally the same asthe bosonic part of the former but their precise relation needs to be clarified further. O ( n ) sigma models.This motivates the study of the reduced theory at the quantum level even regardless itsrelation to the quantum GS superstring theory: it appears to be a remarkable finite integrablemodel with several unique features.Below in section 2 we shall start with a review of the reduced theory action using an explicitparametrization of the fermionic variables and clarifying on the way several important featuresof this theory. As was already mentioned, the construction of Pohlmeyer-reduced theory (see[2] and also [25] and references therein) involves several steps:(i) start with the GS equations (and the Maurer-Cartan equations) written in terms of thecomponents of the b FG = P SU (2 , | SO (1 , × SO (5) supercoset current;(ii) solve the conformal gauge constraints introducing a new set of field variables directly(algebraically) related to the currents, fixing the residual conformal diffeomorpisms and κ -symmetry gauge in the process;(iii) reconstruct an action for the remaining field equations in terms of the new (physical)variables.The resulting reduced theory action defines a massive integrable two-dimensional field theory.Its construction thus involves a non-local map between the original coset coordinate fields andcurrent variables that preserves the integrable structure and allows the reconstruction of theclassical solutions of the GS superstring action from classical solutions of reduced theory action,i.e. the solitonic solutions in the two models are in direct correspondence. The bosonic fields of the reduced theory are g ∈ G = Sp (2 , × Sp (4) ⊂ P SU (2 , |
4) and thetwo-dimensional gauge field A µ taking values in the algebra of H = SU (2) × SU (2) × SU (2) × SU (2) ⊂ G . In addition, there are fermionic fields Ψ R , Ψ L (directly related to fermionic currentsof the GS superstring) which are two-dimensional Majorana spinors with the standard kineticterms transforming under both Sp (2 ,
2) and Sp (4) and thus linking together the two sets ofbosons (corresponding effectively to the “transverse” string fluctuations in AdS and S ). Inthe special case when
AdS × S is replaced by AdS × S the corresponding reduced theory isequivalent [2] to the N = 2 super sine-Gordon model (there H is trivial).At the level of the equations of motion of the reduced theory it is possible to fix the A µ = 0gauge; the equations then become equivalent to a fermionic generalization of non-abelian Todaequations. The linearization of the equations of motion in the gauge A µ = 0 around the trivialvacuum g = 1l gives 8+8 bosonic and fermionic degrees of freedom with mass µ and suggeststhat the symmetry of resulting relativistic S-matrix should be H = [ SU (2)] .The potential term is multiplied by the “built-in” classical scale parameter µ which is aremnant of gauge-fixing the conformal diffeomorphisms at the classical level. Consistency thenrequires that the reduced theory be also UV finite, i.e. while a priori the µ -dependent terms in This correspondence was used in [27]. This model is kind of “hybrid” of a WZW model based on a supercoset (where fermions are in “off-diagonal”blocks of a supermatrix field but have non-unitary second-derivative kinetic terms) and a two-dimensionalsupersymmetric version of a
G/H gWZW model where fermions have the standard first-order kinetic terms buttake values in the coset part of the algebra of the group G . AdS × S case (i.e. in the N = 2 super sine-Gordonmodel). As we shall see in section 4 below, this is also true in the general AdS × S case: weshall demonstrate the cancellation of UV divergences at the 1-loop and 2-loop orders in thenatural dimensional reduction regularization scheme. We believe that similar cancellationsshould extend to all orders in perturbation theory. Then the theory is UV finite and µ remainsan arbitrary conformal symmetry gauge fixing parameter at the quantum level. The cancellationof divergences is presumably related to a hidden symmetry that should have its origin in κ -symmetry of the original GS action that relates the coefficients of the “kinetic” and the WZterms in the action (which, under the reduction, become the potential and the Yukawa termsin the reduced action).There are several conceptual issues that remain to be clarified before one would be able toclaim that the quantum reduced theory is indeed directly relevant for solving the quantum AdS × S superstring theory. These include the precise mapping between observables and con-served charges (cf. [4]) and understanding the relation between massive S-matrices computedby expanding near the respective vacua. The ultimate motivation for the study of the reducedtheory is the hope that it may be more straightforward to define as a quantum integrable theoryand thus easier to solve than the original AdS × S GS superstring model. To demonstratethis remains a program for the future.
AdS × S superstring In this section we shall review the structure of the reduced theory action.Our starting point is the
AdS × S superstring action [5] written in terms of currents forthe supercoset b FG = P SU (2 , | Sp (2 , × Sp (4)The currents take values in the superalgebra b f = psu (2 , |
4) which is a quotient of su (2 , |
4) byelements proportional to unit matrix.Let us first discuss the explicit parametrization of the corresponding supermatrices.
An element of su (2 , |
4) can be written as an 8 × M = (cid:18) A XX † Σ B (cid:19) , Str M = tr A − tr B = 0 , A ∈ u (2 , , B ∈ u (4) . (2.1) The same scheme was used in [14] where the 2-loop finiteness of the
AdS × S GS superstring was verified. The bosonic part of the
P SU (2 , |
4) group is SU (2 , × SU (4) or SO (2 , × SO (6) and an equivalent formof the subgroup is SO (1 , × SO (5). × K (we follow the notation of [28, 2, 11]; I denotes aunit matrix of an appropriate dimension)Σ = (cid:18) I − I (cid:19) , K = (cid:18) J J (cid:19) , J = (cid:18) −
11 0 (cid:19) , [Σ , K ] = 0 , Σ = I, K = − I (2.2)The superalgebra su (2 , |
4) admits a Z automorphism [29], i.e. its elements can be split intofour orthogonal subspaces b f ⊕ b f ⊕ b f ⊕ b f , with [ b f i , b f j ] = b f i + j ( mod in the following way: M , = (cid:18) A , B , (cid:19) , A , = ( A ± KA t K ) , B , = ( B ± KB t K ) , (2.3) M , = (cid:18) X , X † , Σ 0 (cid:19) , X , = [ X ± iK ( X † Σ) t K ] = ( X ± i Σ KX ∗ K ) . (2.4)Here A ∈ sp (4) , B ∈ sp (4), i.e. M belongs to sp (2 , ⊕ sp (4), while M is in the bosonicpart of the coset subspace of the algebra. M and M are expressed in terms of the realand imaginary parts of the complex 4 × X . This split is a “reality decomposition”implemented by the projectors applied to X: X , = P ± X ≡ ( X ± i Σ KX ∗ K ) , P ± = P ± . (2.5)Thus the elements from b f and b f should satisfy the following conditions X ∗ = − i Σ KX K , X ∗ = i Σ KX K , (2.6)which can be solved explicitly in terms of 4 × X , with independent real Grassmannelements X = X + i Σ K X K , X = X − i Σ K X K . (2.7)The
AdS × S GS action [5, 29, 31] is constructed by starting with an element f of b F = P SU (2 , | f − df and then splitting the current according to the Z decomposition of b f ( µ, ν = (0 , µ = f − ∂ µ f = A µ + Q µ + P µ + Q µ , A ∈ b f , Q ∈ b f , P ∈ b f , Q ∈ b f . (2.8)Here A µ belongs to the algebra of the subgroup G defining the b F /G coset, i.e. G = Sp (2 , × Sp (4) (isomorphic to SO (1 , × SO (5)), P is in the bosonic coset (i.e. AdS × S ) component,and Q , Q are the fermionic currents. The Lagrangian in conformal gauge is L = Str (cid:2) P + P − + ( Q Q − − Q − Q ) (cid:3) , (2.9)which should be supplemented by the Virasoro (conformal gauge) constraintsStr( P + P + ) = 0 , Str( P − P − ) = 0 . (2.10)5s already reviewed in the introduction, the idea behind the construction of the reduced action[2, 3] is to express the corresponding equations in terms of currents only, solve the confor-mal conformal gauge constraints algebraically introducing a new set of field variables directlyrelated to the currents, then choose a κ -symmetry gauge and finally reconstruct the actioncorresponding to the resulting field equations in terms of current variables. This constructionimplies the classical equivalence of the original and “reduced” sets of equations; in particular,the reduced theory is also integrable [2].The Virasoro constraints can be solved by fixing a special G -gauge and residual conformaldiffeomorphism gauge such that P + = µ T , P − = µ g − T g , µ = const . (2.11)Here µ is an arbitrary scale parameter (the scale corresponding to fixing the residual conformaldiffeomorphisms, similar to p + in light-cone gauge) and T is a special constant matrix chosenin [2] to be T = i (cid:18) Σ 00 Σ (cid:19) , T = − I, Str T = 0 . (2.12)Here Σ is defined in (2.2) and we also introduced a new bosonic field variable g which belongsto G = Sp (2 , × Sp (4), i.e. to the subgroup whose Lie algebra is b f .Having chosen T , we may define a subgroup H in G that commutes with T , [ T, h ] = 0 , h ∈ H :in the present case we get H = SU (2) × SU (2) × SU (2) × SU (2). Using the gauge freedomand the equations of motion one can choose g ∈ G and A + , A − taking values in the algebra h = su (2) ⊕ su (2) ⊕ su (2) ⊕ su (2) of H and defined by A + ≡ g A + g − + ∂ + g g − , A − ≡ ( A − ) h (2.13)as the new independent bosonic variables [32, 2].Next, one can impose a partial κ -symmetry gauge Q − = 0 , Q = 0 , (2.14)and then define the new independent fermionic variablesΨ = Q , Ψ = gQ − g − . (2.15)Similarly to Q and Q − , the new variables Ψ and Ψ belong to b f and b f , respectively. Indeed,the adjoint action of g ∈ G separately maps the subspaces b f and b f into themselves since thealgebra of G is b f and according to the Z decomposition [ b f i , b f j ] = b f i + j mod . i.e. [ b f , b f ] = b f . Note also that Ψ and Ψ are completely independent being related to different components ofthe fermionic current. In general, we may introduce two different parameters µ + and µ − in P + and P − ; the resulting expressionfor the reduced action will then be obtained by replacing µ → √ µ + µ − . The choice of normalization of T is of course arbitrary and can be changed by rescaling µ . Note that there is a natural arbitrariness in the choice of g in eq. (2.11) since P − is invariant under g → hg if h ∈ H ; that implies an additional H gauge invariance of the resulting equations of motion for g . κ -symmetry can be fixed by further restricting Ψ , by demanding that theyanticommute with T , { Ψ , , T } = 0. Namely, we may introduce the projector from Ψ , to Ψ k , Ψ k ≡ ΠΨ = (Ψ + 4 T Ψ T ) , Π = Π , (2.16)Ψ k T = − T Ψ k , [ T, [ T, Ψ k ]] = − Ψ k , Ψ k = [ T, b Ψ] , b Ψ = − T Ψ . (2.17)Note that since according to (2.2) [Σ , K ] = 0 the projector Π commutes with the “realitycondition” projectors P ± in (2.5), so that it can be imposed in addition to the constraints (2.6)or (2.7).The Z decomposition implied by Π can be represented explicitly as follows:Ψ = (cid:18) XX † Σ 0 (cid:19) , X = X k + X ⊥ , X k = − Σ X k Σ , X ⊥ = Σ X ⊥ Σ . (2.18)Writing X in terms of 2 × X ≡ (cid:18) α βγ δ (cid:19) , X k = (cid:18) βγ (cid:19) , X ⊥ = (cid:18) α δ (cid:19) . (2.19)We may then define the new fermionic variables as [2]Ψ R = √ µ Ψ k , Ψ L = √ µ Ψ k , (2.20)so that Ψ R and Ψ L are expressed in terms of “off-diagonal” matrices X R and X L as X k in(2.19). The “reality” constraints (2.7) on Ψ R ∈ b f and Ψ L ∈ b f in (2.7) then imply that thecorresponding 2 × real Grassmann 2 × ξ and η ( J = − I , see (2.2)) β R,L = ξ R,L ± iJ ξ R,L
J , γ
R,L = η R,L ∓ iJ η R J . (2.21)Explicitly, in terms of 2 × R = ξ R + iJ ξ R J η R − iJ η R J − η t R − iJ η t R J ξ t R − iJ ξ t R J , (2.22)Ψ L = ξ L − iJ ξ L J η L + iJ η L J − η t L + iJ η t L J ξ t L + iJ ξ t L J . (2.23)Thus each of Ψ R and Ψ L are parametrized by 2 × R → L is equivalent to i → − i , i.e.Ψ R ( ξ R , η R ) = Ψ ∗ L ( ξ L → ξ R , η L → η R ) . (2.24)7 .2 Lagrangian of the reduced theory The reduced theory Lagrangian that reproduces the classical equations of the reduced theory(obtained from first-order equations corresponding to the GS Lagrangian (2.9)) is given by theleft-right symmetrically gauged WZW model for GH = Sp (2 , SU (2) × SU (2) × Sp (4) SU (2) × SU (2)supplemented by the following integrable bosonic potential and the fermionic terms [2]: L tot = L B + L F = L gWZW ( g, A ) + µ Str( g − T gT )+ Str (cid:0) Ψ L T D + Ψ L + Ψ R T D − Ψ R + µ g − Ψ L g Ψ R (cid:1) . (2.25)Here all fields are represented by 8 × su (2 ,
2) and su (4) parts. The covariant derivative is D ± Ψ = ∂ ± Ψ + [ A ± , Ψ] , A ± ∈ h . Given that [ T, h ] = 0, h ∈ H , the Lagrangian L tot is invariant under H gauge transformations g ′ = h − gh, A ′± = h − A ± h + h − ∂ ± h, Ψ ′ L,R = h − Ψ L,R h . (2.26)The µ -dependent terms in (2.25) are essentially the original GS Lagrangian after the substitu-tion of (2.11),(2.14), (2.15) and (2.20); one may conjecture that L gWZW ( g, A ) plus free fermionicterms should originate from the change of variables (from fields to currents) in the original GSstring path integral [2, 4].Similarly to the original closed string GS action, the reduced theory action is defined on a 2dcylinder (i.e. the fields are 2 π periodic in σ ) and should also have the string tension in front ofit. In discussing UV (short distance) behavior of the theory the compactness of the σ directionis not relevant; likewise the masses of fields are also unimportant. In that discussion we shalltherefore formally replace the cylinder with coordinates ( τ, σ ) by a plane and consider the massterms as part of the interaction potential. In that case the parameter µ (which, as we shall seewill not be renormalized) can be set to 1 by rescaling the worldsheet coordinates; we will preferhowever not to do that explicitly.The dimension of the bosonic target space in (2.25) is the same as the dimension of the G/H coset, i.e. 4+4=8. The fermionic fields having “standard” two-dimensional fermionic kineticterms are represented by the 8 × Z grading conditions discussedabove, so that they are describing eight left-moving and eight right-moving Grassmann degreesof freedom. Remarkably, the reduced action is only quadratic in fermions, in contrast to originalGS action which is at least quartic in fermions in a generic real κ -symmetry gauge.Another way of writing the fermionic terms, which takes into account the constraint T Ψ L,R = − Ψ L,R T , follows from introducing an explicit projector in the fermion kinetic term, as was donein [2]: Ψ T D Ψ → Ψ T Π D Ψ. The resulting action is L F = Str (cid:0) Ψ L [ T, D + Ψ L ] + Ψ R [ T, D − Ψ R ] + 2 µ g − ΠΨ L g ΠΨ R (cid:1) . (2.27) The projectors in the interaction term may be omitted as they will be implemented in perturbation theorythrough the fermionic propagator factors. Another equivalent way of writing the action is solve the constraint { T, Ψ } = 0 as Ψ = [ T, b Ψ]. Z split may also be implemented byinsertion of the corresponding projectors.One may also write the action in terms of the independent real Grassmann variables enteringthe explicit solution (2.22),(2.23) of the constraints. Using (2.22),(2.23) fermionic kinetic termin L F then takes the standard simple form (upon integration by parts) L F = Str(Ψ L T ∂ + Ψ L + Ψ R T ∂ − Ψ R ) = − i tr( ξ t L ∂ + ξ L + η t L ∂ + η L + ξ t R ∂ − ξ R + η t R ∂ − η R ) . (2.28)The gauge connection in D ± which belongs to h = su (2) ⊕ su (2) ⊕ su (2) ⊕ su (2) can be easilyincluded. If A = diag( A , A , A , A ) , A i ∈ su (2) then we get terms like tr[ β † ( A β − βA ) − γ † ( A γ − γA )]. Then the action can be rewritten in terms of independent 2 × The “Yukawa” interaction term in (2.25) can be written in more explicit form by using that g = (cid:18) g (1) g (2) (cid:19) , g (1) ∈ Sp (2 , , g (2) ∈ Sp (4) (2.29)Str( g − Ψ L g Ψ R ) = tr( g (1) − X L g (2) X † R Σ − g (2) − X † L Σ g (1) X R ) , (2.30)where X R = (cid:18) ξ R + iJ ξ R Jη R − iJ η R J (cid:19) , X L = (cid:18) ξ L − iJ ξ R Jη L + iJ η L J (cid:19) . (2.31)This fermionic interaction term is the only one that that mixes the bosonic fields g (1) ∈ Sp (2 , g (2) ∈ Sp (4) of the reduced models (based on gWZW models for Sp (2 , SU (2) × SU (2) and Sp (4) SU (2) × SU (2) )for the AdS and S parts of the original GS coset model. The fermions carry representationsof both Sp (2 ,
2) and Sp (4) and thus intertwine the two bosonic sub-theories. It is this interaction that is responsible for making the reduced model UV finite, i.e. con-formally invariant modulo the built-in scale parameter µ (which is the remnant of gauge-fixingthe conformal diffeomorphisms at the classical level).At the level of the equations of motion the H gauge field A ± can be gauged away; the resultis the following fermionic generalization of the non-abelian Toda equations [2] (see also [50]) ∂ − ( g − ∂ + g ) + µ [ g − T g, T ] + µ [ g − Ψ L g, Ψ R ] = 0 , (2.32) ∂ − Ψ R − µT ( g − Ψ L g ) k = 0 , ∂ + Ψ L − µT ( g Ψ R g − ) k = 0 , (2.33)( g − ∂ + g − T Ψ R Ψ R ) h = 0 , ( g∂ − g − − T Ψ L Ψ L ) h = 0 , (2.34) Note that (up to a total derivative) Str(Ψ
T d
Ψ) = − i tr[ X † ( dX − Σ dX Σ)], where we used eq. (2.5) and thefact that the fermionic matrices anticommute under the ordinary trace. Expanding near the trivial solution A = 0 , g = 1 the fermionic action then takes the form equivalent tothe quadratic fermionic action in the near - pp-wave or BMN limit in eqs. (5.6),(5.7) in [10]. A similar term in the original GS action reflects the presence of the RR 5-form coupling. This feature resembles more a WZW models based on a supergroup rather than a supersymmetric extensionof WZW model. At the same time, the fermions here have first-order kinetic term, so we obtain a kind ofhybrid model. In the special case of
AdS × S the resulting reduced model does have 2d supersymmetry and isequivalent to the N = 2 supersymmetric extension of the sine-Gordon model. In this case G = SO (1 , × SO (2)so the fermions are in the singlet representation. A less trivial case of the reduced model for AdS × S wasworked out explicitly in [4]; there the existence of the 2d supersymmetry in the resulting model is not obviousand remains an open question. A ± and we used that Ψ L ,R anticommute with T (see (2.16),(2.20)) as well as that T = − I .One may also eliminate the gauge fields from the fermionic terms in (2.25) as usual in 2dimensions – by writing A + = u∂ ± u − , A + = ¯ u∂ ± ¯ u − and performing a local rotation ofthe fermions. The bosonic gWZW part of the Lagrangian written in terms of h ± becomes L WZW ( u − g ¯ u ) − L WZW ( u − ¯ u ) and the potential term can also be written in terms of e g = u − g ¯ u since T commutes with u, ¯ u .Alternatively, one may fix an H gauge on g and integrate the fields A ± out [2] leading to abosonic sigma model with 4+4 dimensional target space coupled to 8 fermions (with quadraticand quartic fermionic terms). Since the fermions are transforming in different representation than bosons, the reducedLagrangian (2.25) is not of a familiar supersymmetric gWZW theory (deformed by a bosonicpotential and Yukawa-type terms) and thus more difficult to analyze. It is nevertheless a simplewell-defined theory intimately connected to the
AdS × S GS superstring. It is therefore ofinterest to study its quantum properties. Finiteness of
AdS × S superstring (checked directlyto the two-loop order [14]) suggests, assuming the relation via the reduction should hold beyondthe classical level, that this theory should also be UV finite. In contrast to the GS superstring,here it should be much easier to verify the finiteness since the reduced theory is power countingrenormalizable.Indeed, the reduced theory is obviously UV finite for µ = 0 (since gWZW model coupled tofermions is). Also, the structure of the µ -dependent interaction terms in (2.25) is constrained bysymmetries, and it seems possible that bosonic and fermionic contributions to renormalizationof the potential terms may cancel each other (as they do in the reduced model for AdS × S superstring which is the N = 2 supersymmetric sine-Gordon theory). Our aim below will beto present evidence that this model is indeed UV finite. To get an idea about the structure of possible UV divergences in reduced theory (2.25) let usfirst consider its bosonic part. We shall first review the form of the sigma model that appearsas a result of choosing a specific parametrization of the basic field g ∈ G and integrating outthe H gauge field A a . That assumes that the H -gauge is fixed by choosing a particular formof the group element g .In the case of the string on R t × S n or sigma model on the sphere F/G = S n the reducedtheory is based on the gWZW model for G/H = SO ( n ) /SO ( n − g in terms of the coordinates of the G/H coset and integratingout the H gauge field A a . We end up with an integrable theory represented by an ( n –1)- As in the supersymmetric WZW model, the corresponding Jacobian may lead to a shift of the coefficientof the bosonic term. A disadvantage of this gauge is that the resulting action does not allow a straightforward expansion nearthe g = 1 point. For this purpose it seems necessary to choose a “intermediate” gauge, where both A ± and g are partially fixed. L = G mk ( x ) ∂ + x m ∂ − x k − U ( x ) . (3.1)Here x m represent the n − G − dim H ) independent components of g left after fixingthe H gauge. The potential term (or “tachyon coupling” in string sigma model language)in (3.1) originates directly from the µ term in the action. It is a relevant (in the case of acompact group F such as for the sphere) or irrelevant (in the case of a non-compact group F such as for AdS n ) perturbation of the gWZW model and thus also of the “reduced” geometry,i.e. it should satisfy 1 √ Ge − ∂ m ( √ Ge − G mk ∂ k ) U − M U = 0 , (3.2)where Φ is the dilaton resulting from integrating out A a . An explicit parametrization of g inthe case of G = SO ( n ) in terms of Euler angles is found by choosing g = g n − ( θ n − ) ...g ( θ ) g (2 ϕ ) g ( θ ) ...g n − ( θ n − ) , (3.3)where g m ( θ ) = e θR m and R m ≡ R m,m +1 are generators of SO ( n + 1). Thus ϕ ≡ θ , and θ p ( p = 2 , ..., n −
1) are n − n − , with ϕ playing adistinguished role. Then the potential U has a universal form for any dimension n : it is simplyproportional to cos 2 ϕ as in the sine-Gordon model ( n = 2) [2]. The metric and the dilatonresulting from integrating out the H gauge field A a satisfy ds = G mk dx m dx k = dϕ + g pq ( ϕ, θ ) dθ p dθ q , √ G e − = (sin 2 ϕ ) n − , (3.4)so that the equation (3.2) is indeed solved by U = − µ ϕ , M = − n − , (3.5)i.e. L = ∂ + ϕ∂ − ϕ + g pq ( ϕ, θ ) ∂ + θ p ∂ − θ q + µ ϕ . (3.6)The explicit form of the Σ n − metric (3.4) with n = 2 , , S and S , i.e. for G/H = SO (2) and G/H = SO (3) /SO (2) we have ds n =2 = dϕ , ds n =3 = dϕ + cot ϕ dθ . (3.7) In contrast to the metric of the usual geometric (or “right”) coset SO ( n ) /SO ( n −
1) = S n − the metric G mk in (3.1) found from the symmetrically gauged G/H = SO ( n ) /SO ( n −
1) gWZW model will genericallyhave singularities and no non-abelian isometries. The corresponding space may be denoted as Σ n − . Whilethe gauge A a = 0 preserves the explicit SO ( n −
1) invariance of the equations of motion, fixing the gauge on g and integrating out A a breaks all non-abelian symmetries (the corresponding symmetries are then “hidden”,cf. [53]). Instead of R mk = a G mk for a standard sphere the metric G mk satisfies R mk + 2 ∇ m ∇ k Φ = 0 whereΦ is the corresponding dilaton resulting from integrating out A a . G/H = SO (4) /SO (3) [34] ds n =4 = dϕ + cot ϕ ( dθ + V dθ ) + tan ϕ dθ sin θ , V = cot θ tan θ , (3.8)or after a change of variables x = cos θ cos θ , y = sin θ ds n =4 = dϕ + cot ϕ dx + tan ϕ dy − x − y . (3.9)From G/H = SO (5) /SO (4) gWZW we get [33] ds n =5 = dϕ + cot ϕ ( dθ + V dθ + W dθ ) + tan ϕ (cid:0) dθ cos θ + dθ sin θ (cid:1) , (3.10) V = tan θ sin 2 θ cos 2 θ + cos 2 θ , W = cot θ sin 2 θ cos 2 θ + cos 2 θ . (3.11)Together with the cos 2 ϕ potential (3.5) the latter metric thus defines the reduced model forthe string on R t × S .One can similarly find the reduced Lagrangians for F/G = AdS n = SO (2 , n − /SO (1 , n −
1) coset sigma models which are related to the above ones by an analytic continuation. A“mnemonic rule” to get the
AdS n counterparts of S n reduced Lagrangians is to change ϕ → iφ and to reverse the overall sign of the Lagrangian. In general, that will give the G/H = SO (1 , n − /SO ( n −
1) counterpart of (3.6) of the form L = ∂ + φ∂ − φ + e g pq ( φ, ϑ ) ∂ + ϑ p ∂ − ϑ q − µ φ , (3.12)where e g pq ( φ ) = − g pq ( iφ ) (i.e. cot ϕ → coth φ in (3.7), etc.).The reduced model for bosonic strings on AdS n × S n can then be obtained by formallycombining the reduced models for strings on AdS n × S and on R × S n [2]. For example, in thecase of a string in AdS × S we find the sum of the sine-Gordon and sinh-Gordon Lagrangians L = ∂ + ϕ∂ − ϕ + ∂ + φ∂ − φ + µ ϕ − cosh 2 φ ) , (3.13)while for a string in AdS × S we get (see [2, 4]) L = ∂ + ϕ∂ − ϕ + cot ϕ ∂ + θ∂ − θ + ∂ + φ∂ − φ + coth φ ∂ + ϑ∂ − ϑ + µ ϕ − cosh 2 φ ) . (3.14)Similar bosonic actions are found for a string in AdS × S and in AdS × S using (3.8) and(3.10).Next, let us discuss the quantum properties of the above bosonic sigma models. Since theseare deformations of conformal gWZW models, we should not expect infinite renormalization12f the resulting sigma model metrics, but the potential terms may get renormalized. Whilethe cos 2 ϕ potential is a relevant perturbation of the coset CFT in the compact S n case, thecosh 2 φ is an irrelevant perturbation of the corresponding coset CFT in the AdS n case (i.e. thesign of the mass term M in (3.5) is opposite). Thus the coefficients of the two terms in thepotential in (3.14) (and in similar higher-dimensional models) “run” in the opposite directions.As a result, the bosonic reduced theory like (3.13) or (3.14) is not renormalizable already atthe leading one-loop order: one would need to introduce two different bare coefficients in frontof the cos 2 ϕ and the cosh 2 φ terms in the potential to cancel the divergences.A simple way to see that different renormalization is to note that the one-loop correctiongiven by log det ∆ terms is not sensitive to a change of sign of the classical action which shouldbe done while going from S n to AdS n reduced model via ϕ → iφ . Thus if in the S n model we geta divergence c cos 2 ϕ ln Λ, then in the AdS n model it should be given simply by the same with ϕ → iφ , i.e. by c cosh 2 φ ln Λ. Hence the total divergence will be c (cos 2 ϕ + cosh 2 φ ) log Λ. Itwill thus have a different structure than the classical potential in (3.13),(3.14), and so cannotbe absorbed into renormalization of the single parameter µ .More generally, the supertrace symbol in Str( g − T gT ) in (2.25) means that the potentialterms for the
AdS and S parts of the reduced theory are taken with the opposite signs (i.e. ascos 2 ϕ − cosh 2 φ in the Euler angle parametrization (3.3)). Since the anomalous dimensions of the corresponding two terms are opposite (which is related to the opposite signs of curvatureof AdS and S ), the logarithmically divergent term coming from the bosonic part of (2.25) isactually the sum, not the difference, i.e. defined in terms of g in the product of the two groupsit contains tr instead of Str L − loop = a tr( g − T gT ) ln Λ . (3.15)One expects that in the full reduced theory (2.25) corresponding to the AdS × S superstringthe fermionic terms will make the whole theory UV finite, i.e. (3.15) will be canceled by thefermionic contributions, i.e. the potential can be considered as an exactly marginal perturbation(with the value of its coefficient µ being finite and arbitrary).This is indeed what happens in the AdS × S case where the reduced theory is equivalentto the (2,2) supersymmetric sine-Gordon theory [2]. For this to happen in the general theory(2.25) the contribution to the divergences coming from the fermionic Yukawa interaction termshould also be proportional to (3.15), i.e. to the sum of the bosonic potentials instead of theirdifference entering the classical action.It is possible to argue that indeed the fermionic part is invariant under the analytic continu-ation ϕ → iφ , so that its one-loop contribution to the renormalization of the bosonic potentialshould also be even, i.e. proportional to the sum of the potential terms as in (3.15). For exam-ple, the explicit form of the fermionic terms in the AdS × S case given in [4] is invariant under ϕ → iφ, φ → − iϕ . In the next section we shall give a general argument of why that shouldhappen and check explicitly that the resulting divergent coefficient indeed cancels against thebosonic one. On dimensional grounds, the deformation terms cannot contribute to the renormalization of the two-derivative terms. It is useful to recall that tr( g − T gT ) is a primary field of the WZW theory [43]. S = 14 πα ′ Z d σ (cid:2) G mn ( x ) ∂ µ x m ∂ µ x n + ǫ µν B mn ( x ) ∂ µ x m ∂ ν x n − U ( x ) (cid:3) . (3.16)The renormalization of U is governed by the β -function (see, e.g., [35, 36, 37, 38]) β U = − γU − U , (3.17) γ = Ω mn D m D n + O ( α ′ ) , (3.18)Ω mn = α ′ G mn + p α ′ R mn + p α ′ H mkl H nkl + O ( α ′ ) . (3.19)Here we follow the notation of [36, 39]. The 2-loop coefficients p , p are scheme dependent (theycan be changed by redefining G mn ). In dimensional regularization with minimal subtraction[35, 36] p = 0 while p , in principle, still depends on how one treats ǫ µν in dimensionalregularization (cf. [39, 40, 37, 41]). In a scheme where ǫ µν is considered as being 2-dimensionalone [40] (which also corresponds to the f = − p = − . Inthis case the dilaton and tachyon 2-loop β -functions take the form β φ = − γφ + 16 (cid:2) D − α ′ H mkl H mkl + O ( α ′ ) (cid:3) , (3.20) β U = − γU − U , γ = α ′ (cid:2) G mn − α ′ H mkl H nkl + O ( α ′ ) (cid:3) D m D n . (3.21)In the case of a WZW model (i.e. when the group space is a target space and H mkl is theparallelizing torsion) these expressions are then in agreement with the WZW central charge( C = 6 β φ , φ = const) and the anomalous dimension of the field tr g ( σ ) as found in [43] (seealso [40, 44]): C = kdk + c G = d (1 − c G k + ... ) , γU = c r k + c G U = c r k (1 − c G k + ... ) U , (3.22)where α ′ = k , R mn = H mkl H kln = Rd G mn , c G = Rd , G mn G mn = d and c r and c G are thevalues of the Casimir operator in, respectively, the fundamental and adjoint representations.More explicitly, if we consider the renormalization of a potential term in a WZW model L = L WZW ( g ) − U ( g ) , (3.23)as we shall do in the next section, then, as follows from the above general results, the renormalization of U will originate only from the vertices in the WZ term in the action (andwill be, in general, scheme-dependent).Such a 2-loop shift in the anomalous dimension is absent in 2d supersymmetric WZW modelsdue to an additional contribution of the fermions that are chirally coupled to g . That can be The corresponding operator γ enters also the dilaton β -function considered in [39]. See also the discussionaround eq.(5.10) in the second reference in [47]. k ofthe WZW term k → k ′ = k − c G and thus eliminates all higher than 1-loop contributions tothe anomalous dimension of U : the corresponding dimension in (3.22) is then c r k ′ + c G = c r k .The case of the reduced theory which we shall consider below is different from the case2d supersymmetric WZW theory with a bosonic potential in that here there is an additionalfermionic interaction term that contributes to the renormalization of the bosonic potential andcompletely cancels out also the 1-loop anomalous dimension.An apparent consequence of the above general expression for β U (3.21) is that in the sigmamodels like (3.14) obtained by integrating out the gauge field A where there is no WZ-type B mn coupling ( H mnk = 0) there will be no non-trivial renormalization of the potential at the2-loop order. There is a caveat that since this sigma model is obtained from a conformal gWZWmodel its classical metric will be conformal only in a special scheme [38]; in a standard (minimalsubtraction) scheme the metric will be deformed by α ′ = k corrections starting from the 2-looporder [45, 46, 47]. As a result, expressed in terms of the “tree-level” metric, the anomalousdimension will receive an effective 2-loop contribution coming from the 1-loop term after oneuses there the 1-loop corrected metric. This subtlety would be absent in a 2d supersymmetricgWZW model where, as recalled above, the fermions produce a compensating shift of the level k and thus the expressions for the central charge, anomalous dimension and the effective sigmamodel metric obtained by integrating out the A gauge field remain essentially the 1-loop ones(see [47] and refs. therein).Though there is no apparent 2d supersymmetry in our reduced Lagrangian (2.25) one maysuspect that the effect of fermions there may be similar to the one in the 2d supersymmet-ric gWZW case. If we assume that the fundamental quantum variables are actually the GSfermionic currents Q and Q in (2.9) then (2.15) which defines Ψ L and Ψ R is similar to arotation that decouples fermions from bosons and produces the level shift k → k ′ = k − c G in the 2d supersymmetric WZW model. The above remark does not, however, directly applyto our case since the fermionic kinetic term in (2.25) contains the matrix T which does not ingeneral commute with g so after the rotation of Ψ L we will be left with a non-trivial g − T g coupling in its kinetic term.As was already stressed above, compared to WZW theory coupled to fermions, we have inaddition a fermionic counterpart of the potential term in (2.25) that may also contribute to therenormalization of the bosonic potential. This “Yukawa” interaction term originated from thefermionic WZ term in the original GS action (2.9) and thus its contribution (beyond the 1-loop In WZW model written in a manifestly supersymmetric form the fermions are Majorana spinors coupledto g as tr( ¯ ψγ γ µ [ ∂ µ gg − , ψ ]), and their rotation ψ L → g − ψ L g, ψ R → gψ L g − that decouples them from g produces a non-trivial jacobian that shifts the coefficient of the WZW term [48]. Indeed, the standard relation [49] for a fermionic determinant implies det( ∂ + + Adj g − ∂ + g ) det( ∂ − +Adj e g − ∂ − e g ) = exp[ c G I WZW ( g e g − )] det ∂ + det ∂ − . Here we assumed that fermions are in adjoint representation;otherwise c G should be replaced by the corresponding quadratic Casimir of the representation, T a T a = c r I . Thisexpression can be factorized into separate chiral determinant contributions using Polyakov-Wiegmann identity,and then I WZW ( g ) (or I WZW ( e g − )) can be interpreted as the effective action for a Dirac fermion with purelyright (left) coupling to the corresponding current. L and Ψ R havingcanonical kinetic terms and will show that all 1-loop divergent contributions to the potentialterms cancel, while the 2-loop contributions which are, in general, scheme-dependent, alsovanish in a natural regularization scheme. In this section we shall study the divergences of the reduced model (2.25) for strings in
AdS × S without first integrating out the H gauge field. This allows us to utilize explicitly the conformalinvariance of the gWZW model so that the only possible renormalization that needs to beanalyzed is that of the potential terms. To study the quantum properties of reduced model it is useful to reorganize its action anddecouple the H gauge field as was already mentioned below eq.(2.34), i.e. following the samepattern as in the bosonic gauged WZW models. Namely, we can always choose the two-dimensional gauge fields to be of the form A ( i )+ = u ( i ) ∂ + u ( i ) − , A ( i ) − = ¯ u ( i ) ∂ + ¯ u ( i ) − , (4.1)where i = 1 , SO (4) algebra in the algebra of H isomorphic to SO (4) × SO (4). Then, the coupling between g and the gauge field may be eliminated byredefining g = diag( g (1) , g (2) ) ∈ Sp (2 , × Sp (4) as follows e g ( i ) = u ( i ) − g ( i ) ¯ u ( i ) . (4.2)This redefinition may be written more compactly as e g = u − g ¯ u by introducing the “superma-trices” e g = (cid:18)e g (1) e g (2) (cid:19) , u = (cid:18) u (1) u (2) (cid:19) , ¯ u = (cid:18) ¯ u (1)
00 ¯ u (2) (cid:19) . (4.3)We can also redefine the fermionic fields in (2.25) as e Ψ L = u − Ψ L u , e Ψ R = ¯ u − Ψ R ¯ u . (4.4) The supertrace of such matrices is defined as a difference of traces of diagonal blocks. Note that since u, ¯ u are from H and thus commute with T the rotated fermionic fields also satisfy theconstraints in (2.16),(2.20), i.e. they anticommute with T . L = L ( G ) WZW ( e g ) − k ′ L ( H ) WZW ( u − ¯ u ) + µ Str (cid:0)e g − T e gT (cid:1) + Str (cid:0) e Ψ L T ∂ + e Ψ L + e Ψ R T ∂ − e Ψ R (cid:1) + µ Str (cid:0)e g − e Ψ L e g e Ψ R (cid:1) . (4.5)We used that u ∈ H commutes with T . Here the factor k ′ in the second term indicates the shiftof the overall coefficient (or the level k , that we formally set to 1) coming from the Jacobiansof the above change of variables from A ± to u, ¯ u and from the rotations of the fermions (4.4) asin the usual 2d supersymmetric gWZW case [30]. Here the shift is k ′ = k + (1 − ) c so (4) where c so (4) is the quadratic Casimir of H (1) = SO (4). The shift by c so (4) is coming from the bosonicJacobian and by − c so (4) from the chiral fermionic Jacobians regularized in a vector-like fashionso that their contributions combine into L ( H ) WZW ( u − ¯ u ).This redefinition is very useful for the purpose of studying the UV properties of the theory:we can ignore the decoupled WZW term for the subgroup H (i.e. the term multiplied by k ′ in (4.7)) since it is conformally invariant on its own. The fermions in (4.7) have free kineticterms. By formally assuming that T transforms under G = Sp (2 , × Sp (4) in an appropriateway we may then treat the remaining terms in the action as being invariant under G .Let us note that in general one can not, of course, completely decouple L WZW ( u − ¯ u ) term:the gauge-invariant observables in the original theory may depend on u and ¯ u . Indeed, theaction (4.7) – even written in an apparently factorized form – still exhibits the following gaugeinvariance e g h e gh − , e Ψ L,R h e Ψ L,R h − , u huh − , ¯ u h ¯ uh − , (4.6)where h = diag( h (1) , h (2) ) ∈ SO (4) × SO (4). The observables of this theory must be invariantunder these transformations. Clearly, traces of products of powers of e g and T are invariant.However, partial derivatives of e g must be promoted to covariant derivatives of e g . Thus, u and¯ u must necessarily enter the observables. We are interested in understanding the UV finiteness properties of the theory (2.25) or, equiv-alently, of (4.5). To simplify the notation in what follows we shall omit tildes on g and Ψ in(4.5), i.e. study the UV properties of the following theory L = L ( G ) WZW ( g ) + µ Str (cid:0) g − T gT (cid:1) + Str (cid:0) Ψ L T ∂ + Ψ L + Ψ R T ∂ − Ψ R (cid:1) + µ Str (cid:0) g − Ψ L g Ψ R (cid:1) , (4.7)where g ∈ Sp (2 , × Sp (4).This theory is power counting renormalizable but it is not clear a priori that divergenceswill preserve the specific structure of the potential terms. Indeed, as was discussed in the One may define this transformation as follows. The fixed matrix T identifies an SO (4) × SO (4) subgroupof Sp (2 , × Sp (4). Then, Sp (2 , × Sp (4) transformations of T amount to choosing different (but equivalent)embeddings SO (4) × SO (4) ⊂ Sp (2 , × Sp (4). At the level of the original action, a realization of this symmetryrequires transformations of the gauge field. This is not surprising, given that one gauges different SO (4) × SO (4)subgroups of Sp (2 , × Sp (4). g (1) ∈ Sp (2 ,
2) and g (2) ∈ Sp (4) with the potential terms “running” in the opposite directions. Thusrenormalizability of the bosonic theory a priori would require us to add also the coupling (see(3.15)) e µ tr (cid:0) g − T gT (cid:1) or introduce two independent couplings for the two bosonic potentials.Moreover, fermionic coupling constant in (4.7) need not be equal (in the absence of explicit2d supersymmetry) to the square of the coupling in the bosonic potential, i.e. it may be some µ ′ that may “run” differently than µ . Our analysis below shows that the corresponding 1-loop renormalization group equations admit a fixed point µ ′ = µ, e µ = 0, i.e. with this choiceall 1-loop divergences (including the ones depending on fermions) cancel. As for the 2-loopdivergences, their coefficients happen, in general, to be scheme dependent and there exists ascheme where they are absent, providing strong evidence of the finiteness of the theory (4.7).We will study the divergent part of the effective action Γ[ g ] for the bosonic field g obtainedby expanding the fields around some generic background g (solving the classical equations ofmotion) g → g e ζ , g − → e − ζ g − , (4.8)and integrating out the fluctuation field ζ (taking values in the algebra of G ) and the fermions.Let us discuss the expected structure of this effective action. It should be consistent with allthe global symmetries which are:( a ) manifest G = Sp (2 , × Sp (4) symmetry assuming that one treats T as a field trans-forming in the bifundamental representation. As mentioned above, this symmetry is manifestat the level of the classical action (4.7).( b ) symmetry under formal rescaling g ag which simply means that each term in theclassical action contains an equal number of factors of g and of g − . ( c ) invariance under g ↔ g − , Ψ L ↔ Ψ R combined with the world-sheet σ + ↔ σ − transfor-mation.( d ) g ( i ) ( − a i g ( i ) , Ψ L,R ( − b L,R Ψ L,R , with a , a , b L , b R = 0 , a + a + b L + b R = 2.( e ) g (1) ↔ g (2) , Ψ L ↔ Ψ R (interchanging the off-diagonal blocks in the fermionic matricesin (2.22),(2.23)) together with changing the sign of the Lagrangian, i.e. the sign of the overallcoupling constant.The contributions to the effective action depend on either j ± = g − ∂ ± g if they come fromthe WZW action or explicitly g if they come from the µ -dependent (or “deformation”) termsin (4.7). Two-dimensional Lorentz invariance requires that all factors of the vector j ± appearin pairs. The structure of the action (4.7) (in particular, the chiral symmetry of the WZWmodel) and the fact that j has dimension 1 imply that the coefficient of the j term must be It is easy to see on dimensional grounds that quartic fermionic terms (which are a priori possible to putinto the bare action) are not actually induced here with UV divergent coefficients and thus their coefficientscan be set to zero. Since g = diag( g (1) , g (2) ) is an element of Sp (2 , × Sp (4) this formal rescaling takes us outside the domainof definition of g so we will understand this rescaling only in the sense of counting the numbers of g and g − factors. g .The symmetries ( a ) and ( b ) above imply that at each loop order the effective action Γ[ g ] isa combination of the tr and Str of polynomials in g − n T g n T . The symmetry (c) implies that ineach monomial g always appears raised to the same power as its inverse. The symmetry (d)implies that the number of factors of g plus the number of factors g − in each term is even.Finally, the symmetry ( e ) together with the fact that g = diag( g (1) , g (2) ) is block-diagonal implythat the contribution to the effective action from diagrams with an even number of loops isthe supertrace of a polynomial in g − n T g n T while the contribution from diagrams with an odd number of loops is the trace of a polynomial in g − n T g n T (cf. (3.15)).Since the only bare g factors may come from the potential terms, having more than twofactors of g and g − requires having more than two vertices from the µ -dependent terms. Thenumber of factors of µ produced this way equals the total number of factors of g plus thenumber of factors of g − . Then the only way to obtain the correct dimension of the effectiveaction is to ensure that the coefficients of such terms are given by (two-dimensional) momentumintegrals with negative mass dimension; such integrals are finite in the UV.From the arguments above it follows that the only potentially divergent contributions to thebosonic part of the effective action must be proportional to µ before the momentum integralsare evaluated. Divergences of this type may be proportional to either the bosonic potentialterm in (2.25), i.e. Str[ g − T gT ] in (4.7), or to tr[ g − T gT ]. Such contributions may come fromthe two types of diagrams: diagrams with one vertex from the bosonic potential and diagramswith two vertices from the boson-fermion (“Yukawa”) interaction term in (4.7). In the following all integrals will be defined with an implicit IR regulator which is differentfrom the UV regulator. This is needed since we are interested only in UV divergences. In thisregime, masses of particles are irrelevant. In other words, we can expand in powers of the massparameter of the world sheet fields or in powers of µ .A special trick that we shall use below to simplify the calculation of the UV divergences isto treat the field g (and the fluctuation field ζ ) as unconstrained matrices rather than elements(of the algebra) of Sp (2 , × Sp (4). This is possible to do by assuming that the matrixmultiplication in the action contains factors of the symplectic Sp (2 ,
2) and Sp (4) metrics. Suchfactors project out the non- Sp (2 , × Sp (4) parts of the fields in each term of the action.Effectively, the contraction with the symplectic metric introduces the appropriate projectors invertices and propagators.To define the perturbation theory we will need the propagators for the bosonic fluctuationfields ζ in (4.8) and the fermionic fields that can be parametrized as ( χ L,R and λ L,R are 4 × ξ L,R and η L,R , see (2.12),(2.22),(2.23))[ T, Ψ L,R ] = (cid:18) λ L,R χ L,R (cid:19) . (4.9) An equivalent argument can be given of course by starting directly with the action (2.25). Depending onthe number of loops one may have additional vertices arising from the expansion of the action. Due to its gaugeinvariance, the gauge field in the gauged WZW action can only contribute through its field strength, so ondimensional grounds it cannot contribute to the UV-divergent terms proportional to µ .
19e will use ( a, b, . . . ) for the Sp (2 ,
2) indices and (¯ a, ¯ b, . . . ) for the Sp (4) indices and introducethe corresponding symplectic metricsΩ ac Ω bc = δ ba , Ω ¯ a ¯ c Ω ¯ b ¯ c = δ ¯ b ¯ a . (4.10)Then the bosonic propagator is h ζ ab ζ cd i = a b p (Ω ac Ω bd + Ω ad Ω bc ) , h ζ ¯ a ¯ b ζ ¯ c ¯ d i = − a b p (Ω ¯ a ¯ c Ω ¯ b ¯ d + Ω ¯ a ¯ d Ω ¯ b ¯ c ) , (4.11)and the fermionic one is ( p ± = p ± p ) h λ L a ¯ b χ L ¯ cd i = i a f p + ( T ad Ω ¯ b ¯ c − T ¯ b ¯ c Ω ad ) , h χ R ¯ cd λ R a ¯ b i = i a f p − ( T ¯ c ¯ b Ω da − T da Ω ¯ c ¯ b ) . (4.12)Here a b and a f are normalization constants a b = − , a f = 12 , (4.13)which we shall sometimes keep arbitrary for generality. The 1-loop contribution to the effective action Γ[ g ] is given simply by the logarithm of the ratioof the determinants of the bosonic and fermionic kinetic operators in the g -background. To testits finiteness it is enough to show the cancellation of the first two terms in the µ -expansion ofthe logarithm of these determinants.The leading ( µ -independent power-like divergent) term in the expansion simply counts thedifference between the number of bosonic and fermionic degrees of freedom and thus cancelsautomatically. To demonstrate the cancellation of the subleading (logarithmic) divergencerequires a short calculation. The relevant Feynman diagrams are shown in figure 1. (b)(a) Figure 1: One-loop diagrams contributing to the logarithmic divergences. Bosonic propagatorsare denoted by solid lines and fermionic ones by dashed lines. Black dots denote vertices comingfrom the bosonic and the bosonic-fermionic potential term in the classical action (4.7).These diagrams represent the next-to-leading order in the mass µ expansion of the trace oflogarithm of the bosonic and fermionic kinetic operators. Their cancellation tests the mass sumrule for the fluctuation fields X i ( − f i m i = 0 . (4.14)20he vertices in figure 1 arise from the expansion of the bosonic and the fermionic terms in theaction (4.7) ( g here is the background field) L ( b )2 = µ Str (cid:2)(cid:0) ζ T + T ζ − ζ T ζ (cid:1) g − T g (cid:3) , (4.15) L ( f )2 = µ Str (cid:2) Ψ R g − Ψ L g (cid:3) . (4.16)We shall formally assume that the fields have Sp ( n − , × Sp ( n )-valued indices (we will set n = 4 at the end). Then the relevant contribution of the bosonic diagram to the effective actionis L ( b )1 − loop = µ a b (cid:0) n +12 + n +12 − (cid:1) I tr[ g − T gT ] , I = Z d p (2 π ) p , (4.17)where tr is the trace over Sp ( n − , × Sp ( n ) indices and in the integral I we assume thepresence of both UV and IR cutoffs. In what follows we shall use dimensional ( d = 2 − ε )UV regularization, and the IR divergences can be subtracted as, e.g., in [55, 41] by replacingthe massless propagators by p → p + πε δ (2) ( p ).The three terms in the bracket in (4.17) came from the three terms in L ( b )2 in (4.15). Weused that (cf. (4.11)) h ζ ad i ≡ h ζ ab Ω bc ζ cd i = a b p (1 + n )Ω ad , h ζ a ¯ d i ≡ h ζ ¯ a ¯ b Ω ¯ b ¯ c ζ ¯ c ¯ d i = − a b p (1 + n )Ω ¯ a ¯ d , h ( ζ T ζ ) ah i ≡ h ζ ab Ω bc T cd Ω de ζ eh i = − a b p T ah , (4.18)and that T with two lower indices (i.e. with one index lowered by Ω) is an antisymmetricmatrix. The fermionic contribution is L ( f )1 − loop = µ a f (cid:0) n + n (cid:1) I tr[ g − T gT ] , (4.19)where in the denominator of the integral we used that − p + p − = p and the overall came formthe expansion of the logarithm of the kinetic operator to the second order. To arrive at (4.19)we noted that decomposing each vertex in 4 × Sp ( n − , × Sp ( n ) one finds two terms for each vertex. Each term in one vertex contractswith exactly one term in the second vertex and each contraction yields one of the two terms inthe bracket in (4.19).Adding L ( b )1 − loop (4.17) and L ( f )1 − loop (4.19) one observes that they cancel out (since accordingto (4.13) a b = − a f = − ). This implies that (4.14) is indeed satisfied and thus the 1-loopeffective action for a generic classical background g is finite. Thus the 1-loop bosonic anomalous dimension of the operator tr[ g − T gT ] in G = Sp ( n ) WZW theory isproportional to n . This coefficient is different from the dimension of tr g which is proportional to n + 1 (see(3.22), c r ( Sp ( n )) = n + 1). From the general perspective of the sigma model anomalous dimension in (3.21) thisdifference can be attributed to the difference of eigenvalues of the Laplace operator on the group space whenacting on the corresponding operators. To compute the action of the Laplacian on tr[ g − T gT ] one may follow[44] and use that ∂ a g = gE ma T a and T a T a = c r
1l as well as a relation for T a [ T, T a ] similar to the one appearingin (4.18) (this additional contribution leads to the subtraction of 1 from c r = n + 1). AdS × S case the presence of two-dimensional supersymmetry in the reduced action[2] makes this calculation redundant, but in general we do not know which symmetry (if any)relates the bosonic and the fermionic potential terms in the reduced Lagrangian (2.25). Sincethese two terms appeared (after gauge fixing and field redefinition) from the original GS action(2.9) where their coefficients were related by κ -symmetry this non-renormalization effectivelychecks the consistency of the reduction procedure at the quantum level.To check that there is no renormalization of the fermionic potential in (4.7) we should con-sider the diagram containing a single bosonic loop and an interaction vertex coming from theexpansion of the fermionic interaction term to second order in the bosonic fluctuations: L ( f )int = µ Str h g − Ψ L g (cid:0) ζ Ψ R − ζ Ψ R ζ + Ψ R ζ (cid:1) i . (4.20)The bosonic propagators (4.11) and the fact that the fermions transform in the bifundamentalrepresentation of Sp (2 , × Sp (4) imply that the expectation value of the second term in thebracket in (4.20) vanishes identically. Finally, the sign difference between the expectation valuesin the first line of equation (4.18) implies that the contributions of the remaining two termscancel each other. Indeed, we get( g − Ψ L g ) a ¯ b (cid:0) h ( ζ ) ¯ b ¯ c i Ψ ¯ cd R Ω da + Ω ¯ b ¯ c Ψ ¯ cd R h ( ζ ) da i (cid:1) − ( g − Ψ L g ) ¯ ab (cid:16) h ( ζ ) bc i Ψ c ¯ d R Ω ¯ d ¯ a + Ω bc Ψ c ¯ d R h ( ζ ) ¯ d ¯ a i (cid:17) (4.21)where each line represents one of the two terms of the supertrace, and then the sign differencebetween h ( ζ ) ¯ b ¯ c i and h ( ζ ) da i in (4.18) implies that each parenthesis vanishes identically. Let us now proceed to analyzing the 2-loop divergent contributions to the action in (4.7). Weshall ignore the power divergences. The ln Λ (or double-pole) divergences should cancel(according to the standard argument) due to the cancellation of the logarithmic divergencesat the 1-loop order established above. The main issue will thus be the ln Λ (or single pole)divergences. We shall first consider corrections to bosonic potential and then discuss possibledivergent contributions to the fermionic Yukawa term.
The relevant diagrams (that may produce potentially divergent order µ contributions) containone µ -vertex from the bosonic potential or two µ -vertices from the bosonic-fermionic interactionterm; they are shown in figure 2. They are absent in dimensional regularization and in any case should cancel due to the balance of degreesof freedom, the mass sum rule (4.14) or under an appropriate choice of the path integral measure. µ µ (a) µ µ (d)(b) (c) Figure 2: Two-loop diagrams at order µ . Bosonic propagators are denoted by solid lines andfermionic ones by dashed lines.The first diagram contains one parity-even 4-point vertex from L WZW in (4.7) (we shall sup-press the overall k π factor) L WZW (4) = − η µν Str [[ ∂ µ ζ , ζ ] , [ ∂ ν ζ , ζ ]] (4.22)and an insertion of a 2-point vertex from the bosonic potential (“mass insertion”). As in (4.15)in eq.(4.22) ζ is assumed to be a matrix in the algebra of Sp ( n − , × Sp ( n ) (we will again set n = 4 at the end). Namely, it is a symmetric matrix when written with both lower indices (i.e.with the upper index contracted with the symplectic metric Ω). The corresponding contributionto the effective action is proportional to the tadpole integrals: L ( a )2 − loop = ~ µ a b n ( n + 2) [ I ( ε )] Str[ g − T gT ] , (4.23) I ( ε ) ≡ Z d d p (2 π ) d p , d = 2 − ε . (4.24) ~ ≡ πk is the inverse of the coefficient in front of the classical WZW action. We assumeddimensional regularization. The second diagram, containing one vertex from the bosonic potential, also yields only tad-pole integrals. The bosonic 4-vertex arising from the expansion of the bosonic potential is L pot (4) = µ Str (cid:16)h
14! ( ζ T + T ζ ) −
13! ( ζ T ζ + ζ T ζ ) + 1(2!) ζ T ζ i g − T g (cid:17) , (4.25)where the multiplication of matrices is assumed with the symplectic metric. Also, the propa-gator in (4.11) enforces the condition that ζ belongs to the algebra of Sp ( n − , × Sp ( n ). Aswas already mentioned above, this implies that we may formally treat ζ as an unconstrainedmatrix rather than an element of the algebra of Sp ( n − , × Sp ( n ).The contribution of each of the three terms in (4.25) to the divergent part in the case whenthe group is Sp ( n ) is proportional to2 ×
14! ( n + 1)(2 n + 1) − ×
13! [ − (2 n + 1)] + 1(2!) [( n + 1) − ( n + 1) + 1] . (4.26) As we are interested in isolating the UV divergence, we understand this integral as having an implicit IRcutoff separate from the dimensional regulator, e.g., one may carry out an IR subtraction at the level of thepropagators as was already mentioned above. Sp ( n ) by Sp ( n − , L ( b )2 − loop = ~ µ a b n (5 n −
2) [ I ( ε )] Str[ g − T gT ] . (4.27)As was already mentioned above, while at odd number of loops the divergent contributionsfrom individual diagrams are proportional to µ tr[ g − T gT ], at even number of loops the diver-gent contributions are proportional to µ Str[ g − T gT ], i.e. have the same form as the classicalpotential.Next, there is a divergent contribution from a diagram (c) with two cubic vertices from theWZ term in the WZW Lagrangian (4.7) and with a µ insertion from the potential. Up to anormalization factor ~ − = k π common to the parity-even part of the WZW Lagrangian, thecubic interaction term is L WZW (3) = 23 ǫ µν Str[ ζ ∂ µ ζ ∂ ν ζ ] . (4.28)It then yields L ( c )2 − loop = 16 ~ µ a b n ( n + 2) I Str[ g − T gT ] , (4.29) I ≡ − Z d d pd d q (2 π ) d ( ǫ µν p µ q ν ) p q [( p + q ) ] . (4.30)Here we again assumed continuation d = 2 − ε but we need to decide how to treat ǫ µν indimensional regularization. This is a well-known issue (see, e.g., [42, 39, 40, 57, 41]). Ingeneral, different regularization prescriptions may lead to different results – the coefficient ofthe 2-loop logarithmic divergences may be scheme-dependent, with different results related byredefinitions of the coupling constants [39, 60].Similarly to the original GS action (2.9) containing the fermionic WZ term, the reducedaction (2.25) or (4.5) does not admit a straightforward d -dimensional generalization. This isanalogous to (chiral) supersymmetric theories (see, e.g., [52, 57, 59]) where it is natural touse the version of dimensional regularization by dimensional reduction [51]. We shall discussalternative regularization schemes in Appendix A and draw an analogy with the case of 2dsupersymmetric sigma models in Appendix B.Under this prescription we shall do all Lorentz (and spinor) algebra in 2 dimensions andcontinue to d dimensions only scalar momentum integrals. In particular, we shall use the2-dimensional relation ǫ µν ǫ µ ′ ν ′ = − η µ ′ µ η ν ′ ν + η ν ′ µ η µ ′ ν , (4.31)where in the Minkowski signature notation η µν = ( − , − ( ǫ µν p µ q ν ) = p q − ( p · q ) , (4.32)24nd thus continuing to d = 2 − ε dimensions we find I = Z d d pd d q (2 π ) d p q − ( p · q ) p q [( p + q ) ] = 14 [ I ( ε )] . (4.33)The contribution of the diagram (c) in (4.29) is then given by L ( c )2 − loop = 4 ~ µ a b n ( n + 2) [ I ( ε )] Str[ g − T gT ] . (4.34)Adding together (4.23),(4.27) and (4.29) and using that a b = − we find that the contributionof the bosonic 2-loop diagrams to the UV singular part of 2-loop effective Lagrangian is L bose2 − loop = ~ µ n [ I ( ε )] Str[ g − T gT ] , (4.35)where [ I ( ε )] = h πε + O (1) i = 1(4 π ) ε + ... . (4.36)The coefficient of the most singular term is consistent with the expected renormalization groupbehavior of the bosonic theory, i.e. it is related to the square of the coefficient of the 1-loopsingle-pole in (4.17). The coefficient of the 2-loop subleading ε pole is, in general, scheme de-pendent; in the standard minimal subtraction scheme we then get no genuine 2-loop divergence(i.e. the 2-loop anomalous dimension coefficient vanishes).Let us introduce the renormalization constant Z ( i ) , i = 1 ,
2, for the two bosonic operators U ( i ) corresponding to two factorized parts (related to the two subgroups of Sp ( n − , × Sp ( n ))in µ Str[ g − T gT ], i.e. U ( i ) = Z ( i ) U ( i )bare Z = µ − ε h ~ γ ε + ~ (cid:0) γ ε + γ ε (cid:1) + ... i , (4.37)where we suppressed the index i and we have chosen µ to be the renormalization scale parameter.Then the corresponding anomalous dimension is γ = dZ − d ln µ = 2 ε + ~ γ + ~ γ + ... . (4.38)From (4.17) it is easy to see that γ (1 , = ± π n which, when squared, reproduces the coefficientof the ε pole in (4.35).Let us now consider the fermionic contributions to the 2-loop divergent part of the bosoniceffective action. There are several types of µ terms which arise from bose-fermi interactionterm in (4.7) and they correspond to the diagrams 2( d ) and 2( e ). They can be representedsymbolically as coming from the square of the interacting terms in the action:2 × h Z d σ Str[ g − Ψ L g Ψ R ] Z d σ Str h g − Ψ L g (cid:0) ζ Ψ R − ζ Ψ R ζ + Ψ R ζ (cid:1) i i
25 12 h (cid:16) Z d σ Str[ g − Ψ L g ( ζ Ψ R − Ψ R ζ )] (cid:17) i . (4.39)The terms in the first line, diagram 2( d ), lead to vanishing contributions to the logarithmicdivergences either because of impossibility of proper Wick contractions (as in the second termin the brackets) or because of Str = 0 (as in the case of the first and the third term). Theremaining non-trivial contribution comes from the term in the second line of (4.39), i.e. diagram2( e ) L ( e )2 − loop = ~ µ a b a f ×
12 [ n ( n + 1) − n ] I Str[ g − T gT ] , (4.40) I = Z d p (2 π ) d q (2 π ) p + q − p q ( p + q ) , (4.41)where we took into account the minus signs due to the fermionic loop, due to the supertrace ineq.(4.39) and due to the factors of i in the fermionic propagators. The factor is inherited fromthe last line of eq.(4.39) and the overall factor of 2 is present because the relevant contributioncomes from the cross term in the square.Here again there is an ambiguity in defining the integral I , i.e. in extending the factor p + q − in the integrand (which has its origin in the chiral nature of the fermion coupling in (4.7)) to d dimensions. In the GS action, the fermionic current components were 2d vectors and theywere reinterpreted as 2d Weyl spinors in the reduced theory. The fermionic interaction term inthe reduced theory (2.25) originated from the WZ term in the GS action (2.9), which suggeststhat chiral fermions should be treated as if they were 2-dimensional fields. An analogy withthe 2d supersymmetric gWZW model suggests again to use the regularization by dimensionalreduction.Explicitly, that means that we shall first use that in 2 dimensions p + q − = ( p + p )( q − q ) = − ( η µν + ǫ µν ) p µ q ν . (4.42)Equivalently, interpreting Ψ L and Ψ R in (4.7) as upper/lower components of left/right MW 2dspinor and rewriting the fermionic terms using the 2-component notation with the explicit 2d γ -matrix factors we observe that p + q − in I in (4.40) arises from p + q − = − tr[ p / q / (1 + γ )] = − p · q − ǫ µν p µ q ν , (4.43)where γ = γ γ and we assumed that all spinor algebra is done in 2 dimensions. Observing that the term with a single factor of the antisymmetric tensor ǫ µν can not con-tribute to the integral and continuing the scalar integrand to d dimensions we end up with I = − Z d d pd d q p · qp q ( p + q ) = − Z d d pd d q ( p + q ) − p − q p q ( p + q ) This is essentially the same calculation which implies the non-renormalization of the fermionic potential at1-loop order. Same result for the parity-even term is found if we extended momenta and γ -matrices to d dimensions byassuming that p / = ¯ p µ ¯ γ µ + b p µ b γ µ , { ¯ γ µ , γ } = 0 , [ b γ µ , γ ] = 0, where ¯ µ are 2-dimensional and b µ are − ε dimensional indices, i.e. µ = (¯ µ, b µ ).
26 12 [ I ( ε )] . (4.44)Then finally (using (4.13)) L fermi2 − loop = − ~ µ n [ I ( ε )] Str[ g − T gT ] . (4.45)Combining this with the bosonic contribution in (4.35) we conclude that the two contributionscancel each other, i.e. the bosonic part of the 2-loop effective action is UV finite, L (bos . pot . )2 − loop = L bose2 − loop + L fermi2 − loop = finite . (4.46)As already mentioned above, this is just a reflection of the cancellation of the 1-loop logarithmicdivergences as all simple ε poles in both the bosonic and the fermionic contributions computedin the dimensional reduction scheme come together with a ε pole which is controlled by the1-loop divergences. The above observation, that the 2-loop correction to renormalization of the bosonic potential isscheme dependent, may seem to contradict the standard lore: in view of the cancellation of theone-loop renormalization of the potential, one could expect that the two-loop renormalizationshould be scheme independent being the first non-vanishing correction. However, as discussedin section 3 and below eq.(4.7), the reduced theory, when viewed as a power-counting renor-malizable model, is actually a multi-coupling theory (with the level k and several µ -parametersas its couplings, with the action (4.7) corresponding to a fixed-point choice). In such a case the2-loop anomalous dimension coefficients may still be scheme-dependent.As was already mentioned, several a priori distinct parameters in the action were set to beequal as required by the reduction procedure starting from the GS action where they wererelated by symmetries. In the bosonic part of the theory these were the couplings of thetwo potential terms corresponding to Sp ( n − ,
2) and Sp ( n ). With fermions included, thecoefficients of the bosonic and the fermionic potential terms, Str[ g − T gT ] and Str[ g − Ψ L g Ψ R ],were also related. It is then necessary to ensure that such relations survive quantum corrections.As we have found above, the corrections to the bosonic potential are finite in a specialdimensional reduction scheme. Finiteness of the full theory then requires that corrections tothe fermionic potential be finite in that same scheme. In the apparent absence of worldsheetsupersymmetry which would relate the bosonic and the fermionic potentials (and thus theirrenormalization, assuming one uses a supersymmetry-preserving regularization scheme) this isnot a priori guaranteed. It is, however, important to recall again that the bosonic and the fermionic potentials are closely connectedto the kinetic and WZ terms in original Green-Schwarz action where the relation between their coefficients is aconsequence of the κ -symmetry. It is possible that a global remnant of the κ symmetry that may be survivingin the gauge (2.14) offers a sufficient protection to guarantee this relation to all orders in perturbation theoryin the reduced model.
27t is therefore crucial to test the finiteness of the corrections to the fermionic potential in(4.7) U f = µ Str (cid:0) g − Ψ L g Ψ R (cid:1) (4.47)in the same dimensional reduction scheme.On dimensional grounds, to (logarithmically) renormalize U f we need terms with a singlepower of µ . Since all the fermionic interactions in (4.7) are proportional to µ and the bosonicpotential is proportional to µ , it follows that this renormalization is entirely governed by thebosonic Sp ( n − , × Sp ( n ) WZW model with fermions treated as background fields.The relevant diagrams are shown in figure 3. µ µ µ (c)(b)(a) Figure 3: Two-loop diagrams contributing to renormalization of the fermionic potential. Solidlines are bosonic propagators and external fermionic legs at each µ -vertex are suppressed.The computation of their divergent parts is formally similar to that of the renormalizationof the bosonic potential in (4.7), assuming one treats T as a background field. There are,however, certain differences related to the different algebraic structure of T and Ψ, whichprevent the bosonic results from being immediately used here. Nevertheless, the mere fact thatthe calculation is effectively governed by the undeformed Sp ( n − , × Sp ( n ) WZW modelguarantees already that the same scheme dependence which entered the bosonic calculationwill enter here as well.Upon using the fact that Ψ L,R are off-diagonal (transforming in bi-fundamental representationof G , see (2.22),(2.23),(4.9)) and that g is diagonal (cf. (4.3)), it is easy to see that the fermionicpotential may be written as U f = µ (cid:16) tr[ g (1) − λ L g (2) χ R ] − tr[ g (2) − χ L g (1) χ R ] (cid:17) , (4.48)where g (1) ∈ Sp ( n − ,
2) and g (2) ∈ Sp ( n ). Since the Sp ( n − ,
2) and Sp ( n ) WZW models arecoupled only through the µ -dependent fermionic terms, it follows that, for the purpose of therenormalization of U f , we may treat g (1) and g (2) separately. Thus, in a diagram of topology3(a) the fields propagating in the two loops must be of the same type since the quartic vertexcoming from the WZW action involves fields of only one type (there are two distinct diagramsin this class). In a diagram of topology 3(b) the fields propagating in the two loops may beeither of the same type or of different types (there are three distinct diagrams in this class). Ina diagram of topology 3(c) the fields propagating in the two loops must be of the same type(there are two distinct diagrams in this class).The diagrams of these three topologies contribute as follows to the 2-loop effective La-grangian: L ( a )2 − loop = ~ µ (cid:2) a b ( n + 1)( n + 2) + ( −
1) 13 ( − a b ) ( n + 1)( n + 2) (cid:3) [ I ( ε )] Str[ g − Ψ L g Ψ R ]28 ( b )2 − loop = − ~ µ a b
12 ( n + 1)( n + 2)[ I ( ε )] Str[ g − Ψ L g Ψ R ] (4.49) L ( c )2 − loop = ~ µ (cid:2) a b ( n + 1)( n + 2) + 8( − a b ) ( n + 1)( n + 2) (cid:3) I ( ε ) Str[ g − Ψ L g Ψ R ]= ~ µ (cid:2) a b ( n + 1)( n + 2) + 2( − a b ) ( n + 1)( n + 2) (cid:3) [ I ( ε )] Str[ g − Ψ L g Ψ R ] , where the integrals I ( ε ) and I ( ε ) were defined in eqs. (4.24) and (4.30), respectively, and inthe last line we used eq. (4.33) relating I and ( I ) .It is interesting to note that each one of the above three contributions is proportional to( n + 1)( n + 2). This factor may be understood on the group theory grounds as being theproduct of the two quadratic Casimirs, in the fundamental and the adjoint representationsof Sp ( n − ,
2) or Sp ( n ). This n dependence is different from that of the corrections to thebosonic potential because, on the one hand, in the bosonic calculation one uses that (see (2.12)) T = −
1l while here the analogous quantities are Ψ L or Ψ L Ψ R do not have similar properties,and, on the other hand, some Wick contractions here are forbidden as the fields belong todifferent algebras.Adding together the above three singular contributions in (4.49) we conclude, in completeanalogy with the bosonic potential case, that they cancel out, i.e. the result is UV finite, L (fermi . pot . )2 − loop = finite . (4.50) The reduced model (2.25) [2, 3] we discussed above is naturally associated, through the Pohlmeyerreduction, to the
AdS × S GS superstring action (2.9) and has certain unique features.Its construction is based on first-order or phase space formulation of superstring dynamics interms of supercoset currents, with the Virasoro constraints explicitly solved in terms of a newset of variables related locally to currents and thus non-locally to the original GS
AdS × S supercoset coordinates. Although various steps in the reduction do not appear to manifestlypreserve 2d Lorentz invariance, the resulting reduced Lagrangian describes the dynamics of thephysical number of degrees of freedom in a manifestly Lorentz invariant way. Being formulatedin terms of left-invariant currents, the reduced theory is apparently “blind” to the originalglobal P SU (2 , |
4) symmetry; however, being integrable (the Lax pairs of the original andthe reduced theory are gauge-equivalent), it still has an infinite number of commuting chargesassociated to hidden symmetries, some of which are implicitly related to the global symmetriesof the original GS theory.In general, the Pohlmeyer reduction procedure, utilizing the classical conformal symmetryof a 2d sigma model, is expected to lead to an equivalent theory only at the classical level; forexample, the original and reduced theory are obviously not equivalent at the quantum level ifthe original sigma model has a running coupling. In the present case of
AdS × S superstringsigma model, which is a conformal 2d theory at the quantum level, the relation between theoriginal and the reduced theory has a perfect chance to hold also at the quantum level. Thenecessary condition for that is that the reduced theory is also UV finite.As we have demonstrated in the present paper, the reduced theory associated to the AdS × S superstring model is indeed free of 2d UV divergences in a certain renormalization scheme. An29dvantage of the reduced theory compared to the GS model is that here the main “kinetic” partof the action is based on a gauged WZW theory and thus is guaranteed to be finite; then whatremains to check is only the absence of divergent contributions to the derivative-independent“potential” part of the action. We explicitly checked that at the 1-loop and 2-loop order butmost likely this should be true to all orders and should be due to a hidden 2d (super)symmetry ofthe reduced theory. The cancellation of divergences is due to a very special balance betweenthe bosonic potential term and the fermionic interaction term in (2.25). These two termsoriginated from the “kinetic” P and the fermionic WZ Q terms in the GS action (2.9) wherethey were related by κ -symmetry. This suggests that some (global) remnant of the κ -symmetrystill present after fixing the κ -symmetry gauge in the reduced action may be responsible for itsUV finiteness.This opens up a possibility of solving the quantum AdS × S superstring theory in terms ofthe the quantum reduced theory. The precise prescription for translating observables betweenthe two theories remains to be understood. The most optimistic scenario is to find a pathintegral version of the reduction procedure based on changing the variables from coordinatesto currents and solving the conformal gauge constraints as delta-function conditions T ++ =0 , T −− = 0 in the path integral.To test the equivalence of the two partition functions one may consider comparing their valuesfor equivalent classical solutions. We leave the study of this problem for the future. Amongother open problems let us mention the construction of the (2d Lorentz-invariant) S-matrix forscattering of the massive elementary excitations in the reduced theory and the determinationof its relation to the BMN (magnon) S-matrix in the AdS × S string theory in a light-conegauge.Let us finish with few comments on the role of the µ parameter in the reduced theory. Theoriginal GS string theory in conformal gauge has a residual part of the 2d diffeomorphism group– conformal reparametrizations – being preserved by quantum corrections. In the process ofconstructing the reduced theory we fix this residual symmetry by a gauge choice (cf. (2.11))that introduces the constant parameter µ . This parameter is a fiducial scale, similar to theconstant p + in the standard light-cone gauge. Thus µ is similar to a gauge-fixing parameterand physical observables should not depend on it. For example, the expression for the energyof a particular string state expressed in terms of conserved charges of the reduced theory (or,e.g., Casimirs of the original GS global symmetry group) should not depend on µ , i.e. µ canbe eliminated by re-expressing it in terms of the charges. At the same time, the S -matrix ofelementary excitations with mass µ (which, by itself, is not a physical observable) will dependon µ . If the reduced theory does not actually have a standard global 2d supersymmetry, this finiteness propertysuggests that there may be other similar models without 2d supersymmetry that are still UV finite. It wouldbe interesting to classify them. Indeed, the condition P + = µT in (2.11) is reminiscent of the relation ∂ + x + ∼ p + in the light-cone gauge.Compared to standard 2d conformal theories where the infinite-dimensional conformal group is interpreted asa global symmetry imposed through conditions on physical states, in the context of string theory this is part ofthe 2d diffeomorphism gauge symmetry and one is allowed to fix it by a gauge choice. One may draw an analogy with quantization of strings in plane wave background. In conformal gauge onehas a sigma model with target space metric like ds = dx + dx − + ax i x i dx + dx + + dx i dx i and certain globalsymmetry group. One may, in principle, develop a covariant quantization and find the spectrum of states which cknowledgments We are grateful to M. Grigoriev, R. Metsaev, G. Papadopoulos and A. Vainshtein for usefuldiscussions. AAT acknowledges the support of the STFC rolling grant. Part of this workwas done while AAT was a participant of the 2008 workshop “Non-Perturbative Methods inStrongly Coupled Gauge Theories” at the Galileo Galilei Institute for Theoretical Physics inArcetri, Florence. RR acknowledges the support of the US National Science Foundation undergrant PHY-0608114, the US Department of Energy under contract DE-FG02-90ER40577 (OJI)and of the A. P. Sloan Foundation.
Appendix A: Comments on regularization scheme ambiguity
Regularization scheme dependence of the 2-loop corrections to the bosonic and fermionic poten-tials implies that while apparently different results may be obtained under different choices ofregularization (and, in particular, of treatment of fermions and Levi-Civita tensors), all of themare related by suitable redefinitions of the coupling constants of the theory. The most naturalregularization scheme should be consistent with the symmetries of the theory, and we believethe dimensional reduction regularization used in the main text is such a scheme, though thatseems non-trivial to demonstrate explicitly. For completeness, in this Appendix we discussthe 2-loop results in some alternative regularization schemes.A version of dimensional regularization prescription (which does not , however, preserve the d -dimensional Lorentz invariance) is to continue momenta to d = 2 − ε from the very beginningwhile still treating the Levi-Civita tensor ǫ µν as if it is defined only in 2-dimensions [40] (i.e. ǫ µν → ¯ ǫ µν ≡ ǫ ¯ µ ¯ ν , ¯ µ, ¯ ν = 1 , − (¯ ǫ µν p µ q ν ) = ¯ p ¯ q − (¯ p · ¯ q ) = [( p − b p )( q − b q ) − ( p · q − b p · b q ) ]= [ p q − ( p · q ) ] − (cid:2) p b q + q b p − p · q b p · b q (cid:3) + [ b p b q − ( b p · b q ) ] . (A.1)Here ¯ p and b p are the 2-dimensional and − ε -dimensional components of the momentum p in d = 2 − ε dimensions, p µ = ( p ¯ µ , p b µ ). The contribution of the first square bracket in the lastline to the integral in (4.30) is then the same as in (4.33), while the second bracket leads to − Z d d pd d q (2 π ) d [ p b q + q b p − p · q b p · b q ] p q [( p + q ) ] = − Z d d pd d q (2 π ) d b p · b qp q ( p + q ) = − b η µν η µν d Z d d pd d q (2 π ) d p · qp q ( p + q ) = − ǫd [ I ( ε )] , (A.2) will be classified by charges of that symmetry. We may instead fix the light-cone gauge x + = p + τ and obtaina model containing free bosons (and fermions, as in the pp-wave model [23, 56, 24] associated to AdS × S background) with mass µ = p + . Then the spectrum will depend on that µ , but we may re-interpret thatdependence as that on one of the global charges which has a fixed value (proportional to µ ) in that light-conegauge. An intuitive reason is that the reduced model is related to the
AdS × S GS superstring where the κ -symmetry should be preserved. The 2-loop finiteness of the AdS × S superstring demonstrated in [14] in thisscheme is a strong indication in this direction. d = 2 − ε one has b η µν η µν = − ε . The integral of the remaining squarebracket in the last line of (A.1) may be written as( b η µν b η ρσ − b η µρ b η νσ ) Z d d pd d q (2 π ) d p µ p ν q ρ q σ p q [( p + q ) ] , (A.3)and produces a finite O ( ε )[ I ( ε )] contribution. As a result, the expression for I in (4.30) inthis regularization scheme is given by the sum of (4.33),(A.2) and (A.3), i.e. I = h − ε O ( ε ) i [ I ( ε )] . (A.4)The contribution of the diagram (c) in (4.29) is then L ( c )2 − loop = 4 ~ µ a b n ( n + 2) h − ε + O ( ε ) i [ I ( ε )] Str[ g − T gT ] , (A.5)where b = 0 in the dimensional reduction regularization used in section 4.4.1 with I given by(4.33) and b = 1 in the second regularization prescription where I is given by (A.4). The totalbosonic contribution is then L bose2 − loop = ~ µ (cid:2) n − b n ( n + 2) ε + O ( ε ) (cid:3) [ I ( ε )] Str[ g − T gT ] , (A.6)where b = 0 corresponds to (4.35).Thus, unlike what happened in the regularization by dimensional reduction, the bosoniccontribution to the 2-loop anomalous dimension does not vanish in this ( d -dimensional Lorenz-violating) scheme. The resulting value for the 2-loop anomalous dimension is, however, inagreement with the standard expression for the two-loop anomalous dimension in a sigma modelwith a WZ coupling (see discussion below eq.(3.17)) and, in particular, with the expression forthe anomalous dimension of the primary field tr g in WZW theory [43] in (3.22). Similarly to the treatment of I there are several options of how to define the integral I (4.41), i.e. of how to extend it to d dimensions. Instead of using the dimensional reductionscheme we may choose to extend momenta to d dimensions from the start but treat the indicesof the integrand factor p + q − in (4.41) as 2-dimensional ones. Then instead of (4.42) we have(¯ µ, ¯ ν = 1 , p + q − = ( p + p )( q − q ) = − η ¯ µ ¯ ν p ¯ µ q ¯ ν − ǫ ¯ µ ¯ ν p ¯ µ q ¯ ν , (A.7) In the bosonic theory with the group Sp ( n − , × Sp ( n ) we have the kinetic and potential terms foreach factor decoupled, so that for, e.g., G = Sp ( n ) we get for the two anomalous dimensions, cf.(4.37),(4.38)( c G = c Sp ( n ) = n + 2) γ ( Sp ( n − , c k ( − c G k + ... ) , γ ( Sp ( n )) = c k (1 + c G k + ... ) , where c = c r = n + 1 (=Casimir of the fundamental representation of Sp ( n )) in the case of the tr g operator and c = n in the present case of the tr( g − T gT ) operator (cf. [54, 62]; for comparison, in the case of tr( g − T a gT b )where T a are generators of G one has c = c G [43]). Going from one group factor to another is thus equivalentto k → − k (notice that we had Str in the WZW kinetic term in (2.25) and (4.7)). I = − Z d p (2 π ) d q (2 π ) (¯ p · ¯ q ) p q ( p + q ) = 12 (cid:0) εd (cid:1) [ I ( ε )] . (A.8)Then L fermi2 − loop = − ~ µ n (1 + f ε ) [ I ( ε )] Str[ g − T gT ] , (A.9)where f = 0 corresponds to the dimensional reduction prescription used in (4.45) and f = 1corresponds to the above prescription leading to (A.8).Combining this with the bosonic contribution in (A.6) we conclude that the leading ε sin-gularity cancels out between the bosonic and the fermionic terms, just as the corresponding ε singularity did at one loop, and we are left with L (bos . pot . )2 − loop = L bose2 − loop + L fermi2 − loop = − ~ µ h [b n ( n + 2) + f n ] ε + O ( ε ) i [ I ( ε )] Str[ g − T gT ]= − ~ µ π ) ε [b n ( n + 2) + f n ] Str[ g − T gT ] + finite . (A.10)This remaining divergent term is clearly regularization-scheme dependent and may be set tozero by an appropriate finite redefinition of the couplings (in particular, the level of the WZWmodel). Appendix B: Analogy with 2d supersymmetric sigma models with potentials
It is important to note that the dimensional reduction scheme in which the reduced model is2-loop finite is also the scheme that would preserve 2d supersymmetry, if it were present at theclassical level.It is useful to draw analogy with a general analysis of 2-loop renormalization of ( p, q ) super-symmetric models deformed by potentials [58] carried out in [59]. A special case of the modelconsidered in [59] is the (1,1) supersymmetric theory generalizing a supersymmetric WZWmodel to the presence of a potential term [58] (cf. (3.16)) S = 14 πα ′ Z d σ h ( G mn ( x ) + B mn ( x )) ∂ + x m ∂ − x n + iG mn ( x ) ψ m L D (+)+ ψ n L + iG mn ( x ) ψ m R D ( − ) − ψ n R + 2 µD ( − ) m W n ( x ) ψ m L ψ n R − µ G mn ( x ) W m ( x ) W n ( x ) i . (B.1)Here G mn and B mn correspond to a group space G , x m are coordinates on G , D ( ± ) are covariantderivatives with respect to the two “flat” connections Γ mnk ( G ) ± H mnk ( B ), and a vector W m defines the bosonic potential. One more option is to use the straightforward dimensional regularization where h p µ q ν i = d η µν h p · q i andthus h p + q − i = − d h p · q i . In this case I = d [ I ( ε )] leading to (1 + 2 ε ) π ) ε divergent term. As is well known, the kinetic terms of the fermions can be decoupled from bosons by defining the tangentspace components like ψ a = E am ( x ) ψ m and “rotating” ψ a .
33n general [58], W m = U m − V m , where D ( m V n ) = 0 (i.e. V m is a Killing vector), ∂ [ m U n ] = H mnk V k , U m V m = 0. The condition of 1-loop (and, in fact, 2-loop) finiteness of such modelis [59] D m W m = const.In the simplest case W m = ∂ m W where W is real (1,1) superpotential. In that case theaction (B.1) can be written in the superfield form: S = 14 πα ′ Z d σd θ h ( G mn ( X ) + B mn ( X )) b D + X m b D − X n − W ( X ) i , (B.2)where X m = x m + θ + ψ m L + θ − ψ m R + θ + θ − F m and b D are spinor derivatives. In the 2d theory (B.1) the bosonic and the fermionic potential terms renormalize simulta-neously, i.e. the β -functions of the corresponding couplings are related by a supersymmetryWard identity. As was shown in [59], the 2-loop correction to this β -function vanishes in thedimensional reduction scheme similar to the one used here in section 4.4.1. Thus in the (1,1)supersymmetric theory (B.1) and the reduced theory (4.7) both treated in the dimensionalreduction scheme there are no genuine 2-loop simple-pole UV divergences, all of them beingaccompanied by a double-pole counterpart related to single-pole 1-loop divergences as dictatedby the renormalizability of the theory.The model (4.7) based on G = G × G bosonic WZW model with a potential coupled tofermions in bi-fundamental representations does not admit the standard version of (1,1) 2dsupersymmetry: the standard supersymmetric extension of its bosonic part would be of theform (B.2), i.e. having the same number of the fermionic degrees of freedom but transformingin the adjoint representation of G . The corresponding G and G supersymmetric modelswould be mutually non-interacting and the divergences in their potential terms will not cancel,precluding finiteness.The non-trivial property of the reduced model observed here is the cancellation of the 1-loop divergences, which makes the theory (at least) 2-loop finite. Such finiteness propertyis also characteristic of (2 ,
2) supersymmetric models [59]. The existence of a finite (2 , G which is a complexmanifold) perturbed by a potential appears to be subtle and we are not aware of its discussionin the literature. The (1,1) supersymmetric WZW action can also be written explicitly in terms of a superfield generalizing thegroup element g field [61]. Explicitly, we may replace g = e x by b g = e X , X ( σ, θ ) = x + θ + ψ L + θ − ψ R + θ + θ − F .Then to supersymmetrize the potential tr( g − T gT ) we need to find the corresponding real superpotential W .This step is straightforward for coset sigma models of the type (3.14) whose potential depends on only twospecial fields ϕ and φ such that the (1,1) superpotential may be written as cosh b φ + cos b ϕ or as Re[cos( b ϕ + i b φ )].Note that the holomorphic superpotential of the (2,2) sine-Gordon model found [2] in the special case of themodel (3.13) is W = cos( b ϕ + i b φ ), but more general models like (3.14) do not admit a straightforward (2,2)extension as ϕ and φ enter separately in the two factors of the target space metric. The existence of a (2,2) superpotential deformation for the supersymmetric WZW models discussed inthis appendix is, to some extent, questionable. Indeed, the relevant superpotential should be a holomorphicfunction on the target space. 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