E7(7) symmetry in perturbatively quantised N=8 supergravity
aa r X i v : . [ h e p - t h ] D ec AEI-2010-118 E symmetry in perturbativelyquantised N = 8 supergravity Guillaume Bossard ∗ , Christian Hillmann † and Hermann Nicolai ‡ ∗‡ AEI, Max-Planck-Institut f¨ur GravitationsphysikAm M¨uhlenberg 1, D-14476 Potsdam, Germany † Institut des Hautes Etudes Scientifiques35, route de Chartres, 91440 Bures-sur-Yvette, France
Abstract
We study the perturbative quantisation of N = 8 supergravity in a formulationwhere its E symmetry is realised off-shell . Relying on the cancellation of SU (8)current anomalies we show that there are no anomalies for the non-linearly realised E either; this result extends to all orders in perturbation theory. As a conse-quence, the e Ward identities can be consistently implemented and imposed atall orders in perturbation theory, and therefore potential divergent countertermsmust in particular respect the full non-linear E symmetry. ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] ontents N = 8 supergravity with off-shell E invariance 8 N = 8 supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 The classical E current . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Transformations in the symmetric gauge . . . . . . . . . . . . . . . . . . 22 SU (8) anomaly at one loop 27 e master equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2 The SU (8)-equivariant cohomology of e . . . . . . . . . . . . . . . . . 50 E with gauge invariance 58 Maximally extended N = 8 supergravity [1, 2] is the most symmetric field theoreticextension of Einstein’s theory in four space-time dimensions. Although long thought todiverge at three loops [3, 4], spectacular computational advances have recently shownthat, contrary to many expectations, the theory is finite at least up to and including fourloops [5, 6], and thereby fuelled speculations that the theory may actually be finite to all orders in perturbation theory. It appears doubtful whether maximal supersymmetryalone could suffice to explain such a far reaching result [7], if true. Rather, it seemsplausible that the possible finiteness of N = 8 supergravity will hinge on known or1nknown ‘hidden symmetries’ of the theory. Indeed, already the construction of the N = 8 Lagrangian itself was only possible thanks to the discovery of the non-linearduality symmetry E of its equations of motion [1]. This symmetry is expected tobe a symmetry of perturbation theory, and to be broken to an arithmetic subgroup of E by non-perturbative effects when the theory is embedded into string theory (see e.g. [8, 9] for a recent update, and also the comments below). Nevertheless, the status of thenon-linear duality symmetry at the level of quantised perturbation theory has remainedrather unclear, because E is not a symmetry of the original N = 8 Lagrangian andthe corresponding non-linear functional Ward identities therefore have not been workedout so far.Inspired by earlier work devoted to the definition of an action for self-dual form fields[10], one of the authors recently was able to set up a formulation of N = 8 supergravityin which the Lagrangian is manifestly E -invariant [11]. The main peculiarity ofthe formalism is to replace the 28 vector fields A m µ of the original formulation by 56 =28 + 28 vector fields A m i ≡ ( A m i , A ¯ m i ) with spatial components only , whose conjugatemomenta are determined by second class constraints in the canonical formulation, insuch a way that they represent the same number of physical degrees of freedom as theoriginal 28 vector fields in the conventional formulation of the theory. Although notmanifestly diffeomorphism invariant, the theory still admits diffeomorphism and localsupersymmetry gauge invariance [11]. By virtue of its manifest off-shell E invariance,the theory possesses a bona fide E Noether current, unlike the covariant formulation[13], and this is the feature which permits to write down functional Ward identities forthe non-linear duality symmetry.In this paper we will consider the perturbative quantisation of N = 8 supergravityin this duality invariant formulation. As our main result, we will prove that there existsa renormalisation scheme which maintains the full non-linear (continuous) E dualitysymmetry at all orders in a perturbative expansion of the theory in the gravitationalcoupling κ . A key element in this proof is the demonstration of the absence of linear SU (8) and non-linear E anomalies.As is well known [14], the proper definition of any quantum field theory relies onthe quantum action principle , according to which the ultra-violet divergences of the 1PIgenerating functional are always local functionals of the fields. Only thanks to thisproperty can one carry out the renormalisation program by consistently modifying the The formalism had been applied earlier to the definition of a manifestly SL (2 , R ) bosonic action for N = 4 supergravity [12]. N = 8 supergravity,and its lack of manifest Lorentz invariance in particular, the validity of the quantumaction principle is however not automatically guaranteed.To deal with this problem, we will in a first step prove that the duality invariantpath integral of the theory is equivalent to the conventional formulation by means of aGaussian integration. In order to ensure the validity of the quantum action principle,we will require the existence of a local regularisation scheme in the two formulationsof the theory, which are equivalent modulo a Gaussian integration (but note that theGaussian integration reduces the manifest E invariance to an on-shell symmetry). Thevalidity of the quantum action principle in the conventional formulation of the theory thenensures its validity in the duality invariant formulation. We will define a Pauli–Villarsregularisation of the theory satisfying these criteria. Although this regularisation wouldbreak Lorentz invariance in the covariant formulation as well, it is local and invariantwith respect to abelian gauge invariance in the two formulations. We will exhibit theconsistency of this regularisation in the explicit computation of the one-loop vector fieldcontribution to the su (8)-current anomaly.With a consistent duality invariant formulation at hand, we can address and answerthe question of whether the e current Ward identities are anomalous or not in per-turbation theory. According to [15], the local su (8) gauge invariance in the version of N = 8 supergravity with linearly realised E is anomalous at one-loop. However, asshown in [16] this anomaly can be cancelled by an SU (8) Wess–Zumino term which inturn breaks the manifest E invariance, whereby the local SU (8) anomaly is convertedinto an anomaly of the global E — unless there appear new contributions to the latter,as happens to be the case for N = 8 supergravity. According to [16] one thus has theoption of working either with the locally SU (8) invariant version of N = 8 supergravity,or with its gauge-fixed version where E is realised non-linearly. Here we prefer thesecond option, that is, we will consider an explicit parametrisation of the scalar manifold E /SU c (8) in terms of 70 scalar fields Φ ∈ e ⊖ su (8) which coordinatise the cosetmanifold. A consistent anomaly must then be a non-trivial solution to the Wess–Zuminoconsistency condition. We will prove that the associated cohomology problem reducesto the cohomology problem associated to the current su (8) Ward identities. It followsfrom this result that, although the non-linear character of the e symmetry is such Where throughout the notation SU c (8) will be used as a shorthand for the quotient of SU (8) by the Z kernel of the representations of even rank. su (8) anomaly coef-ficient — thereby saving us the labour of having to determine an infinitude of anomalousdiagrams! Now, thanks to a crucial insight of [17], it is known that for N = 8 super-gravity, the anomalous contributions to the current (rigid) su (8) Ward identities fromthe fermions cancel against the contributions from the vector fields because the latter arealso chiral under SU (8). Therefore the non-linear e Ward identities are likewise freeof anomalies. Moreover, the cohomological arguments of section 3 show that this resultsextends to all loop orders.The fact that the consistent e anomalies are in one-to-one correspondence with theset of consistent su (8) anomalies can also be understood more intuitively, and in a waythat makes the result almost look trivial. Namely, in differential geometric terms, thiscorrespondence is based on the homotopy equivalence E ∼ = SU c (8) × R , (1.1)which implies that the two group manifolds have the same De Rham cohomology. Wewill show how to extend the algebraic proof of this property by means of equivariantcohomology to the cohomology problem of classifying the e anomalies in N = 8supergravity, and in this way arrive at a very explicit derivation of the non-linear e anomaly from the corresponding linear su (8) anomaly. N = 8 supergravity is a gauge theory, and its first class constraints (associated todiffeomorphisms, local supersymmetry, abelian gauge invariance, and Lorentz invariance)must be taken care of by means of the BRST formalism. This likewise requires the ex-plicit parametrisation of the coset manifold E /SU c (8), such that there are no firstclass constraints associated to SU (8) gauge invariance in the formulation. For the va-lidity of the proof of the E invariance of the theory, one must therefore establish thecompatibility of the latter with the BRST invariance. We will demonstrate in the lastsection that the theory can be quantised in its duality invariant formulation within theBatalin–Vilkovisky formalism, as it does in the ordinary formulation. It is not difficultto see that one can define a consistent E -invariant fermionic gauge fixing-functional(or ‘gauge fermion’). We will explain how the E Noether current can be coupledconsistently to the theory, despite its lack of gauge invariance.In summary, the proof of the duality invariance of the quantised perturbation theoryrelies on establishing the following results: 4. Existence of a local action Σ depending on the physical fields and sources, wellsuited for Feynman rules, and satisfying consistent functional identities associatedto both e current Ward identities and BRST invariance.2. Existence of a regularisation prescription consistent with the quantum action prin-ciple; as dimensional regularisation appears unsuitable in the present formulation,we will employ a Pauli–Villars regulator.3. Existence of a unique non-trivial solution to the E Wess–Zumino consistencycondition associated to the one-loop anomaly.4. Vanishing of the coefficient of the unique anomaly, which implies the absence ofany obstruction towards implementing the full nonlinear E symmetry at eachorder in perturbation theory via an associated e master equation.However, our exposition will not follow these steps in this order, i.e. as a successiveproof of each of these points. Instead, we chose to postpone the discussion of the firstpoint, i.e. the consistency with BRST invariance, to the end and to first discuss othercomponents of the proof that we consider to be more interesting (and perhaps also moreeasily accessible). As one of our main results we separately derive the master equations(or “Zinn–Justin equations”) for both N = 8 supersymmetry and non-linear E . Usingstandard textbook results (see e.g. [18, 19]) readers may then directly deduce from theseany (non-linear) Ward identity of interest if they wish.Our results confirm the expectation that any divergent counterterm must respect thefull non-linear E symmetry. They may thus be taken as further evidence that di-vergences of N = 8 supergravity, if any, will not make their appearance before sevenloops. The strongest evidence so far of the 6-loop finiteness was the absence of loga-rithm in the string effective action threshold [20]. The chiral invariants associated topotential logarithmic divergences at three, five and six loops are only known in the lin-ear approximation [21], and if they are invariant with respect to the linearised dualitytransformations, there is no reason to believe that their non-linear completion would beduality invariant. Indeed, it has recently been exhibited through the study of on-shelltree amplitudes in type II string theory that the 1/2 BPS invariant corresponding to thepotential 3-loop divergence is not E invariant [22, 23]. The same argument appliesto the invariants associated to potential 5 and 6-loop divergences. The manifestly E invariant 7-loop counterterm is the full superspace integral of the supervielbein determi-nant. This is known to vanish for lower N supergravities, suggesting that the first E eight loops. As a corollary of ourresults, we may also point out that N ≤ K possesses a U (1) factor, do exhibit anomalies, and therefore possible divergences neednot respect the non-linear duality invariance.It is important to emphasise that the preservation of the continuous duality sym-metry in perturbation theory is not in contradiction with the string theory expectationthat only its arithmetic subgroup remains a symmetry at the quantum level. Withinsupergravity, we expect that only E ( Z ) will be preserved by non-perturbative cor-rections in exp( − κ − S Instanton ). Although the status of instanton corrections in N = 8supergravity is not clear by any means, we will provide some evidence relying on the clas-sical breaking of the E current conservation in non-trivial gravitational backgrounds,see section 2.4. On the other hand, considering N = 8 supergravity as a limit ℓ s → r i (to be taken → g ≡ ℓ s g s Q i =1 r i , (1.2)while keeping the gravitational coupling constant κ = 8 πg ℓ s fixed, since necessar-ily g → ∞ in this limit. It is therefore clear that the supergravity limit of stringtheory must involve string theory non-perturbative states [24], and thus defines somenon-perturbative completion of the supergravity field theory. If the supergravity limitof the string theory effective action is the effective action in field theory, the latter mustnecessarily include non-perturbative contributions associated to field theory instantons.The E ( Z ) ‘Eisenstein series’ that multiplies the Bel–Robinson square R term in thestring theory effective action is defined in string theory as an expansion in exp( − /g )[8, 9]. This expansion diverges as g in the supergravity limit g → ∞ [9], see also[20] for an explicit resummation of the eight-dimensional SL (2 , Z ) × SL (3 , Z ) invariantthreshold in the supergravity limit. The result of the present paper suggests that if thislimit makes sense in field theory, it should be defined as an expansion in e − /κ , and thatthe perturbative contribution would vanish.The paper is organised as follows. We will first recall the duality invariant formulationof the classical theory defined in [11], and exhibit its equivalence with the conventionalformulation of the theory [1, 2] by means of a Gaussian integration. Then we will recall thedefinition of the E Noether current. In order to deal with the non-linear realisation of6he E symmetry in the symmetric gauge, it will be convenient to define the non-lineartransformations in terms of formal power series in Φ in the adjoint representation. Wederive such formulas in Section 2.5, and we exhibit the commutation relations betweenlocal supersymmetry and the e symmetry. More generally, we show that the BRSToperator commutes with the non-linear e symmetry , cf. (2.91), hence is E invariant.Section 3 exhibits the well definedness and consistency of the formalism (and par-ticular the validity of the quantum action principle), through the explicit computationof the one-loop vector field contribution to the su (8) anomaly. It will therefore provideanswers to both 2 and 4. In this section we discuss the Feynman rules for the vector fieldsin detail, exhibiting the equivalence with the conventional formulation in terms of freephotons. It has been shown in [25] that self-dual form fields contribute to (gravitational)anomalies, just like chiral fermion fields, by means of a formal Fujikawa-like path integralderivation. This result can be understood geometrically from the family’s index theorem[26], and it has been used in [17] to establish the absence of anomalies for the su (8)current Ward identities in N = 8 supergravity. Here we will exploit the duality invariantformulation to provide a full fledged Feynman diagram computation of the vector fieldcontribution which confirms the expected result, and therefore the absence of anomaliesin the theory. In this section we also set up the Pauli–Villars regularisation for the vectorfields, and exhibit its (non-trivial) compatibility with the quantum action principle.Section 4 is also very important: it will provide the definition of the non-linear e Slavnov–Taylor identities for the current Ward identities, and define and solve the Wess–Zumino consistency condition, incidentally answering 3.The last section finally provides an answer to the first point of the above list. Wethere discuss the solution of the Batalin–Vilkovisky master equation in the duality invari-ant formulation, including the coupling to the E Noether current. Using the propertythat the BRST operator commutes with the e symmetry and considering a dualityinvariant gauge-fixing, we are able to define consistent and mutually compatible masterequations for BRST invariance and e symmetry. In this section we also discuss the‘energy Coulomb divergences’ in the one-loop insertions of E currents, which consti-tute a special subtlety of the formalism. We will exhibit that these divergences can beconsistently removed within the Pauli–Villars regularisation.As this paper is rather heavy on formalism, we here briefly summarise our nota-tional conventions for the reader’s convenience. (Curved) space-time indices are µ, ν, ... ,(curved) spatial indices are i , j , k , ... , and space-time Lorentz indices are a, b, c, .... .Indices in the fundamental representation of E are m, n, ... = 1 , ...,
56; when7plit into 28+28 they become m , n , .. and ¯ m , ¯ n , .. . Rigid SU (8) indices are I, J, K, ... such that the E adjoint representation decomposes as ⊕ with generators X IJ KL ≡ δ [ I [ K X J ] L ] , X IJKL ≡ X [ IJKL ] + ε IJKLP QMN X P QMN , etc. Local SU (8)indices are i, j, k ... = 1 , ...,
8, and raising or lowering them corresponds to complex con-jugation. Space-time indices are lowered with the metric g µν , and the tensor densities ε ij k and ε µνρσ are normalised as ε = ε = 1. Finally, we will use the letters S forthe classical action, Σ for the classical action with sources, ghost and antifield termsincluded. While both S and Σ are local, the full quantum effective action Γ is not, butobeys Γ = Σ + O ( ~ ). N = 8 supergravity with off-shell E invariance We start from the usual ADM decomposition of the 4-metric g µν dx µ dx ν = − N dt + h ij ( dx i + N i dt )( dx j + N j dt ) , (2.1)with the lapse N and the shift N i ; h ij is the metric on the spatial slice. The vector fields A m i of the theory appear only with spatial indices, and are labeled by internal indices m, n, ... which transform in a given representation of the internal symmetry group G with maximal compact subgroup K (for N = 8 supergravity G ∼ = E and K ∼ = SU c (8),with the vector fields transforming in the of E ). In comparison with the usualon-shell formalism this implies a doubling of the vector fields, such that the multiplet A m i comprises both the (spatial components of the) electric and their dual magnetic vectorpotentials. To formulate an action we also need the field dependent G -invariant metric G mn on the vector space on which the electromagnetic fields are defined ( i.e. the E invariant metric on R for N = 8 supergravity; this metric is explicitly given in (2.30)below). In addition we need the symplectic invariant Ω mn = − Ω nm = Ω mn , which isalways present, because the generalised duality symmetry is generally a subgroup of asymplectic group acting on the electric and magnetic vector potentials [13] (the group Sp (56 , R ) ⊃ E for N = 8 supergravity). Duality invariance implies the followingrelation for the inverse metric G mn G mn = Ω mp Ω nq G pq , ( G mp G pn = δ mn ) . (2.2) Hence, with our conventions Ω mp Ω pn = − δ nm . J mn ≡ G mp Ω pn ⇒ J mp J pn = − δ mn . (2.3)Note that J mn depends on the scalar fields via the metric G mn . The maximal compactsubgroup K can be characterised as the maximal subgroup in G which commutes with J mn (˚Φ) (for some background value ˚Φ of the scalar fields).After these preparations we can write down the part of the action containing thevector fields S vec = 12 Z d x (cid:16)
12 Ω mn ε ij k (cid:0) ∂ A m i + N l F m i l (cid:1) F n j k − N √ h G mn h i k h j l F m ij F n kl − N √ hh i k h j l F m ij W kl m − N √ h G mn h i k h j l W ij m W kl n (cid:17) . (2.4)Here W ij m is a bilinear function of the fermion fields, which will be discussed in moredetail shortly (see (2.36) below). We also consider the W term which define the non-manifestly diffeomorphism covariant quartic terms in the fermions. For quantisation, theabove action must be supplemented by further terms depending on the ghost fields aswell as the anti-fields; this will be discussed in more detail below.As shown in [11], the main advantage of the above reformulation is that it incorporatesboth the electric and the dual magnetic vector potentials off-shell , at the expense ofmanifest space-time diffeomorphism invariance. In particular, the equation of motion ofthe 56 vector fields A m i can be expressed as a twisted self-duality constraint [1] for thesupercovariant field strength ˆ F mµν (see [11] for further details)ˆ F mµν = − √ - g ε µν σρ J mn ˆ F nσρ , (2.5)where the tensor J takes the place of an imaginary unit. We briefly explain this procedureand why the time-components A m of the vector fields naturally enter this equation,although they are absent in the original Lagrangian (2.4). The variation of the actionfunctional S vec (2.4) with respect to the 56 vector fields A m i leads to the second orderequation of motion ε ij k ∂ j E m k = 0 , (2.6)with the abbreviation E m i ≡ ∂ A m i + N j F m ij − N √ h h ij ε j kl (cid:16) J mn F n kl + Ω mn W kl n (cid:17) . (2.7) Do not confuse the equation of motion function E m i with the electric potential E m i introduced in[10, 11]. E mk dx k is closed. On anycontractible open set of the d = 4 space-time manifold, every closed form is exact byPoincar´e’s lemma, which implies the existence of a zero-form A m satisfying E m i = ∂ i A m . (2.8)It is straightforward to verify that this equation of motion is completely equivalent tothe twisted self-duality constraint of equation (2.5). Furthermore, only the identificationof the zero-form with the time-component A m gives rise to an equation of motion that isdiffeomorphism covariant in the usual sense.Before we prove that the action functional (2.4) and the usual second order form ofthe action are equivalent, and related by functional integration, we briefly explain therealisation of the diffeomorphism algebra on the vector fields. To this aim we recall thatthe Lie derivative on the vector field in the covariant formulation can be rewritten as δA mµ = ∂ µ ξ ν A mν + ξ ν ∂ ν A mµ = ∂ µ ( ξ ν A mν ) + ξ ν F mνµ . (2.9)Considering the vector fields A m as abelian connections, the geometrical action of diffeo-morphism is defined via the horizontal lift of the vector ξ µ to the principle bundle, andis modified by a gauge transformation. We will consider this covariant (or ‘horizontal’)diffeomorphism δA mµ = ξ ν F mνµ . (2.10)Splitting indices into time and space indices, we get δA m i = ξ F m i + ξ j F m ji . (2.11)The recipe for obtaining the correct formula in the present formulation then consistssimply in replacing F m i → ∂ A m i − E m i (2.12)everywhere according to (2.7), such that (2.11) becomes δ ξ A m i ≡ ξ µ ∂ µ A m i − ξ j ∂ i A m j − ξ E m i . (2.13)We note that the recipe (2.12) also yields the correct formulas for all other transformationsin the manifestly duality invariant formalism, including the modified supersymmetrytransformations and the BRST transformations of the ghosts.10he non-standard representation of the diffeomorphism algebra (2.13) on the vectorfields is consistent, because it closes off-shell up to a gauge transformation with parameterΛ m , which cannot be separated from the diffeomorphism action: (cid:2) δ ξ , δ ξ (cid:3) A m i = δ [ ξ ,ξ ] A m i + ∂ i Λ m with Λ m = ξ i ξ j F m ij + ( ξ ξ j − ξ ξ j )( ∂ A m j − E m j ) . (2.14)The gauge transformation Λ m can be obtained from the one that would appear in thecovariant formulation by the substitution (2.12).To sum up: although the equations of motion are covariant under the diffeomorphismaction in both formulations of maximal supergravity, the representations of the diffeo-morphism algebra on the vector fields do not coincide off-shell . Agreement can a priori be achieved only on-shell , if we impose the equations of motion in their first order form(2.8) with the introduction of the time-component of the 56 vector fields. Nevertheless,the two formulations are also formally equivalent at the quantum level, as we are goingto see. To establish the link with the manifestly diffeomorphism covariant formalism, we mustin a first step decompose the electromagnetic fields into Darboux components associatedto the symplectic form Ω mn = Ω ¯ m ¯ n = 0 Ω m ¯ n = − Ω ¯ n m = δ m ¯ n (2.15)where the indices m, n, ... are split into pairs ( m , ¯ m ) each running over half the range of m, n . For the vector fields this entails the split A m i → ( A m i , A ¯ m i ) (2.16)into electric and magnetic vector potentials. With the above split, the manifest off-shell E symmetry will be lost after the Gaussian integration to be performed below, andis thus reduced to the on-shell symmetry of the standard version. Extending (2.4) by agauge-fixing term, the action functional becomes S vec = 12 Z d x (cid:16)
12 Ω mn ε ij k (cid:0) ∂ A m i + N l F m i l (cid:1) F n j k − N √ h G mn h i k h j l F m ij F n kl − N √ hh i k h j l F m ij W kl m − N √ h G mn h i k h j l W ij m W kl n + 2 b m ∂ i A m i (cid:17) . (2.17)11ums over repeated indices are understood even when they are both down, which onlyreflects the property that the corresponding terms are not invariant with respect todiffeomorphisms. Performing the split, and up to an irrelevant boundary term, we arriveat the following Lagrange density S vec = 12 Z d x (cid:18)(cid:16) δ m ¯ n ε ij k (cid:0) ∂ A m i + N l F m i l (cid:1) − N √ h G m ¯ n h i k h j l F m ij − N √ h h i k h j l W ij ¯ n (cid:17) F ¯ n kl − N √ h G ¯ m ¯ n h i k h j l F ¯ m ij F ¯ n kl − N √ h G mn h i k h j l F m ij F n kl − N √ h h i k h j l F m ij W kl m − N √ h G mn h i k h j l W ij m W kl n + 2 b m ∂ i A m i + 2 b ¯ m ∂ i A ¯ m i (cid:19) . (2.18)Integrating out the auxiliary field b ¯ m enforces the constraint ∂ i A ¯ m i = 0, and the La-grangian only depends on A ¯ m i through F ¯ m ij = ∂ i A ¯ m j − ∂ j A ¯ m i (note that this is the caseeven when considering the ghost field terms that we neglect in this discussion). One hasthen an isomorphism between the square integrable fields A ¯ m i satisfying ∂ i A ¯ m i = 0, andthe square integrable fields Π i ¯ m satisfying the same constraint ∂ i Π i ¯ m = 0, throughΠ i ¯ m = ε ij k ∂ j A ¯ m k , A ¯ m i = − [ ∂ l ∂ l ] − ε ij k ∂ j Π k ¯ m , (2.19)where repeated indices are summed (and appropriate boundary conditions assumed).This change of variables leads to a non-trivial functional Jacobian, but the latter doesnot depend on the fields and can therefore be disregarded. Introducing a Lagrangemultiplier A m for the constraint ∂ i Π i ¯ m = 0, one has the action S vec = 12 Z d x (cid:18)(cid:16) δ m ¯ n (cid:0) ∂ A m i − ∂ i A m + N l F m i l (cid:1) − N √ h ε i lh h l j h hk (cid:0) G m ¯ n F m j k + W j k ¯ n (cid:1)(cid:17) Π i ¯ n − N √ h G ¯ m ¯ n h ij Π i ¯ m Π j ¯ n − N √ h G mn h i k h j l F m ij F n kl − N √ h h i k h j l F m ij W kl m − N √ h G mn h i k h j l W ij m W kl n + 2 b m ∂ i A m i (cid:19) , (2.20)where we normalised A m such that it can be identified as the time component of thevector field, and F m i = ∂ A m i − ∂ i A m . (2.21) Note that this is only true in the specific metric independent Coulomb gauge we used, in whichthe ghosts decouple. For a metric dependent gauge, the functional Jacobian would depend non-triviallyon the metric, but this field dependence would be exactly compensated by the functional determinantgenerated by the Gaussian integration over the ghosts ¯ c ¯ m and c ¯ m , as is ensured by BRST invariance. ¯ m i define the momentum conjugate to the vector fields A m i . One then sees that(2.19) actually corresponds to a second class constraint , as one would expect in a firstorder formalism. We also emphasise that, when the equations of motion are satisfied, theLagrange multiplier field A m in the path integral can be identified with the correspondingcomponent of A m appearing in (2.8), which is the classical field resulting from rewritinga given expression E m i as a curl.One can now integrate the momentum variables Π i ¯ m through formal Gaussian inte-gration, the remaining action is S vec = 12 Z d x (cid:18) √ hN δ m ¯ m δ n ¯ n H ¯ m ¯ n h ij (cid:0) F m i + N k F m i k (cid:1)(cid:0) F n j + N l F n j l (cid:1) − N √ h (cid:0) G mn − G m ¯ m H ¯ m ¯ n G ¯ nn (cid:1) h i k h j l F m ij F n kl − ε ij k δ m ¯ m H ¯ m ¯ n G ¯ nn (cid:0) F m i + N l F m i l (cid:1) F n j k − ε ij k δ m ¯ m H ¯ m ¯ n (cid:0) F m i + N l F m i l (cid:1) W kl ¯ n + H ¯ n ¯ m G ¯ mm N √ h h i k h j l F m ij W kl ¯ n − N √ h h i k h j l F m ij W kl m + 12 N √ h H ¯ m ¯ n h i k h j l W ij ¯ m W kl ¯ n − N √ h G mn h i k h j l W ij m W kl n + 2 b m ∂ i A m i (cid:19) , (2.22)where H ¯ m ¯ n is the inverse of G ¯ m ¯ n (not to be confused with the component G ¯ m ¯ n of theinverse metric G mn ). We will discuss the functional determinant afterward. First notethat, by (2.2), duality invariance implies G mn − G m ¯ m H ¯ m ¯ n G ¯ nn = δ m ¯ m δ n ¯ n H ¯ m ¯ n , δ m ¯ m H ¯ m ¯ n G ¯ nn = δ n ¯ m H ¯ m ¯ n G ¯ nm , (2.23)and therefore the bosonic component is manifestly diffeomorphism invariant S vec = 14 Z d x (cid:18) −√− gδ m ¯ m δ n ¯ n H ¯ m ¯ n g µσ g νρ F m µν F n σρ − ε µνσρ δ m ¯ m H ¯ m ¯ n G ¯ nn F m µν F n σρ + O ( W ) (cid:19) . (2.24)The formal Gaussian integration over the momentum variables Π i ¯ m also produces afunctional determinantDet − (cid:20) N √ h G ¯ m ¯ n h ij δ ( x − y ) (cid:21) = Y x (cid:16) det − [ G ¯ m ¯ n ] N − h (cid:17) , (2.25)which defines a one-loop local divergence quartic in the cutoff ∼ Λ . This determinantdefines in particular the modification of the diffeomorphism invariant measure of themetric field from the duality invariant formulation to the conventional one [27], andrespectively for the E invariant scalar field measure. This kind of volume divergenceis in fact a general property of (super)gravity theories [28].13 .3 N = 8 supergravity The discussion was rather general so far, and we now turn to the specific case of maximal N = 8 supergravity, where the formalism developed in the foregoing section leads to aformulation of the theory with manifest and off-shell E invariance. Here we show thatthe formalism reproduces the vector Lagrangian as well as the couplings of the vectorfields to the fermions and the scalar field dependent quartic fermionic terms in the formgiven in [2] (the remaining quartic terms in the Lagrangian are manifestly E invariant).In this case the choice of Darboux coordinates amounts to decomposing the 28 complexvector fields A IJ i into imaginary and real (or ‘electric’ and ‘magnetic’) components A m i ˆ= Im[ A i IJ ] , A ¯ m i ˆ= Re[ A i IJ ] . (2.26)For the coset representative E /SU c (8), this corresponds to the passage from the SU (8)basis in which V ˆ= u ijIJ v ijKL v klIJ u klKL (2.27)to an SL (8 , R ) basis in which e V = √ √ − i √ i √ V √ i √ √ − i √ , (2.28)or, written out in components, e V ˆ= Re (cid:0) u ijIJ + v ijIJ (cid:1) Im (cid:0) − u ij KL + v ijKL (cid:1) Im (cid:0) u klIJ + v klIJ (cid:1) Re (cid:0) u klKL − v klKL (cid:1) . (2.29) With our usual convention A IJ i = ( A i IJ ) ∗ . Recall that the standard formulation of N = 8 super-gravity has 28 real vectors, for which there is no need to distinguish between upper and lower indices. This transformation is analogous to the M¨obius transformation mapping the unit (Poincar´e) disk tothe upper half plane, and relating SU (1 ,
1) to SL (2 , R ). G = e V T e V ˆ= (cid:0) u ijIJ + v ijIJ (cid:1)(cid:0) u ijKL + v ijKL (cid:1) (cid:2)(cid:0) u ijIJ + v ijIJ (cid:1) u ijKL (cid:3) (cid:2) u ijIJ (cid:0) u ijKL + v ijKL (cid:1)(cid:3) (cid:0) u ijIJ − v ijIJ (cid:1)(cid:0) u ijKL − v ijKL (cid:1) = (cid:16) Re (cid:2) S − (cid:3)(cid:17) − (cid:16) Re (cid:2) S − (cid:3)(cid:17) − Im (cid:2) S (cid:3) Im (cid:2) S (cid:3)(cid:16) Re (cid:2) S − (cid:3)(cid:17) − (cid:16) Re (cid:2) S − (cid:3)(cid:17) − + Im (cid:2) S (cid:3)(cid:16) Re (cid:2) S − (cid:3)(cid:17) − Im (cid:2) S (cid:3) (2.30)where we used Im (cid:2)(cid:0) u ijIJ + v ijIJ (cid:1)(cid:0) u ijKL + v ijKL (cid:1)(cid:3) = 0 , (2.31)to compute the first matrix, and where the symmetric matrix S is defined such that (cid:0) u ijIJ + v ijIJ (cid:1) S IJ,KL = u ij KL . (2.32)To prove the equality of the two matrices in (2.30), one uses again (2.31) to show that2 Im (cid:2) u ijIJ (cid:0) u ijKL + v ijKL (cid:1)(cid:3) = Im [2 S ] IJ,P Q (cid:0) u ijP Q + v ijP Q (cid:1)(cid:0) u ijKL + v ijKL (cid:1) , (2.33)and Re (cid:2) S − (cid:3) IJ,P Q (cid:0) u ijP Q + v ijP Q (cid:1)(cid:0) u ijKL + v ijKL (cid:1) == Re (cid:2)(cid:0) u ijIJ − v ijIJ (cid:1)(cid:0) u ijKL + v ijKL (cid:1)(cid:3) = δ KLIJ , (2.34)which establishes the equality for the first column in (2.30). The equality in the secondcolumn then follows by using the property that the matrix is symmetric and symplectic.Identifying H ¯ m ¯ n ˆ= Re (cid:2) S − ] IJ,KL , H ¯ m ¯ n G ¯ nm ˆ= Im (cid:2) S ] IJ,KL , (2.35)one recovers the conventional form of the action (2.24) as given in [2].To investigate the couplings of the vectors to the fermions, we recall from [11] thatthe fermionic bilinears W ij m in (2.18) are determined by W IJ ij = e a i e b j (cid:16) u ijIJ O + abij + v ijIJ O − ab ij (cid:17) , (2.36) Readers should keep in mind the different meanings of the letters i , j , ... and i, j, ... in this and otherequations of this section (with apologies from the authors for the proliferation of different fonts!). W ij m ˆ= Im[ W IJ ij ] , W ij ¯ m ˆ= Re[ W IJ ij ] . (2.37)Here, O + abij and its complex conjugate O − ab ij are the fermionic bilinears defined in [2] O + abij = ¯ ψ ic γ [ c γ ab γ d ] ψ jd −
14 ¯ ψ kc γ ab γ c χ ijk − ε ijklmnpq ¯ χ klm γ ab χ npq . (2.38)modulo normalisations (our coefficients here are chosen to agree with [11]). By complexself-duality they satisfy O + abij = i ε abcd O + cdij , O − ab ij = − i ε abcd O − cd ij . (2.39)These relations allow us to express the ‘timelike’ components W IJ i in terms of the purelyspatial components W IJ ij , and thereby to recover the full fermionic Lagrangian of thecovariant formulation in terms of just the purely spatial components W IJ ij .After these preparations we return to the Lagrangian (2.22), from which we read offthe couplings of the vector fields to the fermions ε ij k Im (cid:2) F IJ i + N l F IJ i l (cid:3) Re (cid:2) S − (cid:3) IJ,KL Re (cid:2) W KL kl (cid:3) + N √ h h i k h j l Im [ F IJ ij ] (cid:16) Im (cid:2) W IJ kl (cid:3) − Im (cid:2) S (cid:3) IJ,KL Re (cid:2) W KL kl (cid:3)(cid:17) . (2.40)Using the properties of S IJ,KL one computes thatRe (cid:2) S − (cid:3) IJ,KL Re (cid:2) u ijKL O + abij + v ijKL O − ab ij (cid:3) = Re (cid:2) (2 S − ) IJ,KL (cid:0) u ijKL + v ijKL (cid:1) O − ab ij (cid:3) + Im[2 S (cid:3) IJ,KL Im (cid:2)(cid:0) u ijKL + v ijKL (cid:1) O + abij (cid:3) = Re (cid:2)(cid:0) u ijIJ − v ijIJ (cid:1) O + abij (cid:3) + Im[2 S (cid:3) IJ,KL Im (cid:2)(cid:0) u ijKL + v ijKL (cid:1) O + abij (cid:3) . (2.41)Invoking the complex self-duality of O + abij one recovers the manifest diffeomorphism in-variant coupling e e a µ e b ν Im[ F IJµν ] (cid:16) Im (cid:2)(cid:0) u ijIJ − v ijIJ (cid:1) O + abij (cid:3) − Im (cid:2) S (cid:3) IJ,KL Re (cid:2)(cid:0) u ijKL + v ijKL (cid:1) O + abij (cid:3)(cid:17) = e e a µ e b ν Im[ F IJµν ] Re (cid:2) S − (cid:3) IJ,KL Im (cid:2)(cid:0) u ijKL + v ijKL (cid:1) O + abij (cid:3) = e e a µ e b ν Im [ F IJµν ] Im (cid:2) S IJ,KL ( u − ) KLij O + abij (cid:3) . (2.42)Next we consider the quartic terms in the fermions. They read12 N √ h H ¯ m ¯ n h i k h j l W ij ¯ m W kl ¯ n (2.43)= 12 e h i k h j l e a i e b j e c k e d l Re (cid:2)(cid:0) u ijIJ + v ijIJ (cid:1) O + abij (cid:3) Re (cid:2) S − (cid:3) IJ,KL Re (cid:2)(cid:0) u klKL + v klKL (cid:1) O + cdkl (cid:3) − N √ h G mn h i k h j l W ij m W kl n = − e h i k h j l e a i e b j e c k e d l O − ab ij O + cdij , (2.44)where in the last equation the dependence of W ij m on scalar fields in (2.36) is eliminatedthrough the contraction with G mn . Using (2.34) andRe (cid:2) S − (cid:3) IJ,KL (cid:0) u ijIJ + v ijIJ (cid:1)(cid:0) u klKL + v klKL (cid:1) = ( u − ) IJ ij (cid:16) S IJ,KL + u pqIJ v pqIJ (cid:17) ( u − ) KLkl , (2.45)we obtain12 N √ h H ¯ m ¯ n h i k h j l W ij ¯ m W kl ¯ n = 14 e h i k h j l e a i e b j e c k e d l (cid:18) O − ab ij O + cdij + 12 h O + abij ( u − ) IJ ij (cid:16) S IJ,KL + u pqIJ v pqKL (cid:17) ( u − ) KLkl O + cdkl + c.c. i(cid:19) . (2.46)The first term in parentheses cancels the (manifestly E invariant) expression (2.44)— as must be the case because any Lorentz invariant extension of type O + ij O − ij mustnecessarily vanish because of the opposite duality phases. Altogether we have shownthat the relevant part of the Lagrangian agrees with the corresponding one from [2]which reads, in the present notations and conventions L VF = e (cid:18) − (cid:2) S − (cid:3) IJ,KL
Im[ F IJµν ] − Im[ F µν KL ] − − ie aµ e bν O + abij ( u − ) IJ ij S IJ,KL
Im[ F KLµν ]+ 18 O + abij ( u − ) IJ ij (cid:0) S IJ,KL + u pqIJ v pqKL (cid:1) ( u − ) KLkl O + abkl + c.c. (cid:19) . (2.47)Because the vector fields only appear through the field strength F IJ ij in the BRSTtransformations of the fields, the Gaussian integration can be carried out for the completeBatalin–Vilkovisky action which will be discussed in the last section. The validity of theBRST master equation all along the process of carrying out the Gaussian path integralsto pass from one formalism to the other ensures the validity of the above formal argument,by fixing all possible ambiguities associated to the regularisation scheme. The conventions of [2] are recovered with the identifications A [2] IJµ ≡ √ A IJµ ], ψ [2] iµ ≡ √ ψ iµ , χ [2] ijk ≡ χ ijk . The charge conjugation matrix of [2] is related to ours by, C [2] ≡ i C , such that, forinstance O + abij = − i √ O [2] + abij and the complex self-duality convention is reversed. .4 The classical E current A main advantage of the present formulation is that the E current can be derived asa bona fide Noether current [11]. It consists of two pieces J µ = J (1) µ + J (2) µ . (2.48)Here the first piece J (1) does not depend on the vector fields and has the standard form asin any σ -model with fermions (see also [29]). The more important piece for our discussionhere is the second term J (2) , which depends on the 56 electric and magnetic vector fieldsand is of Chern-Simons type; this part of the current does not exist off-shell in the usualformulation [13], where it would be given by a non-local expression on-shell. The current J µ is an axial vector which defines the current three-form J = J (1) + J (2) ≡ ε µνρσ J µ dx ν ∧ dx ρ ∧ dx σ , (2.49)in terms of which the classical current conservation simply reads dJ = 0.Following the standard Noether procedure, the E -current J µ was computed in [11]by an infinitesimal displacement along Λ ∈ e . Under the SU c (8) subgroup of E , J µ decomposes into 63 components ( J µ ) I K and 70 components ( J µ ) IJKL : J µ (Λ) = ( J µ ) I K Λ I K + ( J µ ) IJKL Λ IJKL . (2.50)The easiest way to write the first piece J (1) is in terms of matrices: J (1) µ (Λ) = −
124 tr (cid:0) V − R µ V Λ (cid:1) . (2.51)Here, we are using the matrix form of the scalar coset V (2.27) and the matrices Λ and R µ in e that are defined as usualΛ ˆ= δ [ M [ I Λ J ] N ] Λ IJOP Λ KLMN − δ [ K [ O Λ P ] L ] ! , R µ ˆ= − δ [ m [ i R µ n ] j ] R µijop R µ klmn δ [ k [ o R µ l ] p ] ! . (2.52)The components R µ ij and R µijkl = ε ijklmnop R µ mnop have the form R µ ij ≡ iε µνσρ (cid:16) ¯ ψ iν γ σ ψ ρj − δ ij ¯ ψ kν γ σ ψ ρk (cid:17) + √− g (cid:16) ¯ χ ikl γ µ χ jkl − δ ij ¯ χ klm γ µ χ klm (cid:17) R µijkl ≡ √− g ˆ A µijkl − i ε µνσρ (cid:16) ¯ χ [ ijk γ σρ ψ νl ] − ε ijklmnop ¯ χ mno γ σρ ψ ν p (cid:17) , (2.53)18here ˆ A ijklµ is the supercovariant derivative of the scalar cosetˆ A ijklµ ≡ u ijIJ ∂ µ v klIJ − v ijIJ ∂ µ u klIJ − ¯ ψ [ iµ χ jkl ] − ε ijklmnop ¯ ψ µ m χ nop . (2.54)Since the second part J (2) µ of the current contains the 56 vector fields A m i , it necessarilylacks manifest covariance. With spatial indices i , j = 1 , ,
3, it has the form: J (2) k = − ε ij k A m i (cid:0) ∂ A n j − N l F n l j (cid:1) Ω pm Λ np + √− gh j k h i l A n i (cid:16) G pm F m j l + W j l p (cid:17) Λ np J (2) = 14 ε ij k A m i F n j k Ω pm Λ np . (2.55)Like for J (1) in eq. (2.51), the independent components of J (2) are provided by the 133independent components of Λ within the 56 × np . For instance, the time-likecomponents in the su (8) basis are given by J (2) I J = i ε ij k (cid:16) A IK i F j k JK + A i JK F JK j k − δ IJ (cid:0) A KL i F j k KL + A i KL F KL j k (cid:1)(cid:17) J (2) IJKL = − i ε ij k (cid:16) A [ IJ i F KL ] j k − ε IJKLMNOP A i MN F j k OP (cid:17) (2.56)The space-like components admit a similar form that can straightforwardly be obtainedfrom (2.55). However, the explicit expressions are rather complicated, and would notprovide any further insight in this discussion.As a next step, we want to rewrite the vector field part (2.55) in a way that allowsa direct comparison with the current constructed in [13]. A simple computation revealsthe identity [11] J (2) µ = (cid:16) ε µνρσ F mνρ A nσ + 12 δ µ k ε ij k ∂ i (cid:0) A m j A n (cid:1) − δ µ k ε ij k A n i (cid:0) E mj − ∂ j A m (cid:1) (cid:17) Λ np Ω pm . (2.57)where a spurious dependence in the component A m has been introduced, such that allthe A m dependent terms add up to zero. This form of the current decomposes into threeterms:1. The first term in J (2) µ is a Chern–Simons three-form. It is manifestly diffeomor-phism covariant in the usual sense.2. The second term is a ‘curl’, and thus does not affect current conservation. Note that the normalisation of the vector fields here differs from the one in [11] by a factor 2. Although the Noether procedure only determines the current J up to a ‘curl’, this term cannot beavoided in (2.57), because A m is not a fundamental field in the duality invariant formulation.
19. The third term is proportional to the integrated equation of motion of the vectorfield (2.8) with E m k defined in (2.7).Let us now recall the procedure of [13] for obtaining the conserved current associatedto the duality invariance of the equations of motion. The idea is to supplement themanifestly covariant part of the current J (1) µ (Λ) by a further term J (2) µ GZ in such a waythat the complete current (which we will henceforth refer to as the Gaillard–Zuminocurrent ) is conserved ∂ µ (cid:16) J (1) µ (Λ) + J (2) µ GZ (Λ) (cid:17) = 0 , (2.58)if the equations of motion are enforced. Therefore, it is clear that J (2) µ GZ (Λ) is only definedup to a curl, and modulo terms proportional to the equations of motion. From thecomplete Noether current (2.57), we thus deduce J (2) µ GZ (Λ) = 14 ε µνρσ A mν F nρσ Λ mp Ω pn . (2.59)This Chern-Simons three-form exhibits manifest diffeomorphism covariance and it de-pends only on the 56 vector fields, unlike the current (2.55). The explicit form of J (2) µ GZ (Λ)as given in [29] is indeed equivalent to the decomposition of (2.59) in a Darboux basisfor the 56 electromagnetic fields A mµ into A m µ and A ¯ m µ (2.26). The usual covariant formu-lation of [1] contains only the 28 vector fields A m µ off-shell, whereas the dual fields A ¯ m µ are non-local functionals of all the other fields satisfying the equations of motion.In a non-trivial background, the Chern–Simons like component (2.59) is not globallydefined in general. For a non-trivial connection, one must introduce a reference connec-tion ˚ A m , such that the one-form A m − ˚ A m is gauge invariant (and so globally defined), and˚ F m represents a non-trivial cohomology class in the given background. The backgrounddependent extension of (2.59) is given from the Cartan homotopy formula as J (2) GZ (Λ) = − (cid:0) A m − ˚ A m (cid:1) ∧ (cid:0) F n + ˚ F n (cid:1) Λ mp Ω pn . (2.60)By definition of the Cartan homotopy formula, it follows that the globally defined E current then suffers from a classical anomaly dJ (Λ) = 12 ˚ F m ∧ ˚ F n Λ mp Ω pn . (2.61)Even without a general classification of the instanton backgrounds that may occur in N = 8 supergravity, this result by itself already shows how the continuous E symmetry20an be broken in a non-trivial background. When the gravity background is such thatthere is a non-trivial cohomology group H ( Z ) ∧ H ( Z ) → H ( Z ) (2.62)and both ˚ F m and ˚ F m ∧ ˚ F n define non-trivial cohomology classes in H ( Z ) and H ( Z ),respectively, the e Ward identities will be broken in the background. In this casethe 1PI generating functional Γ evaluated on E transformed fields varies as (withappropriate normalisation) Γ[ g ] = Γ[ ] + 2 π Ω mp g pn q m q n , (2.63)with integer charges q m = π R ˚ F m . This ‘classical anomaly’ is not affected by theLegendre transform, and the generating functional W of connected diagrams transformsas Γ with respect to E transformations. As a consequence, the generating functional Z = exp[ iW ] will no longer be invariant under continuous E transformations, butonly with respect to transformations g ∈ E ( Z ). Such backgrounds appear for examplein the classification of [30] as C P and S × S type spaces. One might therefore anticipatethat E gets broken to a discrete subgroup when the path integral also includes a sumover such instanton contributions.However, we should caution readers that the status of ‘instanton solutions’ in N = 8supergravity is not clear by any means. Unlike the usual self-duality constraint (whichrequires a Euclidean metric) the twisted self-duality constraint (2.5) contains an addi-tional ‘imaginary unit’ J , and any E invariant Euclidean theory must therefore in-volve scalar fields parameterising a pseudo-Riemmanian symmetric space E /SU ∗ (8) c or E /SL (8) c such that the representation is real. It is thus doubtful whether a‘Wick rotation’ really makes sense, or whether one should instead look for real saddlepoints in a Lorentzian path integral. The second approach would still require to definethe action in a non-trivial non-globally hyperbolic background. It is rather straightfor-ward to modify the classical action similarly as (2.61) such that the equations of motionare not modified, and such that the Lagrangian density is gauge invariant and trans-forms covariantly with respect to spatial diffeomorphisms. Nonetheless, this Lagrangiandensity transforms covariantly with respect to D = 4 diffeomorphisms only up to termslinear in the equations of motion. This is not the case for dyonic solutions in an asymptotically Minkowskian space-time: even though˚ F m is non-trivial for such solutions of Maxwell’s equations, the product ˚ F m ∧ ˚ F n is trivial. A positive definite ‘kinetic term’ could then be recovered by decomposing E /SU ∗ (8) c ∼ = R ∗ + × E /Sp (4) × R (respectively E /SL (8) c ∼ = SL (8) /SO (8) × R ), and dualising 27 axionic scalars(respectively 35) into 2-forms, in analogy with the type IIB D-instantons [31]. .5 Transformations in the symmetric gauge Under the combined action of local SU (8) and rigid E the 56-bein transforms as V ( x ) → V ′ ( x ) = h ( x ) V ( x ) g − , h ( x ) ∈ SU (8) , g ∈ E . (2.64)For the classical theory, one has the option of either keeping the local SU (8) with lin-early realised E , or fixing a gauge for the local SU (8), retaining only the 70 physicalscalar fields, whereby the rigid E becomes realised non-linearly. However, we are hereconcerned with the quantised theory , where the compatibility and mutual consistency ofthese two descriptions is not immediately evident. Indeed, the SU (8) gauge-invariantformulation of the theory may appear not to be well defined at the quantum level be-cause the gauge su (8) Ward identity is anomalous at one loop due to the contributionfrom the spin- and spin- fermions [15]. On the other hand, as shown by Marcus [17],the rigid SU (8) ⊂ E left after gauge-fixing is non-anomalous , implying the absence ofanomalies for the rigid su (8) current Ward identities in the gauge-fixed formulation of thetheory. This is because the rigid su (8) symmetry acts linearly on the vector fields, whosechiral nature under SU (8) implies that there is an extra contribution to the anomalyfrom the vector fields which precisely compensates the contribution from the fermionfields. From the path integral perspective, the main difference between those two kindsof su (8) Ward identities can be viewed as resulting from a redefinition of the 56 vectorfields as A IJ i → ˇ A ij i ≡ u ijIJ A IJ i + v ij IJ A i IJ (2.65)that is, to the passage between objects transforming under rigid E and local SU (8),respectively. According to the family’s index theorem this change of variables does notleave the path integral measure for the vector fields invariant (because the action of E on the vector fields is chiral), and thus generates an anomaly. The results of [15] and[17] are therefore perfectly consistent with each other, because the associated sets ofWard identities cannot be both free of anomalies. In the following section we will presentan explicit Feynman diagram computation of the vector field contribution to the su (8)anomaly. This explicit computation was not given in [17], which relied on the formulationof N = 8 supergravity with only on-shell E and on arguments based on the family’sindex theorem.We emphasize that the su (8) anomaly for the local SU (8) gauge invariance is some-what artificial because it can be compensated by the addition of an appropriate Wess–Zumino term for the SU c (8) components of the E /SU c (8) vielbein V ( x ) [16]. This22rocedure replaces the gauge su (8) anomaly by a corresponding anomaly of the su (8)current Ward identities (with the same coefficient). While restoring local SU (8), thelatter by itself would break the rigid E symmetry, but for N = 8 supergravity thisanomaly is cancelled in turn by the contribution from the vector fields! Consequentlywe anticipate that our results can be re-obtained for the version of N = 8 supergravitywith local SU (8) and linearly realised E such that both descriptions of the quantisedtheory are consistent, but a detailed verification of this claim remains to be done.In order to set up the perturbative expansion of the quantised theory, we will nev-ertheless parameterise the symmetric space E /SU c (8) with explicit coordinates. Wewill consider as coordinates the scalar fields φ ijkl in the of SU (8), which parameterisea representative V ( x ) in the symmetric gauge , viz. V ( x ) ≡ exp Φ( x ) ˆ= exp φ ijkl ( x ) φ ijkl ( x ) 0 ! (2.66)with Φ ∈ e ⊖ su (8) and the standard convention φ ijkl = ( φ ijkl ) ∗ , (having fixed the SU (8) gauge there is no need any more to distinguish between SU (8) and E indices).After this gauge choice we are left with a rigid E symmetry, whose SU (8) subgroupis realised linearly. The remaining rigid E transformations require field dependent com-pensating SU (8) rotation in order to maintain the chosen gauge (2.66), and are thereforerealised non-linearly on the 70 scalar fields. In this section, we work out these non-lineartransformations in more detail to set the stage for the implementation of the full nonlinear E symmetry at the quantum level. For this purpose we adopt the following notationalconvention: for any two Lie algebra elements X and Y and any function f ( X ) that isanalytic at X = 0, we abbreviate the adjoint action of f ( X ) on Y by f ( X ) ∗ Y ≡ f (ad( X ))( Y ) (2.67)Here, the right hand side is to be evaluated term by term in the Taylor expansion, wherethe n -th order term (ad) n ( X )( Y ) is the n -fold commutator [ X, [ X, ... [ X, Y ] ... ]]. It is easyto check that f ( X ) ∗ g ( X ) ∗ Y = ( f g )( X ) ∗ Y . For the evaluation of the non-lineartransformations the main tool is the Baker–Campbell–Hausdorf formulaexp( X ) exp( Y ) = exp (cid:16) X + Td( X ) ∗ Y + O ( Y ) (cid:17) = exp (cid:16) Y + Td( − Y ) ∗ X + O ( X ) (cid:17) (2.68)with Td( x ) ≡ x − e − x = 1 + 12 x + O ( x ) . (2.69)23ccordingly we now consider an E transformation with parameter Λ in the of SU (8), viz. g ≡ exp Λ ˆ= ijkl ( x )Λ ijkl ( x ) 0 ! . (2.70)Then, by use of (2.68),exp Φ exp( − Λ) = exp (cid:18) Φ − Φ / / ∗ Λ − (cid:2) Φ , Λ (cid:3) + O (Λ ) (cid:19) (2.71)where the odd piece is [Φ , Λ] ≡ Φ ∗ Λ ∈ su (8). Now we must choose the compensating SU (8) transformation from the left so as to cancel the third term in the exponential.Using (2.71) and the second line of (2.68), we obtainexp (cid:0) Φ + δ Φ (cid:1) = exp (cid:16) tanh(Φ / ∗ Λ (cid:17) exp Φ exp( − Λ)= exp (cid:18) Φ − Φtanh Φ ∗ Λ + O (Λ ) (cid:19) (2.72)or δ e Φ ≡ δ e (Λ)Φ = − Φtanh Φ ∗ Λ . (2.73)In the same way one computes the supersymmetry transformation of the scalar fields andthe non-linear modifications due to the compensating SU (8) rotations. Infinitesimally,local supersymmetry acts on the scalar fields by a shift along the non-compact directionswith parameter X = ǫ [ i χ jkl ] + ε ijklmnpq ǫ m χ npq ǫ [ i χ jkl ] + ε ijklmnpq ǫ m χ npq (2.74)Observing that this shift acts on V from the left (unlike the E transformation in (2.64)which acts from the right), one computes that, again using (2.68),exp (cid:16) X + tanh(Φ / ∗ X (cid:17) exp Φ = exp (cid:18) Φ + Φsinh Φ ∗ X + O ( X ) (cid:19) , (2.75)whence the supersymmetry transformation of Φ is δ Susy Φ ≡ δ Susy ( X )Φ = Φsinh Φ ∗ X . (2.76)By elementary algebra, this can be re-written in terms of an E transformation withparameter X , δ Susy ( X )Φ = − δ e ( X )Φ − Φ tanh(Φ / ∗ X . (2.77)24e can now check the commutation rules between supersymmetry and E . Theexpectation is that the commutator of two such transformations gives rise to a supersym-metry transformation whose parameter ǫ ′ is obtained from the original supersymmetryparameter ǫ by acting on it with the compensating SU (8) transformations induced bythe action of e on the fermions, i.e. [ δ e (Λ) , δ Susy ( ǫ )] = δ Susy ( ǫ ′ ) , ǫ ′ ≡ δ su (8) ( − tanh(Φ / ∗ Λ) ǫ . (2.78)At this point it is convenient to modify the supersymmetry variation by requiring thespinor parameter ǫ also to transform with respect to the induced SU (8) transformationas δ e ǫ = δ su (8) (cid:16) tanh(Φ / ∗ Λ (cid:17) ǫ (2.79)so ǫ transforms in the same way as the gravitino field ψ under the compensating SU (8).As a consequence, the parameter X in the adjoint representation simply transforms as δ e (Λ) X = h tanh(Φ / ∗ Λ , X i (2.80)which correctly reproduces the corresponding su (8) action in the . As we will showbelow, with this extra compensating transformation we obtain (cid:2) δ e , δ Susy (cid:3)
Φ = 0 , (2.81)If (2.81) holds on the scalar fields, this commutator will also vanish on functions of Φas well as on all other fields. Indeed, the only transformation that could still appearis a local Lorentz transformation which does not act on Φ. To check the absence ofthe latter we simply evaluate the above commutator on the vierbein field. While δ e does not act on the vierbein, δ Susy produces a term ¯ ǫ i γ a ψ µi + ¯ ǫ i γ a ψ iµ . However, withthe extra compensating transformation (2.79), this expression becomes a singlet underthe induced SU (8) transformation, and therefore the commutator also vanishes on thevierbein. The main advantage of defining the transformation such that (2.81) is satisfiedwill become apparent when we discuss the quantum theory, because with (2.81) theBRST transformation will commute with E , and this enables us to directly formulatethe Ward identities for the full non-linear E symmetry.In the remainder of this section we prove the key formula (2.81). It is more convenientto evaluate the commutator on exp Φ rather than on Φ itself, because then all the non-linear terms appear via the compensating SU (8) transformation δ e (Λ) exp Φ = (cid:0) tanh(Φ / ∗ Λ (cid:1) exp Φ − (cid:0) exp Φ (cid:1) Λ δ Susy ( X ) exp Φ = (cid:16) X + tanh(Φ / ∗ X (cid:17) exp Φ (2.82)25rom which we read off that δ Susy ( X ) exp Φ = δ e ( X ) exp Φ + X exp Φ + (cid:0) exp Φ (cid:1) X , (2.83)a relation that will be useful below. Using (2.80) we get (cid:2) δ e (Λ) , δ Susy ( X ) (cid:3) exp Φ == (cid:0)(cid:0) δ e (Λ) tanh(Φ / (cid:1) ∗ X (cid:1) exp Φ − (cid:0)(cid:0) δ Susy ( X ) tanh(Φ / (cid:1) ∗ Λ (cid:1) exp Φ+ h tanh(Φ / ∗ X , tanh(Φ / ∗ Λ i exp Φ+ (cid:16) tanh(Φ / ∗ h tanh(Φ / ∗ Λ , X i(cid:17) exp Φ . (2.84)To evaluate these terms further we need to make use of the closure property (cid:2) δ e (Λ ) , δ e (Λ ) (cid:3) Φ = h Φ , (cid:2) Λ , Λ (cid:3)i , (2.85)that is, the fact that the commutator of two compensated E transformations mustclose properly into su (8). This formula obviously extends to all functions f (Φ) which areexpandable in a power series. Observe that without the compensating su (8) transforma-tion, the commutator [Λ , Λ ] in (2.85) would only act on Φ from the right (correspondingto the uncompensated E action), while its action from the left is due to the compen-sating su (8). Using (2.82) we now apply this formula to exp Φ to obtain (cid:16)(cid:16) δ e (Λ ) tanh(Φ / (cid:17) ∗ Λ (cid:17) exp Φ − (cid:16)(cid:16) δ e (Λ ) tanh(Φ / (cid:17) ∗ Λ (cid:17) exp Φ == h tanh(Φ / ∗ Λ , tanh(Φ / ∗ Λ i exp Φ − (cid:2) Λ , Λ (cid:3) exp Φ . (2.86)Modulo the difference between δ Susy ( X ) and δ e ( X ), cf. (2.83), this formula allows usto rewrite the right hand side of (2.84) as (cid:2) X , Λ (cid:3) exp Φ + (cid:16) tanh(Φ / ∗ h tanh(Φ / ∗ Λ , X i(cid:17) exp Φ . (2.87)Now exploiting (2.83) in the form δ Susy ( X ) exp( n Φ) = δ e ( X ) exp( n Φ) + X exp( n Φ) + exp( n Φ) X ++ 2 X ≤ m ≤ n − exp( m Φ) X exp(( n − m )Φ) (2.88)and expanding tanh(Φ /
2) as a formal power series in exp Φ we get δ Susy ( X ) tanh(Φ /
2) = δ e ( X ) tanh(Φ /
2) +
X − tanh(Φ / X tanh(Φ / , (2.89)26as can also be checked by expanding the formal power series around Φ = 0). Therefore − δ Susy ( X ) tanh(Φ / ∗ Λ = − δ e ( X ) tanh(Φ / ∗ Λ + − (cid:2) X , Λ (cid:3) + tanh(Φ / ∗ h X , tanh(Φ / ∗ Λ i . (2.90)Acting with this expression on exp Φ we see that these terms cancel the ones in (2.87),which proves the key formula (2.81).Finally, it is straightforward to see that the remaining gauge symmetries triviallycommute with the non-linear action of δ e . Combining all gauge symmetries into asingle BRST transformation with generator s in the usual way we therefore see that therelation (2.81) extends to the more general statement [ δ e , s ] = 0 . (2.91)Consequently, at the classical level, the BRST (gauge) transformations can be completelydisentangled from the non-linear action of E . In the remaining sections it will be ourtask to elevate this statement to the full quantum theory. SU (8) anomaly at one loop As a first application of the formalism developed in the foregoing sections, we now presenta Feynman diagram computation of the SU (8) anomaly considered long ago by verydifferent methods. In [17], N. Marcus pointed out the absence of rigid SU (8) anomaliesfor N = 8 supergravity at one loop; the cancellation is based on the following identity3 × tr X − × tr X + 1 × tr X = (cid:16) × − × × (cid:17) tr X = 0 , (3.1)where X is any su (8) generator, and where the first and third contributions are due tothe eight gravitinos and the 56 spin- fermions of N = 8 supergravity, while the middlecontribution is due to the 28 chiral vectors. In this section we will not consider thefermionic triangle diagrams which can be obtained by standard methods, but concentrateon the vector fields, that is, the middle term in (3.1). The formalism of this paper makespossible (for the first time) a full fledged Feynman diagram calculation because it allowsfor an off-shell realisation of the chiral properties of the vector fields and their interactionsunder SU (8). At the end of this section and in the following sections we will extend theseconsiderations to the full E current, where we will encounter a non-linear variant of For anti-commuting e parameters, this relation becomes an anti-commutator: { δ e , s } = 0. SU (8) current operators obtained by restricting the E current to its SU (8) subgroup.Now it is known (for linearly realised symmetries) that the anomaly involves a trace ofthe form (with Lie algebra generators X , X , X )Tr { X , X } X . (3.2)However, there is no invariant symmetric tensor of rank three in the or the adjoint of E , and hence a priori also none for its SU (8) subgroup (in these representations) soreaders may wonder how one could get an anomaly at all. It is here that the distinctionbetween a linearly realised symmetry and a non-linearly realised one makes all the dif-ference. Namely, as the explicit calculation below will show, the relevant trace involvesthe complex structure tensor J mn as an extra factor, so (3.2) is replaced byTr J { X , X } X . (3.3)This extra factor (which one might think of as being analogous to the insertion of a γ ) breaks the manifest symmetry from E to SU (8), and at the same time allows forthe appearance of chirality, and hence a non-vanishing trace (effectively replacing thevector-like = ⊕ by the chiral in the trace). Nevertheless, the E symmetryis still present, but necessarily non-linear. With these remarks we can now proceed to the actual computation. We first work outthe propagators by starting from the gauge-fixed kinetic term for the vector fields L = 12 Ω mn ε ij k ∂ A m i ∂ j A n k − G mn ( δ i k δ j l − δ i l δ j k ) ∂ i A m j ∂ k A n l + b m ∂ i A m i , (3.4)which is obtained from (2.17) by retaining only the parts quadratic in the fields. Further-more, in the linearised approximation, we set h ij = δ ij in (3.4) and expand the scalar We shall occasionally point out similarities of the present computation with the familiar γ anomaly;readers may therefore find it useful to consult the textbooks [32, 33, 18, 19] for further information onthis well known topic. G mn = G mn (˚Φ) also becomes constant(with ˚Φ = 0 we have G mn = δ mn ). Going to momentum space, the quadratic operator tobe inverted is ∆ − ( p ) = Ω mn ε ij k p p k + G mn ( δ ij p − p i p j ) ip i δ nm − ip j δ mn , (3.5)with p ≡ δ ij p i p j . The vector propagator is therefore∆( p ) = 1 p Ω mn ε ij k p p k − G mn ( δ ij p − p i p j ) p − p + iε ip i δ mn − ip j δ nm ; (3.6)it is a (4 ×
56) by (4 ×
56) matrix, with three spatial directions and the fourth componentcorresponding to the Lagrange multipliers b m which enforce the condition ∂ i A m i = 0.The propagating spin-1 degrees of freedom correspond to the residues of the poles of thepropagator at p = ±| p | . There is no pole in the off-diagonal components mixing b m and A m i , and the residue is given by2 | p | res (∆) (cid:12)(cid:12) p = | p | = Ω mn ε ij k ˆ p k − G mn ( δ ij − ˆ p i ˆ p j ) , (3.7)where ˆ p i ≡ p i / | p | . An important difference between (3.6) and the usual covariant prop-agator in four dimensions is that (3.6) contains terms which are odd under parity (forwhich p i → − p i and p → p ). It is these terms, together with the parity odd verticesto be given below, which introduce the extra factor J mn into the traces, and hence cancontribute to chiral anomalies, even if only vector fields circulate in the loop.We can rephrase these results in canonical language. Consider the free quantum field A m i ( x ) ≡ Z d p (2 π ) p | p | X σ (cid:16) e − ix · p e ∗ m i ( σ, p ) a ( σ, p ) + e ix · p e m i ( σ, p ) a † ( σ, p ) (cid:17) , (3.8)where a † ( σ, p ) and a ( σ, p ) are creation and annihilation operators of asymptotic freeparticles of momentum p and helicity h ( σ ) = ±
1, and 56 SU (8) quantum numbers σ (weanticipate in this notation that σ determines h by (3.14)), (cid:2) a ( σ, p ) , a † ( σ ′ , q ) (cid:3) = δ σσ ′ δ (3) ( p − q ) . (3.9) In this section we write G mn instead of using the (perhaps more appropriate) notation ˚ G mn ≡ G mn (˚Φ), since G mn (Φ) does not appear and the notation is therefore unambiguous. Except in (3.16)and (3.18), we refrain from using boldface latters for the spatial components of four-vectors, as it shouldbe clear from the context which is meant.
29n order for the operator algebra to reproduce the propagator (3.6) D (cid:12)(cid:12)(cid:12) T n A m i ( x ) A n j ( y ) o(cid:12)(cid:12)(cid:12) E = − i Z d p (2 π ) e ip · ( x − y ) p − p + iε (cid:16) Ω mn ε ij k p | p | ˆ p k − G mn (cid:0) δ ij − ˆ p i ˆ p j (cid:1)(cid:17) , (3.10)the polarisation vectors e m i ( σ, p ) and their complex conjugates e ∗ m i ( σ, p ) must satisfy X σ e m i ( σ, p ) e ∗ n j ( σ, p ) = − Ω mn ε ij k ˆ p k + G mn ( δ ij − ˆ p i ˆ p j ) . (3.11)As usual, the polarisation vectors are transverseˆ p i e m i ( σ, p ) = 0 . (3.12)With the convention ε ij k ˆ p k e m j ( σ, p ) = ih ( σ ) e m i ( σ, p ) , (3.13)it follows from (3.11) that the polarisation vectors must satisfy in addition J mn e n i ( σ, p ) = ih ( σ ) e m i ( σ, p ) , (3.14)with the ‘complex structure’ tensor J mn ≡ J mn (˚Φ), see (2.3). With this extra constraint,there are only 56 independent polarisations, so σ runs from 1 to 56. The linearisedequations of motion are then satisfied with a zero gradient ∂ i A m = 0 in (2.8), ∂ A m i = ε ij k J mn ∂ j A n k , (3.15)such that the action of the Lorentz group on A m i is the same as in the standard formulationof the free theory in the Coulomb gauge. It follows that the 56 creation operators a † ( σ, p )are the same as in the standard formulation of the free theory, and the 28 states of helicity h = 1 transform in the of SU (8), whereas the 28 states of helicity h = − of SU (8), as required by (3.14).Note that because of (3.11), the free quantum field A m i ( x ) does not commute withitself at equal time, but satisfies instead (cid:2) A m i ( x , x ) , A n j ( x , y ) (cid:3) = i Ω mn ε ij k ∂∂x k π | x − y | . (3.16)This equal time commutator could be derived alternatively from the Dirac quantisationof the theory in the Coulomb gauge, with the second class constraints Π i m −
12 Ω mn ε ij k ∂ j A n k ≈ , ∂ i A m i ≈ . (3.17) As in the conventional formulation, the Poisson bracket of the first class Coulomb constraint ∂ i Π i m ≈ ∂ i A m i ≈ i m in the Darboux basis only coincideswith the definition (2.19) of the canonical momentum Π ¯ m i in the conventional formu-lation of the theory (2.20) up to a factor 2. Although the canonical Poisson bracketstherefore differ by a factor 2 in the two formulations, the Dirac brackets are equivalent.The commutation relation (3.16) is consistent with causality, because (cid:2) F m ij ( x , x ) , F n kl ( x , y ) (cid:3) = 2 i Ω mn ε ij [ k ∂ l ] δ (3) ( x − y ) , (3.18)as follows directly from (3.16), and therefore gauge-invariant operators commute at space-like separation ( x − y ) < | x − y | .The cubic vertex defining the couplings of the E current to the vector fields canbe obtained from the quadratic action by adding to (3.4) terms with source fields B mµ n coupling to the conserved E current, such that the latter is re-obtained by taking thederivative with respect to the source fields and then setting them equal to zero. Herewe will restrict attention to the su (8) part of the full E current, for which the source B mµ n leaves the background metric G mn invariant: B pµ m G pn + B pµ n G pm = 0 (3.19)(For G mn = δ mn this just means that SU (8) is realised by anti-symmetric matrices inthe real basis of the representation of E ). As is well known, the introduction ofsuch sources corresponds to formally covariantising the action (3.4) with respect to local SU (8), such that (3.4) is replaced by the density L [ B ] = 12 Ω mn ε ij k (cid:0) ∂ A m i + B m p A p i (cid:1)(cid:0) ∂ j A n k + B n j q A q k (cid:1) − G mn ( δ i k δ j l − δ i l δ j k ) (cid:0) ∂ i A m j + B m i p A p j (cid:1)(cid:0) ∂ k A n l + B n k q A q l (cid:1) + b m (cid:0) ∂ i A m i + B m i n A n i (cid:1) (3.20)(in fact, dropping the restriction (3.19) this action becomes covariant with respect tolocal E , as required for a study of the full E current, cf. section 5.2). For thefermion fields, the SU (8) tensor structure factorises out, and the vertex associated toone SU (8) current insertion just has the expected structure ∝ (1 ± iγ ) γ µ . For vectorfields, on the other hand, the Lorentz and su (8) tensor structures are a priori entangledfor the vertices computed from (3.20). Nevertheless, for correlation functions of SU (8) The momentum dependence of the 3-point vertex can be derived in the usual way [32] by writingthe corresponding terms from (3.20) in momentum space and symmetrising in the internal legs involvingthe quantum fields A m i (not forgetting the antisymmetry condition (3.19)). SU (8) invariant tensors δ mn and J mn ;these can be diagonalised according to the decomposition of the E representation ∼ = ⊕ of SU (8). The calculation shows that all the su (8) Lie algebra generators X ’s can be moved to the left such that the vertex for linking an su (8)-current J µ and achiral boson A m i with incoming momenta p and k respectively to a chiral boson A n j withoutgoing momentum p + k J µ p A m i k A n j p + k is effectively given byΥ ( k + p, k ) = i Ω mn ε ij k (cid:0) k k + p k (cid:1)
00 0 , Υ k ( k + p, k ) = i Ω mn ε ij k (cid:0) k + p (cid:1) + iG mn (cid:16) (2 k k + p k ) δ ij − δ k i ( k j + p j ) − δ k j k i (cid:17) − δ k i δ nm δ k j δ mn , (3.21)where the bottom-left component gets a positive sign because of (3.19). The notation weuse here is formally very similar to the one used for the familiar fermionic vertices. Thevertices Υ µ are analogous to the (1 ± iγ ) γ µ matrices that appear in the correspondingcomputation of the anomalous fermionic triangle diagram. This analogy is for instancereflected in the identity − ip µ Υ µ ( k + p, k ) = ∆ − ( k + p ) − ∆ − ( k ) , (3.22)which is analogous to the (trivial) identity /p = ( /k + /p ) − /k , and will be similarly useful tocancel propagators in the diagrams and thereby simplify them. However, in contradis-tinction to the case of fermion fields which are governed by a first order kinetic term,(3.20) is quadratic in B mµ n and thus the insertion of more than one current requiresthe consideration of contact terms absent in the fermionic triangle. The correspondingvertices R µν µ p J ν p A m i k − p A n j k + p do not depend on the momenta: R k = R k = Ω mn ε ij k
00 0 , R kl = G mn (cid:16) δ kl δ ij − δ k j δ l i (cid:17)
00 0 . (3.23)The vertices (3.21) and (3.23) satisfyΥ µ ( k + p, k ) T = − Υ µ ( − k, − k − p ) , R µν = (cid:0) R νµ (cid:1) T , (3.24)where transposition is defined in the matrix notation, and includes the interchange ofthe index pairs ( i , m ) ↔ ( j , n ) of the top left component. Furthermore, ip ν R µν = Υ µ ( q, l + p ) − Υ µ ( q, l ) , (3.25)for any choice of momenta l µ and q µ . The contribution to the vacuum expectation valueof three currents of the one-loop diagrams with vector fields circulating in the loop isencoded in the triangle diagram J µ ( X ) p J ν ( X ) p k k + p k − p J σ ( X ) p + p and in the one with the orientation of the loop momenta reversed as well as in the sixindependent permutations of the bubble diagram J µ ( X , p ) J ν ( X , p ) k + p k − p J σ ( X ) p + p D J µ ( X , p ) J ν ( X , p ) J σ ( X , − p − p ) E vec = i Z d k (2 π ) Tr X X X (cid:18) Υ µ ( k + p , k )∆( k )Υ ν ( k, k − p )∆( k − p )Υ σ ( k − p , k + p )∆( k + p )+ R µν ∆( k − p )Υ σ ( k − p , k + p )∆( k + p ) + Υ µ ( k + p , k )∆( k ) R νσ ∆( k + p )+ ∆( k )Υ ν ( k, k − p )∆( k − p ) R σµ (cid:19) + i Z d k (2 π ) Tr X X X (cid:18) Υ ν ( k + p , k )∆( k )Υ µ ( k, k − p )∆( k − p )Υ σ ( k − p , k + p )∆( k + p )+ R νµ ∆( k − p )Υ σ ( k − p , k + p )∆( k + p ) + Υ ν ( k + p , k )∆( k ) R µσ ∆( k + p )+ ∆( k )Υ µ ( k, k − p )∆( k − p ) R σν (cid:19) . (3.26)Here, X , X , X are su (8) matrices, valued in the ⊕ and the trace is to be takenover (4 × matrices corresponding to components of the vector propagator.Let us compute the divergence of the third current in this expectation value. Usingthe formulas (3.22, 3.25), one computes that i ( p σ + p σ ) D J µ ( X , p ) J ν ( X , p ) J σ ( X , − p − p ) E vec = i Z d k (2 π ) Tr X X X (cid:18) Υ µ ( k + p , k )∆( k )Υ ν ( k, k + p )∆( k + p ) − Υ µ ( k − p , k )∆( k )Υ ν ( k, k − p )∆( k − p ) + R µν ∆( k + p ) − R µν ∆( k − p ) (cid:19) + i Z d k (2 π ) Tr X X X (cid:18) Υ µ ( k, k + p )∆( k + p )Υ ν ( k + p , k )∆( k ) − Υ µ ( k, k − p )∆( k − p )Υ ν ( k − p , k )∆( k ) + R νµ ∆( k + p ) − R νµ ∆( k − p ) (cid:19) . (3.27)The commutator component ∝ Tr [ X , X ] X of (3.27) gives rise to the vector fieldcontribution to the vacuum expectation value of the insertion of two currents, as requiredby the current Ward identity i ( p σ + p σ ) D J µ ( X , p ) J ν ( X , p ) J σ ( X , − p − p ) E = i D J µ ( X , p ) J ν ([ X , X ] , − p ) E + i D J µ ([ X , X ] , − p ) J ν ( X , p ) E . (3.28)34y contrast, the anticommutator component of (3.27) is proportional to Tr J X X X ,and reduces to the difference of divergent integrals with respect to a constant shift ofthe integration variable, as for the one-loop contribution of the fermion fields. The twofirst lines in (3.27) give rise to the difference of linearly divergent integrals12 Tr { X , X } X Z d k (2 π ) (cid:16) I µν ( k, p ) − I µν ( k − p , p ) − I µν ( k, − p ) + I µν ( k + p , − p ) (cid:17) = 12 Tr { X , X } X Z d k (2 π ) (cid:18) p σ ∂I µν ( k, p ) ∂k σ + p σ ∂I µν ( k, − p ) ∂k σ (cid:19) , with Tr X I µν ( k, p ) = Tr X Υ µ ( k + p, k )∆( k )Υ ν ( k, k + p )∆( k + p ) , (3.29)where X can be any SU (8) generator. Because there is no invariant symmetric tensor ofrank three in the adjoint of E by (3.2) the last anticommutator term of (3.27) reducesto a double difference of quadratically divergent integrals12 Tr { X , X } X R µν Z d k (2 π ) (cid:18) ∆( k + p ) − ∆( k − p ) − ∆( k + p ) + ∆( k − p ) (cid:19) = 12 Tr { X , X } X R µν ( p σ + p σ )( p ρ − p ρ ) Z d k (2 π ) ∂ ∆( k ) ∂k σ ∂k ρ . (3.30)In the above derivation, we have made use of standard formulas [33, 18] to express theintegrals as surface integrals which leads to the final expressions with first and secondderivatives on the integrands.Although finite, these integrals are not absolutely convergent, and they are subjectto ambiguities associated to the order of integration of the momentum components k µ .This ambiguity can be fixed in the conventional case (with fermions in the loop) byrequiring Lorentz invariance. However, when photons run in the loop, the integrandsare not Lorentz invariant and this prescription cannot be consistently defined. Thisproblem is in fact general in the theory. Indeed, because the Feynman rules are not man-ifestly Lorentz invariant, and because of the explicit appearance of the Levi-Civit`a tensor ε ij k , one cannot regularise the theory with the dimensional regularisation. Nevertheless,we will now explain how one can perform a consistent computation using Pauli–Villarsregularisation. Because ghosts do not give rise to any term of type (3.3), they do not contribute to the anomaly. We are aware that a consistent dimensional regularisation via an SO (3) invariant prescription hasbeen used successfully in other contexts, such as the post-Newtonian approximation in general relativity,where there are no anomalies (T. Damour, private communication). However, this prescription appearsto give inconsistent results in the present case. .2 Pauli–Villars regularisation The formulation of the theory is defined such that it is formally equivalent to the mani-festly diffeomorphism covariant formulation, up to a Gaussian integration of the 28 vectorfields A ¯ m i as in (2.18, 2.19, 2.20). Therefore, we will require the massive Pauli–Villarsvector fields, to be defined through a local formulation after Gaussian integration. Thisis the case only if the vectors A ¯ m i appear in the mass term through F ¯ m ij up to a totalderivative. The only ‘sensible’ possibility is therefore to introduce a symmetric tensorΓ mn which is off-diagonal in the Darboux basis ( i.e. Γ mn = 0 = Γ ¯ m ¯ n ) L ( M ) = 12 Ω mn ε ij k ∂ A m i ∂ j A n k + i mn ε ij k M A m i ∂ j A n k − G mn ( δ i k δ j l − δ i l δ j k ) ∂ i A m j ∂ k A n l + b m ∂ i A m i . (3.31)We will show next that this Lagrangian gives rise to the standard equations of motionfor the 28 Pauli–Villars vector fields in the Coulomb gauge. Before doing so, note thatthere is no necessity to modify the interaction terms in the Lagrangian (3.31), and thatthe tensor Γ mn necessarily breaks SU (8) to (at most) SO (8). In fact, a manifestly SU (8)regularisation would be in contradiction with the possible existence of chiral anomalies.We define Γ mn such that it reads Γ m ¯ n = Γ ¯ nm = δ m ¯ n , (3.32)in the Darboux basis. Following the procedure of section 2.2, the manifestly covariantaction for the Pauli–Villars vector fields is obtained after a Gaussian integration of the28 (dual) Pauli–Villars vector fields A ¯ m i . This amounts to performing the replacement F m i → F m i + M A m i , (3.33)in all expressions. In particular, the equations of motion read ∂ i (cid:0) F m i + M A m i (cid:1) = 0 , ∂ µ F i µ + M ∂ i A − M A m i = 0 , (3.34)and are manifestly gauge invariant with respect to the modified gauge transformations δA m = ∂ c m + M c m , δA m i = ∂ i c m . (3.35)In the Coulomb gauge ∂ i A i = 0, they reduce to A m = 0 , A m i + M A m i = 0 . (3.36)36he substitution (3.33) breaks diffeomorphism invariance manifestly, which can thereforebe restored only after the regulator is removed (possibly with a non-Lorentz invariantlocal counterterm, see below).With these replacements, the propagator is manifestly massive in the duality invariantformulation. Indeed, one has∆ − ( p, M ) = Ω mn ε ij k p p k + Γ mn ε ij k M p k + G mn ( δ ij p − p i p j ) ip i δ nm − ip j δ mn , (3.37)and the propagator is∆( p, M ) = 1 p Ω mn ε ij k p p k +Γ mn ε ij k Mp k − G mn ( δ ij p − p i p j ) p − p − M + iε ip i δ mn − ip j δ nm , (3.38)where Γ mn is the inverse of Γ mn and satisfiesΓ mp Ω pn = − Ω mp Γ pn , Γ mp G pn = G mp Γ pn , Γ mp Γ pn = δ nm . (3.39)Therefore ( p Ω mp + M Γ mp )( p Ω pn + M Γ pn ) = ( − p + M ) δ nm , (3.40)which permits to check (3.38).To define the associated SU (8)-current vertex one must distinguish the vector andthe axial components, respectively, corresponding to the decomposition → ⊕ ofthe su (8) adjoint under its so (8) subalgebra. One can thus consider a manifestly SO (8)invariant regularisation by considering the coupling of the SO (8) current source B mµ n tothe mass term. So we consider the coupled Lagrangian L [ B ] = 12 Ω mn ε ij k (cid:0) ∂ A m i + B m p A p i (cid:1)(cid:0) ∂ j A n k + B n j q A q k (cid:1) + i mn ε ij k M A m i (cid:16) ∂ j A n k + B n j p A p k (cid:17) − G mn ( δ i k δ j l − δ i l δ j k ) (cid:0) ∂ i A m j + B m i p A p j (cid:1)(cid:0) ∂ k A n l + B n k q A q l (cid:1) + b m (cid:0) ∂ i A m i + B m i n A n i (cid:1) . (3.41)Note however that the mass term does only couple to the axial component of thesource B n j p , because an axial generators X mp satisfies X mp Γ pn − X np Γ mp = 0 . (3.42)For simplicity, we will focus on the contribution of the massive Pauli–Villars vectorfields to the vacuum expectation value of three axial currents (the three of them in the37 ) for which the vertices Υ µ are still defined by (3.21). Using (3.42), one obtains thatthe latter is given by D J µ ( X , p ) J ν ( X , p ) J σ ( X , − p − p ) E PV = i Z d k (2 π ) Tr X X X (cid:18) ∆( k + p , M )Υ µ ( k + p , k )∆( k, − M )Υ ν ( k, k − p )∆( k − p , M )Υ σ ( k − p , k + p )+∆( k + p , M ) R µν ∆( k − p , M )Υ σ ( k − p , k + p )+∆( k + p , M )Υ µ ( k + p , k )∆( k, − M ) R νσ + R σµ ∆( k, − M )Υ ν ( k, k − p )∆( k − p , M ) (cid:19) + i Z d k (2 π ) Tr X X X (cid:18) ∆( k + p , M )Υ ν ( k + p , k )∆( k, − M )Υ µ ( k, k − p )∆( k − p , M )Υ σ ( k − p , k + p )+∆( k + p , M ) R νµ ∆( k − p , M )Υ σ ( k − p , k + p )+∆( k + p , M )Υ ν ( k + p , k )∆( k, − M ) R µσ + R σν ∆( k, − M )Υ µ ( k, k − p )∆( k − p , M ) (cid:19) (3.43)where the propagator ∆( k, − M ) gets an opposite mass through the commutation withthe axial generators (similarly as in the standard fermion triangle). (3.43) is thereforethe analogue of (3.26) for M = 0. In addition to the traces (3.2) and (3.3) there are twomore types of traces, both of which give vanishing contribution becauseTr (ΩΓ) X X X = 0 = Tr ( G Γ) X X X , (3.44)Therefore the resulting integral is an even function of M . The massive generalisation of(3.22) is − ip µ Υ µ ( k + p, k ) = ∆ − ( k + p, M ) − ∆ − ( k, − M ) − M Υ (2 k + p ) , (3.45)where Υ ( p ) = Γ mn ε ij k p k
00 0 , (3.46)again indicating the formal similarity of our computation with the usual fermionic triangle38iagram. Using the latter identity, one computes that i ( p σ + p σ ) D J µ ( X , p ) J ν ( X , p ) J σ ( X , − p − p ) E PV = i Z d k (2 π ) Tr X X X (cid:18) ∆( k + p , M )Υ µ ( k + p , k )∆( k, − M )Υ ν ( k, k + p ) − Υ µ ( k − p , k )∆( k, − M )Υ ν ( k, k − p )∆( k − p , M ) + R µν ∆( k + p ) − R µν ∆( k − p ) (cid:19) + i Z d k (2 π ) Tr X X X (cid:18) Υ µ ( k, k + p )∆( k + p , − M )Υ ν ( k + p , k )∆( k, M ) − ∆( k, M )Υ µ ( k, k − p )∆( k − p , − M )Υ ν ( k − p , k ) + R νµ ∆( k + p ) − R νµ ∆( k − p ) (cid:19) − iM Z d k (2 π ) Tr X X X (cid:18) ∆( k + p , M )Υ µ ( k + p , k )∆( k, − M )Υ ν ( k, k + p )∆( k + p , M ) × Υ (2 k + p − p ) + ∆( k + p , M ) R µν ∆( k − p , M )Υ (2 k + p − p ) (cid:19) − iM Z d k (2 π ) Tr X X X (cid:18) ∆( k + p , M )Υ ν ( k + p , k )∆( k, − M )Υ µ ( k, k − p )∆( k − p , M ) × Υ (2 k − p + p ) + ∆( k + p , M ) R νµ ∆( k − p , M )Υ (2 k − p + p ) (cid:19) (3.47) To compute the anomaly we now follow the standard procedure by subtracting (3.47)(indicated by the subscript “PV”) from (3.27) (indicated by the subscript “vec”), andthen taking the limit M → ∞ . The first two integrals in (3.47) are very similar to themassless case (3.27), and their contribution to the anomaly reduces to a difference oflinearly divergent integrals as well. Because such integrals only depend on the leadingpower in the momentum k , they do not depend on the mass and hence these contributionscancel precisely between (3.27) and (3.47). It follows that the overall contribution of thevector fields to the anomaly is obtained by (minus) the sum of the two last integralsof (3.47) in the limit M → ∞ — just like for the fermionic triangle in the standardcomputation with Pauli–Villars regulators, see e.g. [33].For the computation, it is straightforward to see that for the massive propagators andvertices, the relations (3.24) remain unchanged, and that we have in addition,∆( k, M ) T = ∆( − k, M ) , Υ ( p ) T = Υ ( − p ) . (3.48)These relations permit to prove that the two last integrands in (3.47) are invariant under39he substitution k ↔ − k , p ↔ p , µ ↔ ν . (3.49)It follows that they are equal, and the anomaly being defined as A µν vec ( p , p ) Tr J X X X ≡ i ( p σ + p σ ) D J µ ( X , p ) J ν ( X , p ) J σ ( X , − p − p ) E vec+PV − i D J µ ( X , p ) J ν ([ X , X ] , − p ) E vec+PV − i D J µ ([ X , X ] , − p ) J ν ( X , p ) E vec+PV (3.50)is given by A µν vec ( p , p ) = 2 i lim M → + ∞ (cid:20) M Z d k (2 π ) S µν ( p , p , k ) (cid:21) , (3.51)where the integrand S µν ( p , p , k ) is S µν ( p , p , k ) = −
156 Tr J (cid:18) ∆( k + p , M )Υ µ ( k + p , k )∆( k, − M )Υ ν ( k, k + p )∆( k + p , M ) × Υ (2 k + p − p ) + ∆( k + p , M ) R µν ∆( k − p , M )Υ (2 k + p − p ) (cid:19) . (3.52) S µν includes three propagators, and takes the form S µν ( p , p , k ) = M P µν ( p , p , k ) ( k + p ) ( ( k + p ) − ( k + p ) − M + iε ) k ( k − k − M + iε ) ( k − p ) ( ( k − p ) − ( k − p ) − M + iε )(3.53)where P µν is a sum of monomials of order eight in p , p , k and M . One can neglect allterms of order three and higher in p and p , because they will not contribute to (3.51).Moreover, for the terms of order two in p and p , the denominator can be approximatedas well by k ( k − k − M + iε ) and one can use the usual simplifications k n k m k i k j ∼ δ ij k n +2 k m , k n k m +1 k i ∼ , (3.54)according to the standard integration rules. After a rather tedious computation, weobtain P i ∼ − k ( k − k − M ) ε ij k k j (2 p k + 6 p k ) + 13 k (cid:0) k − k − k − M ) (cid:1) ε ij k p j p k P ij ∼ k ε ij k (cid:18) p + p )( k − k − M ) k k −
83 ( k − k − M ) k ( p k + p k )+ 13 ( k − k − M ) (cid:16) p + p )( p k − p k ) − p p k − p p k ) (cid:17) − k ( p p k − p p k ) (cid:19) (3.55)40nd P i = − P i using the symmetries (3.48, 3.49). Using the formula Z d k (2 π ) k n ( − k ) m ( k − k + M − iε ) l = − i Γ( + m )Γ( + n )Γ( l − m − n − π ) Γ( l ) M l − m − n − , (3.56)this leads to the integrals A i vec = 2 M iε ij k p j p k Z d k (2 π ) (cid:0) k − k + M − iε (cid:1) − k − (cid:0) k − k + M − iε (cid:1) ! = 0 (3.57)and A ij vec = 2 M iε ij k ( p p k − p p k ) Z d k (2 π ) (cid:18) k k − ( k − k − M + iε ) − k − ( k − k + M − iε ) (cid:19) +2 M iε ij k ( p + p )( p k − p k ) Z d k (2 π ) (cid:18) k k − ( k − k − M + iε ) − k − k + M − iε ) − k − ( k − k + M − iε ) (cid:19) = 16 π ε ij k ( p p k − p p k ) − π ε ij k ( p + p )( p k − p k ) . (3.58)The resulting anomaly is not Lorentz invariant, but this is not so surprising since weused a regulator that breaks Lorentz invariance. In order to restore Lorentz invariance,one must renormalise the theory with a finite non-Lorentz invariant counterterm withthe appropriate su (8) tensor structure. The only such SO (3) invariant density is δ L ∝ ε ij k J mn B n p B p i q ∂ j B q k m . (3.59)It follows that the vacuum expectation value of three current insertions is only definedup to a shift δ D J i ( X , p ) J j ( X , p ) J ( X , − p − p ) E = − ia ε ij k ( p k − p k )Tr J X X X , (3.60)and permutations. This shift affects the anomaly factor as δ A i vec = − δ A i vec = a ε ij k p j p k , δ A ij vec = a ε ij k ( p + p )( p k − p k ) , (3.61) Which itself follows from the standard formula [18] Z ∞ x a − dx ( x + s ) b = s a − b Γ( a )Γ( b − a )2Γ( b ) . a = π , one recovers the Lorentz invariant anomaly A µν vec ( p , p ) = 16 π ε µνσρ p σ p ρ . (3.62)We have thus verified that the anomaly coefficient associated to the vector fields is, aspredicted from the family’s index theorem, ( −
2) times the one associated to the Diracfermion fields. Taking into account the fermionic contributions it follows that the totalcoefficient of the anomaly vanishes for N = 8 supergravity, in agreement with (3.1).Within the path-integral formulation of the theory, the variation of the formal inte-gration measure with respect to an infinitesimal su (8) local transformation gives rise toa local functional of the fields linear in the su (8) parameter C k which defines a 1-cocycleover the space of su (8) gauge transformations. This factor can be compensated by aredefinition of the local action if this cocycle is trivial in local cohomology. The trivialityof this cohomology class is equivalent, via a transgression operation, to the triviality of a2-cocycle over the moduli space of framed su (8)-connections B identified modulo su (8)-gauge transformations in local cohomology. The latter can be computed by means of thefamily’s index theorem [26], as the Chern class of the vector bundle defined over an S two parameters family of su (8)-gauge orbits of su (8) framed connections with fibre theindex of the chiral differential operators(1 + iγ ) /D , (1 + J ⋆ ) d B , (1 + iγ ) ⋆ e a ∧ γ a d B , (3.63)acting on the fields of spin 1/2, 1, and 3/2, respectively. A similar construction appliesto gravitational and mixed anomalies. According to the family’s index theorem, thecontribution of the fermion fields and the vectors has been computed in [26], and appliedto various supergravity theories in [17], giving for instance the cancelation (3.1) of the su (8) anomaly in N = 8 supergravity.Let us now turn to the generalisation of these results to E . Unlike the linear SU (8) anomaly, the full E current and the non-linearly realised E symmetry giverise to an infinite number of potentially anomalous diagrams. Namely, for the complete e current Ward identities, one must also take into account the potential anomaliesassociated to the component of the current, as well as diagrams with any number ofscalar field insertions. We will write X , X for su (8) generators, and Y , Y for generatorsin the . Because there is no in the symmetric tensor product ( ⊗ ) sym , theWard identity associated to D J µ ( Y , p ) J ν ( Y , p ) J σ ( X , − p − p ) E (3.64) The antisymmetric product just gives the usual contribution to the non-anomalous Ward identity. J µ ( X )denoting the projection of the current J µ along the Lie algebra element X ). However,further anomalies can appear if one includes scalar field insertions, as e.g. J µ ( Y ) J ν ( Y ) J σ ( X )Φ ΦΦ ΦThis is because the insertion of one scalar field into the diagram does not only addone propagator, but also two derivatives, whence the degree of divergence of the dia-gram remains the same with any number of external scalar fields (the same is true forfermionic loops, where the insertion of an extra fermionic propagator is accompanied byone derivative, as well as for the current vertex including scalar fields legs, which do notcarry derivatives, but do not add propagators either). As a first non-trivial example,consider the vacuum expectation value D J µ ( X , p ) J ν ( X , p ) J σ ( Y , p )Φ( Y , − p − p − p ) E . (3.65)It satisfies the su (8) Ward identity − ip σ D J µ ( X , p ) J σ ( X , p ) J ν ( Y , p )Φ( Y , − p − p − p ) E = i D J µ ([ X , X ] , p + p ) J σ ( Y , p )Φ( Y , − p − p − p ) E + i D J µ ( X , p ) J σ ([ Y , X ] , p + p )Φ( Y , − p − p − p ) E + i D J µ ( X , p ) J σ ( Y , p )Φ([ Y , X ] , − p − p ) E . (3.66)By contrast, the e Slavnov–Taylor identity takes a more complicated form because the43ransformation is non-linear (see (4.17) below for the derivation) − ip σ D J µ ( X , p ) J ν ( X , p ) J σ ( Y , p )Φ( Y , − p − p − p ) E = i D J µ ([ X , Y ] , p + p ) J ν ( X , p )Φ( Y , − p − p − p ) E + i D J µ ( X , p ) J ν ([ X , Y ] , p + p )Φ( Y , − p − p − p ) E + D J µ ( X , p ) J ν ( X , p )Φ A ( p ) Φ( Y , − p − p − p ) EDh
Φtanh(Φ) ∗ Y i A E + i D J µ ( X , p ) J ν ( X , p ) h Φtanh(Φ) ( − p − p ) ∗ Y i A ED Φ A ( p + p + p )Φ( Y , − p − p − p ) E + i D J µ ( X , p ) h Φtanh(Φ) ( − p ) ∗ Y i A ED J ν ( X , p )Φ A ( p + p )Φ( Y , − p − p − p ) E + i D J ν ( X , p ) h Φtanh(Φ) ( − p ) ∗ Y i A ED J µ ( X , p )Φ A ( p + p )Φ( Y , − p − p − p ) E . (3.67)where the index A = 1 to 70 labels an orthonormal basis of the coset component of the E Lie algebra.At one loop, there is a potential anomaly to these Ward identities of the form ∝ ε µνσρ p σ p ρ Tr J X X [ Y , Y ]for the su (8) Ward identity (3.66), and an anomalous contribution to the Slavnon-Tayloridentity (3.67) ∝ ε µνσρ (cid:0) p σ p ρ + ( p σ − p σ ) p ρ (cid:1) Tr J X X [ Y , Y ] . (3.68)Similarly, the Ward identities associated to the vacuum expectations values D J µ (cid:16) X , − N +3 X m =1 p m (cid:17) J ν ( X , p N +3 ) J σ ( Y , p ) N Y n =2 Φ( Y n , p n ) E , D J µ (cid:16) X , − N +4 X m =1 p m (cid:17) J ν ( Y , p ) J σ ( Y , p ) N Y n =3 Φ( Y n , p n ) E , D J µ (cid:16) Y N +6 , − N +5 X m =1 p m (cid:17) J ν ( Y , p ) J σ ( Y , p ) N Y n =3 Φ( Y n , p n ) E , (3.69)are potentially anomalous for all N ≥
0. Computing these anomalies explicitly wouldinvolve an infinite number of Feynman diagrams of increasing complexity. Fortunately, as44e are going to see in the following section, the coefficients associated to these anomaliesare determined by the Wess–Zumino consistency conditions in terms of the su (8) anomalycoefficient. It thus follows from the computation of this section that they all vanish.What about higher loops? Remarkably, for strictly non-renormalisable theories theAdler–Bardeen Theorem is almost trivial in the following sense. By non-renormalisabilityand power counting higher loop anomalies would have a different form and dimension(involving more derivatives) from the one-loop anomaly studied above. However, suchanomalies can be ruled out by the cohomology arguments given in the next section. Inconclusion, with the cancellations exhibited above there are no su (8) or e anomaliesin N = 8 supergravity at any order in perturbation theory. The purpose of this section is to show that ‘non-linear’ e anomaly is completely deter-mined by the ‘linear’ su (8) anomaly. In this way the determination of an infinite numberof potentially anomalous diagrams involving three currents and an arbitrary number ofscalar field insertions can be reduced to the single diagram computed in section 3. Asalready mentioned in the introduction, this result has its differential geometric roots inthe homotopy equivalence (1.1). e master equation The ‘non-linear’ e Ward identities are Slavnov–Taylor identities, which can be sum-marised in a master equation for the 1PI generating functional Γ. To simplify the dis-cussion, we will postpone the discussion of the ghost sector and the compatibility withthe BRST master equation to the next section.Because the discussion of this section does not rely on particular properties of E and applies equally to other supergravity theories coupled to abelian vector fields andscalar fields parametrising a symmetric space, we keep it general by considering a Liealgebra g with decomposition g ∼ = k ⊕ p , (4.1)with maximal ‘compact subalgebra’ k and the ‘non-compact’ part p , and the usual com-mutation relations [ k , k ] ⊂ k , [ k , p ] ⊂ p , [ p , p ] ⊂ k . (4.2) ‘Strictly’ in the sense that there are no coupling constant of dimension ≥
45s explained in section 2.5, the transformation δ g in the non-linear realisation acts onthe scalar fields as δ g Φ ≡ δ k Φ + δ p Φ = − [ C k , Φ] + Φtanh Φ ∗ C p , (4.3)where the compact subalgebra k acts linearly with parameter C k , while the remainingtransformations δ p with parameter C p are realised non-linearly . With regard to ourprevious discussion of these transformations in section 2.5, we note two important differ-ences:1. As we wish to treat the theory within the ‘BRST formalism’, we will from nowon take the transformation parameters C k and C p as anti-commuting (which is thereason why we use the letter C rather than Λ for the transformation parameters).2. Although the g symmetry acts rigidly , we will nevertheless take C to be a local parameter, i.e. to depend on x . The corresponding source fields B ≡ B k + B p cou-pling to the conserved G Noether current consequently transform as (non-abelian)gauge fields with these parameters.With regard to the second point we emphasise that the introduction of an artificial local G invariance here is merely a formal device (well known to specialists) which will enableus to derive current Ward identities for g . The sources B are external fields , which arenot part of any supermultiplet and are not integrated over in the path integral. Hence,the symmetries of the physical degrees of freedom of N = 8 supergravity and theirinteractions are still the same as before. Similarly, C k and C p , though x -dependent, arenot quantum fields. Readers might nevertheless find it convenient to consider them asghosts for the fictitious local G symmetry, when the G current is coupled to the sources B p and B k . For instance, we will shortly consider a grading that corresponds to the orderof the functional in these parameters, and that can be thought of as a ghost number(although it must not be confused with the true ghost number associated to the BRSToperator which implements the gauge symmetries of the theory).With these comments, the action of the transformations (4.3) on the other fields isstraightforward to describe. On the fermionic fields (as well as on the supersymme-try ghost or superghosts) the transformations act via an induced k transformation withparameter C k + tanh(Φ / ∗ C p (4.4)while on the vector fields and their ghosts the variations act linearly with parameter C in the corresponding representation (the of E for N = 8 supergravity). Finally,46riting C ≡ C k + C p , we have δ g B = − dC − { B, C } , δ g C = − C (4.5)on the current source B ≡ B k + B p and on the parameter itself, both of which transform inthe adjoint of g (that is, the of E for N = 8 supergravity). The anticommutatorin this formula appears because δ g anticommutes with forms of odd degree.In summary, on all the fields (but C k ) the differential δ g decomposes into a k trans-formation of parameter C k , which we will denote δ k , and the remaining (coset) transfor-mation δ p with parameter C p δ g = δ k + δ p , δ k ≡ δ k ( C k ) , δ p ≡ δ p ( C p ) (4.6)For instance, and as a consequence, (4.5) splits as follows δ k B k = − dC k − { B k , C k } , δ k B p = −{ B p , C k } ,δ p B k = −{ B p , C p } , δ p B p = − dC p − { B k , C k } ,δ k C p = −{ C k , C p } , δ p C p = 0 , (4.7) δ k is a nilpotent differential defined on all the fields, including C k with δ k C k = − C k . (4.8)By contrast, δ p makes sense only on expressions which do not depend on C k . If suchexpressions are moreover k -invariant, the operator δ p is nilpotent as a consequence of(4.2), i.e. δ p ( C p ) ◦ δ p ( C p ) = δ k ( C p ) ≈ . (4.9)We will refer to such k -invariant expressions which do not depend of C k as ‘ k -basic ’; andthe cohomology of the nonlinear operator δ p on the complex of k -basic expressions, asthe equivariant cohomology H • K ( δ p ) (see for example [34] for a mathematical definition).We will write S [ ϕ, B ] for the classical action coupled to G -current sources B , where by ϕ a we designate all the fields of the theory including ghosts. For each field ϕ a we introducea source ϕ g a for the non-linear g transformation δ p ( C p ) of the field ϕ a of anti-commutingparameter C p . We define the action coupled to sources byΣ (cid:2) ϕ, ϕ g , B, C (cid:3) ≡ κ S [ ϕ, B ] − Z d x X a ( − a ϕ g a δ p ( C p ) ϕ a , (4.10)where ( − a is ± ϕ a , and the letter a labels all the fields of the theory. Of course, the parity of the antifields is the reverse47f the corresponding fields, such that the action Σ is bosonic and of zero ghost number.Σ[ ϕ, ϕ g , B, C ] satisfies the linear functional identity δ k Σ = Z d x X a δ k ( C k ) ϕ a δ L Σ δϕ a + X a δ k ( C k ) ϕ g a δ L Σ δϕ g a − (cid:0) dC k + { B k , C k } (cid:1) · δ L Σ δB k − { C k , B p } · δ L Σ δB p − { C k , C p } · δ L Σ δC p (cid:19) = 0 , (4.11)associated to the k -current Ward identities, and the bilinear functional identity Z d x X a δ R Σ δϕ g a δ L Σ δϕ a − (cid:0) dC p + { B k , C p } (cid:1) · δ L Σ δB p −{ C p , B p } · δ L Σ δB k − X a ϕ g a δ k ( C p ) ϕ a ! = 0 , (4.12)associated to the p -current Slavnov–Taylor identities, where the dots stand for the ap-propriately normalised K -invariant scalar products.Here we disentangled the linear and the non-linear Ward identities, however, in orderto discuss possible anomalies it will be more convenient to combine both of them into asingle bilinear G master equation (cid:0) Σ , Σ (cid:1) g = 0 , (4.13)which can be obtained by introducing sources for the sources B and the parameter C [14]. In the absence of anomalies, the above master equation can be elevated to a G master equation for the full effective action, i.e. the 1PI generating functional Γ (cid:0) Γ , Γ (cid:1) g = 0 . (4.14)This, then, is the equation which encapsulates the g invariance of the theory up to anygiven order in perturbation theory.Before discussing the anomalies, let us give an example of Slavnov–Taylor identitiesthat can be obtained from the (to be proved to be) non-anomalous master equation(4.14). For example, one can consider correlation functions involving scalar fields only,with Y n ∈ I X n · δδ Φ( x n ) X · δδC ( x ) (cid:0) Γ , Γ (cid:1) g ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (4.15)48here the notation | means that we set all the classical field ϕ a and sources to zero afterdifferentiation. This gives the Ward identity ∂∂x µ D J µ ( X, x ) Y n ∈ I Φ( X n , x n ) E = X J ⊂ I D Φ A ( x ) Y m ∈ J Φ( X m , x m ) ED(cid:20)
Φtanh(Φ) ( x ) ∗ X (cid:21) A Y n ∈ I \ J Φ( X n , x n ) E , (4.16)where the sum over J ⊂ I is the sum over all odd subsets of indices J inside the odd setof indices I . In the same way (3.67) is the Fourier transform of (cid:18) X · δδB k µ ( x ) (cid:19) (cid:18) X · δδB k ν ( x ) (cid:19) (cid:18) Y · δδC ( y ) (cid:19) (cid:18) Y · δδ Φ( y ) (cid:19) (cid:0) Γ , Γ (cid:1) g (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . (4.17)Let us first briefly recall why the existence of anomalies is equivalent to a cohomologyproblem. It is well known that the master equation (4.14) can in principle be broken bythe renormalisation process at each order n in perturbation theory, such that (cid:0) Γ n , Γ n (cid:1) g = ~ n A n + O ( ~ n +1 ) , (4.18)where Γ n ≡ P p ≤ n ~ p Γ ( p ) is the n -loop renormalised 1PI generating functional, and A n is a local functional of the fields and antifields linear in C . Because of the ‘anti-Jacobi’functional identity (cid:0) Γ , (cid:0) Γ , Γ (cid:1) g (cid:1) g = 0 (4.19)the anomaly nevertheless satisfies the Wess–Zumino consistency condition (cid:0) Γ n , A n (cid:1) g = O ( ~ ) (4.20)and therefore (cid:0) Σ , A n (cid:1) g = 0 , (4.21)where Σ is the classical action. If A n satisfies A n = (cid:0) Σ , Σ ( ♭ n ) (cid:1) g , (4.22)for a local functional of the fields Σ ( ♭ n ) , the anomaly is trivial, because one can simplyadd it to the bare action in order to define a 1PI generating functional which is notanomalous at this order (as we did for example in the last section with the counterterm(3.59) in order to restore Lorentz invariance). The existence of an anomaly therefore49equires that the cohomology of the linearised Slavnov–Taylor operator F → (Σ , F ) onthe set of local functionals {F } of the fields is non-trivial. This cohomology is equivalentto the cohomology H ( δ g ) of the differential operator δ g which generates the non-linear e action on the set of local functionals of the fields identified modulo the equations ofmotion [35].As we already pointed out, the property that A is a local functional is known as thequantum action principle [14]. This principle holds true generally for any well definedregularisation scheme. Because of the rather non-standard character of the duality invari-ant formulation of the theory we are using, it is important to show that such consistentregularisation scheme exists for the theory. Although a fully rigorous proof of the validityof the quantum action principle within the Pauli–Villars regularisation scheme defined inthe preceding section is beyond the scope of the present paper, the one-loop computationof the preceding section provides a strong indication for its validity. SU (8) -equivariant cohomology of e To investigate the general structure of anomalies, we need a basis of local functionals.For this purpose it is convenient to consider functions of the fields and their covariantderivatives, defined as d B Φ ≡ d Φ + [ B k , Φ ] − Φtanh Φ ∗ B p (4.23)for the scalars, and similarly for the other fields. Keeping in mind that δ g and the exteriorderivative anti- commute, we then have δ g (cid:0) d B Φ (cid:1) = −{ C k , d B Φ } − d B (cid:16) Φtanh Φ (cid:17) ∗ C p . (4.24)In deriving this formula, we make use of the closure property (2.85) in the form δ p ( C p ) (cid:18) Φtanh Φ (cid:19) ∗ B p + δ p ( B p ) (cid:18) Φtanh Φ (cid:19) ∗ C p = − (cid:8) B p , C p (cid:9) ∗ Φ , (4.25)which allows us to trade one expression (the variation of Φ / tanh Φ) which we cannotwrite in closed-form in terms of another (the B p covariantisation of the last term in(4.24)) which we also cannot write in closed-form.Given a basis of local functionals, a potential anomaly A decomposes into a termlinear in C k and a term linear in C p A = Z (cid:16)
F · C k + G · C p (cid:17) , (4.26)50ith two local functionals F and G of the fields, the current sources and their derivatives. F and G take values in k and p , respectively. Accordingly, the Wess–Zumino consistencycondition δ g A = 0 decomposes into three components Z δ k (cid:0) F · C k (cid:1) = 0 , Z (cid:16) δ p F · C k + δ k (cid:0) G · C p (cid:1)(cid:17) = 0 , Z (cid:16) − F · C p + δ p G · C p (cid:17) = 0 , (4.27)corresponding to the coefficients of C k , C k C p and C p , respectively. The first equation isthe condition that R F · C k defines a consistent anomaly for the k current Ward identity. A priori , there are therefore two kinds of anomalies, the ones associated to the linearlyrealised subgroup K and determined by R F · C k in H ( δ k ), which would have to beextended to the non-linear representation by an appropriate R G · C p ; and ‘genuinelynon-linear anomalies’ , with R F · C k = 0, associated to the non-linear transformationsonly and given by R G · C p . The latter expression is then a k -invariant functional of thefields and the current sources which is δ p closed by (4.27). If it can be written as the δ g variation of a functional of the fields, the latter must be K invariant, and the action of δ g and δ p on it are identical. Such a functional R G · C p , if non-trivial, defines by definitiona cocycle representative of the equivariant cohomology H K ( δ p ) of δ g with respect to K .This property can be summarised in the following exact sequence0 ֒ → H K ( δ p ) ι −→ H ( δ g ) π −→ H ( δ k ) , (4.28)which states that to each element of H K ( δ p ) there corresponds one element of H ( δ g ),and that all the other elements of H ( δ g ) correspond to elements of H ( δ k ) (although π is not necessarily surjective a priori ).Let us first consider the non-trivial anomalies associated to k current anomalies. Theanomalies associated to a linearly realised group are well known, and are classified bysymmetric Casimirs. A nice way to derive such anomalies is by means of the ‘Russianformula’ [36, 37, 38, 39]( d + δ k ) (cid:0) B k + C k (cid:1) + (cid:0) B k + C k (cid:1) = F (0) k ≡ dB k + B k , (4.29)to derive a ( d + δ k )-cocycle from any symmetric Casimirs by use of the Cartan homotopyformula. In four dimensions, the relevant Casimir is the symmetric tensor of rank three, The superscript on H here refers to the ‘ghost number’. d + δ k )Tr (cid:16) ˜ B k F (0) k −
12 ˜ B k F (0) k + 110 ˜ B k (cid:17) = Tr F (0) k = 0 , (4.30)where we define the extended connection (always indicated by a tilde) as˜ B k ≡ B k + C k , (4.31)and the trace is taken in the complex representation of k . Picking the component of theChern–Simons function of form degree four, one obtains from this equation the conven-tional non-abelian Adler–Bardeen anomaly A k = Z Tr dC k (cid:16) B k F (0) k − B k (cid:17) , (4.32)which satisfies δ k A k = 0 . (4.33)Here we are specifically interested in the case when the rigid symmetry group G does not admit an invariant tensor of rank three, as for G = E . In this case the trace mustbe taken in a complex representation of the subgroup K , and (4.30) cannot be definedfrom the straightforward extension of the ‘linear’ Russian formula (4.29) to the linearformula ( d + δ g ) (cid:0) B + C (cid:1) + (cid:0) B + C (cid:1) = F g = dB + B , (4.34)for the full Lie algebra g , because this formula would only make sense in a linear rep-resentation of E . Instead we must now look for a non-linear variant of the Russianformula. To this aim, we first observe that the closure of the non-linear representationof g on the fermion fields implies δ g (cid:0) C k + tanh(Φ / ∗ C p (cid:1) + (cid:0) C k + tanh(Φ / ∗ C p (cid:1) = 0 . (4.35)in the given complex representation of k . This formula (which is a non-linear analogueof the usual BRST variation, cf. second formula in (4.5)) suggests that one can define anon-linear Russian formula for the g symmetry in the fundamental representation of k .The most natural guess for the (extended) ‘non-linear g connection’ is˜ B ≡ B k + C k + tanh(Φ / ∗ ( B p + C p ) , (4.36)which is indeed valued in the Lie subalgebra k . In turn, this motivates the followingdefinition of the (extended) g field strength, viz. ˜ F g ≡ ( d + δ g ) ˜ B + ˜ B , (4.37)52hich one then computes (using the extended version of (4.35) to B + C ) to be˜ F g = F k + tanh(Φ / ∗ F p + d B (cid:0) tanh(Φ / (cid:1) ∗ ( B p + C p ) , (4.38)with F k ≡ dB k + B k + B p , F p ≡ dB p + { B k , B p } , (4.39)and d B tanh(Φ /
2) defined in terms of the covariant derivative (4.23) similarly as in (4.24).In contradistinction to the conventional Russian formula, the extended field-strength(4.38) is not only the ‘horizontal’ two-form curvature, but has an extra component linearin the parameter C p . Nevertheless, it does not depend on C k , and transforms covariantlywith respect to k in the adjoint representation k , δ k ˜ F g = − (cid:2) C k , ˜ F g (cid:3) . (4.40)With these definitions, the Cartan homotopy formula( d + δ g )Tr (cid:16) ˜ B g ˜ F g −
12 ˜ B g ˜ F g + 110 ˜ B g (cid:17) = Tr ˜ F g (4.41)therefore admits a non-vanishing right-hand-side (whereas for a linear representation of E , the right hand side of (4.41) would simply vanish). But because it is independent of C k and k -invariant, and hence k -basic, it defines a cocycle of the equivariant cohomology H K ( δ p ). Note that it is a cocycle of ‘ghost number’ 2, because the associated 4-formcomponent is of ‘ghost number’ 2. We here tacitly use the corollary of the algebraicPoincar´e lemma, i.e. all d -closed functions of the fields and their derivatives of form-degree p ≤ d -exact, whence the cohomology of a differential δ p in the complex oflocal functionals of ‘ghost number’ n is isomorphic to the cohomology of the extendeddifferential d + δ p in the complex of functions of the fields of form-degree plus ‘ghostnumber’ 4 + n [40].If this cocycle is trivial in H K ( δ p ), i.e. if there exists a k -basic function ˜ M such that( d + δ p ) ˜ M = Tr ˜ F g , (4.42)one can extend the k Adler–Bardeen anomaly to a g anomaly by considering the integralof the 4-form component A = Z (cid:18) Tr (cid:16) ˜ B g ˜ F g −
12 ˜ B g ˜ F g + 110 ˜ B g (cid:17)(cid:12)(cid:12)(cid:12) (4 , − M (4 , (cid:19) (4.43)It follows that the only possible obstruction to extend a k anomaly in H ( δ k ) to a full g anomaly in H ( δ g ) are defined by cohomology classes of the second equivariant cohomol-ogy group H K ( δ p ). 53ne can summarise these properties into the exact sequence0 ֒ → H K ( δ p ) ι −→ H ( δ g ) π −→ H ( δ k ) −→ H K ( δ p ) (4.44)which states that the H ( δ g ) and H ( δ k ) only differ by cocycles associated to equivariantcohomology classes. The last arrow is the map which associates to any K consistentanomaly, the corresponding invariant polynomial in ˜ F g .Now that we have motivated our interest in the equivariant cohomology, we are goingto prove that it is trivial. The intuitive idea is the following, the equivariant cohomologyon the set of local functional of the fields is closely related to the equivariant cohomologyon the set of functions of the scalars only, and the latter is homomorphic to the De Rhamcohomology of the coset space G/K ∼ = R n which is trivial [34].In order to carry out this program, it will turn out to be useful to introduce a filtrationin terms of the order of the functional in naked scalar fields Φ, (considering d B Φ asindependent). The expansion of the variation of Φ and its covariant derivative are δ p Φ = C p + 13 [Φ , [Φ , C p ]] −
145 [Φ , [Φ , [Φ , [Φ , C p ]]]] + O (Φ ) ,d B Φ = − B p + d B k Φ −
13 [Φ , [Φ , B p ]] + 145 [Φ , [Φ , [Φ , [Φ , B p ]]]] + O (Φ ) ,δ g C k (cid:0) d B k Φ (cid:1) = − d B k C p + [Φ , { B p , C p } ] − d B k [Φ , [Φ , C p ]] + O (Φ ) . (4.45)The first order in Φ of the equivariant differential only acts on Φ itself as δ g ( − C k Φ = C p .Any SU (8)-invariant local function of the fields admits an expansion X = X k ∈ N X ( n + k ) (4.46)and δ p X = 0 ⇒ δ p ( − X ( n ) = 0 . (4.47)If X ( n ) depends non-trivially on C p or Φ, then there exist a function Y ( n +1) such that X ( n ) = δ p ( − Y ( n +1) [40]. To see this, let us define the trivialising homotopy σ , which actstrivially on all fields, but C p σC p = Φ , { σ, δ p ( − } X = N X ≡ Z d x (cid:16) C p δδC p + Φ δδ Φ (cid:17) X , (4.48)and [
N, δ p ( − ] = [ N, σ ] = 0 . (4.49)54f X ( n ) depends non-trivially on C p or Φ, N − X ( n ) exists and X ( n ) = { σ, δ p ( − } N − X ( n ) = δ p ( − σN − X ( n ) (4.50)Now with Y ( n +1) = σN − X ( n ) , X − δ p Y ( n +1) = X ( n +1) − δ p (0) Y ( n +1) + O (Φ n +2 ) . (4.51)Using the trivialising homotopy, one proves in the same way that Y ( n +2) exists such that X − δ p (cid:0) Y ( n +1) + Y ( n +2) (cid:1) = X ( n +2) − δ p (1) Y ( n +1) − δ p (0) Y ( n +2) + O (Φ n +3 ) . (4.52)Iteratively one proves that there exist a formal power series Y = X k ∈ N Y ( n +1+ k ) (4.53)in Φ that trivialises X , X = δ p Y . (4.54)This proof extends trivially to functionals [40], and therefore H nK ( δ p ) ∼ = 0 for n ≥ . (4.55)As a direct consequence, the exact sequence (4.44) implies the isomorphism H ( δ g ) ∼ = H ( δ k ) . (4.56)The equivalence of these two cohomology groups is a main result of this paper: it statesthat the e consistent anomalies are in one-to-one correspondence with the su (8) consis-tent anomalies. In particular, it follows that their coefficients are the same, establishingas a corollary that the absence of anomalies for the su (8) current Ward identities im-plies the absence of anomalies for the non-linear e Ward identities. This statementcompletes our proof that the rigid E symmetry of d = 4 N = 8 supergravity is notanomalous in perturbation theory.In the remaining part of this section, we want to illustrate in some more detail how apotential su (8) Adler–Bardeen anomaly would generalise to an e anomaly. The three-form component Tr ˜ F k | (3 , is cubic in C p , and being δ p -closed by construction, thereexists an SU (8)-invariant function M (3 , of the fields quadratic in C p such thatTr ˜ F k | (3 , = δ g M ( F k , F p , B p , d B Φ , Φ , C p ) (3 , . (4.57) Note that although δ p (0) vanishes on Φ and the fermion fields, it acts non-trivially on the electro-magnetic fields and their ghost, and so δ p (0) Y ( n +1) does not vanish in general. F k | (4 , − dM (3 , is itself δ p -closed because of the Bianchi identity, δ p (cid:0) Tr ˜ F k | (4 , − dM (3 , (cid:1) = − d Tr ˜ F k | (3 , + d δ p M (3 , = 0 , (4.58)and being quadratic in C p , there exists a K -invariant function M (4 , of the fields linearin C p such thatTr ˜ F k | (4 , = δ g M ( F k , F p , B p , d B Φ , Φ , C p ) (4 , + dM ( F k , F p , B p , d B Φ , Φ , C p ) (3 , . (4.59)The consistent E anomaly is defined as Z (cid:18) Tr (cid:16) ˜ B k ˜ F k −
12 ˜ B k ˜ F k + 110 ˜ B k (cid:17)(cid:12)(cid:12)(cid:12) (4 , − M ( F k , F p , B p , d B Φ , Φ , C p ) (4 , (cid:19) (4.60)whereTr (cid:16) ˜ B k ˜ F k −
12 ˜ B k ˜ F k + 110 ˜ B k (cid:17)(cid:12)(cid:12)(cid:12) (4 , = Tr (cid:16) C k + tanh(Φ / ∗ C p (cid:17) d (cid:18)(cid:16) B k + tanh(Φ / ∗ B p (cid:17) d (cid:16) B k + tanh(Φ / ∗ B p (cid:17) + 12 (cid:16) B k + tanh(Φ / ∗ B p (cid:17) (cid:19) + Tr d B (cid:0) tanh(Φ / (cid:1) ∗ C p (cid:18)(cid:16) B k + tanh(Φ / ∗ B p (cid:17) d (cid:16) B k + tanh(Φ / ∗ B p (cid:17) + d (cid:16) B k + tanh(Φ / ∗ B p (cid:17) (cid:16) B k + tanh(Φ / ∗ B p (cid:17) + 32 (cid:16) B k + tanh(Φ / ∗ B p (cid:17) (cid:19) . (4.61)One computes perturbatively that M (3 , = 18 Tr (cid:18) − [Φ , B p ] { C p , B p } + 12 (cid:16) { C p , d B k Φ } − [Φ , d B k C p ] (cid:17)n [Φ , B p ] , { C p , B p } o(cid:19) + O (Φ ) , (4.62)and M (4 , = 38 Tr n { C p , B p } − { C p , d B k Φ } + [Φ , d B k C p ] , [Φ , B p ] o(cid:0) F k − B p (cid:1) + 14 Tr n [Φ , B p ] , { C p , B p } o(cid:16) [Φ , F p ] + { B p , d B k Φ } (cid:17) + O (Φ ) . (4.63)There is no difficulty in computing higher order terms in Φ, but the complete solution isnot obvious. Anyway, the important property is that it exists, at least as a formal powerseries in Φ (the issue of convergence being irrelevant in perturbative theory).56he results of this section extend straightforwardly to any consistent K anomaly forany supergravity theory in arbitrary dimensions. For example, the solution for Tr ˜ F k is rather trivial when K admits a U (1) factor, as for lower N -extended supergravitytheories. In that case, Tr ˜ F g ∧ R ab ∧ R ab = 0 and one has the anomaly A u (1) = Z Tr (cid:16) C k + tanh(Φ / ∗ C p (cid:17) R ab ∧ R ab . (4.64)In particular, this anomaly does not vanish when C p is constant, and the current sourcesare set to zero. It follows that the rigid Ward identities are anomalous at one-loop if thecoefficient does not vanish, as is the case for minimal N = 4 supergravity with dualitygroup SL (2 , R ), and more generally for N ≤ G = SL (2 , R ), we can spell out the above formulas in explicitdetail. In this case, the second relation in (4.5) becomes δ sl α = i W ¯ W , δ sl W = − iα W , (4.65)where α ≡ C k and W ≡ C p are real and complex anticommuting numbers, respectively. Ifwe denote by φ the complex scalar parametrising the coset SL (2 , R ) /SO (2) (the analogueof Φ) the formula (4.3) can be worked out as δ sl φ = (cid:16)
12 + | φ | tanh 2 | φ | (cid:17) W − iαφ + (cid:16) | φ | − | φ | tanh 2 | φ | (cid:17) ¯ W φ (4.66)and the anomaly (4.64) reads A u (1) = Z (cid:16) α − i tanh | φ | | φ | (cid:0) ¯ φ W − φ ¯ W (cid:1)(cid:17) R ab ∧ R ab (4.67)Equivalently, within the triangular gauge parametrisation of SL (2 , R ) /SO (2) by the com-plex modulus τ = τ + iτ , and the (hopefully self explanatory) notation C = C h C e C f − C h , (4.68)the algebra reads δ sl C h = − C e C f , δ sl C e = − C h C e , δ sl C f = 2 C h C f ,δ sl τ = − C e − C h τ + C f τ . (4.69)57he consistent anomaly (4.64) then becomes ( C f τ being the parameter of the compen-sating u (1) transformation in the triangular gauge) A u (1) = Z C f τ R ab ∧ R ab . (4.70)Indeed, explicit computation shows that δ sl (cid:0) C f τ (cid:1) = 0 , (4.71)but C f τ itself cannot be written as δ sl F ( τ ): indeed, the vanishing of the C e componentof δ sl F ( τ ) implies that F is a function of τ only, and the vanishing of the C h componentthen entails that F must be constant.In the conventional formulation of N = 4 supergravity, and similarly in N = 2 super-gravity with a semi-simple duality group SL (2 , R ) × SO (2 , n ), we see that the non-linearlyrealised generator f of sl is anomalous at one-loop. More generally, in N = 2 supergrav-ity theories with vector multiplets scalar fields parametrising a symmetric special K¨ahlermanifold, the duality group will be anomalous at one-loop. Indeed, one computes simi-larly as in [17] that the addition of matter multiplets does not permit to cancel the U (1)gravitational anomaly, the anomaly coefficient of (4.64) being proportional to 24 + 12 n V in N = 4 supergravity coupled to n V vector multiplets, and to 102 + 10 n V + 3 n H in N = 2supergravity coupled to n V vector multiplets and n H hypermultiplets. E with gauge invariance Up to this point we have discussed the properties of the E symmetry and its possibleanomalies, irrespectively of its compatibility with gauge invariance. We now extendthis discussion to the full quantum theory, with the aim of deriving the Ward identitiesassociated to the conservation of the E Noether current, thereby corroborating ourmain claim that the non-linear E symmetry is compatible with all gauge symmetriesof the theory, in the sense that it can be implemented order by order in a loop expansionof the full effective action. To this aim, we have to make use of the BRST formalism (seee.g. [19, 40]). Because the algebra of gauge transformations is ‘open’, we have to goone step further by including higher order ghost interactions [41], and ultimately bringin the full machinery of the Batalin–Vilkovisky formalism [42]. In addition to the usualghosts and antighosts (anti-commuting for the bosonic transformations, and commuting That is, the gauge algebra closes only modulo the equations of motion. E with the BRST symmetryis then encoded into two corresponding mutually compatible ‘master equations’.Now, a complete treatment of our E invariant formulation of maximal supergravityalong these lines would be very involved and cumbersome, and certainly unsuitable forpractical computations of the type performed in [5, 6]. Instead, we here focus on thespecific features of the duality invariant formulation in comparison with the conventionalformulation of the theory, and the fact that the cancellation of E anomalies, togetherwith the well admitted absence of diffeomorphism and supersymmetry anomalies in fourspace-time dimensions, eliminates any obstruction towards implementing the BRST and E master equations at any order in perturbation theory. We emphasise again thatthese results do not preclude the appearance of divergent counterterms, but they ensurethat potential divergences must respect the full E symmetry of the theory. Following [19] we will designate by e ‡ µa , A ‡ i m , ψ ‡ µi and χ ‡ ijk the antifields associated to thevierbein, the vector fields, the gravitino and the Dirac fields, respectively, and by ξ µ , Ω ab and c m the anticommuting ghost fields associated to diffeomorphism invariance, Lorentzinvariance, and abelian gauge invariance, respectively; the commuting supersymmetryghost is ǫ i . In addition we also need antifields ξ ‡ , Ω ‡ ab , c ‡ m and ǫ ‡ i for these ghost fields.Regarding gauge-fixing, the e Ward identities can be implemented without furtherado as long as the gauge-fixing manifestly preserves E invariance. Of course, this istrivially the case for any sensible gauge choice for diffeomorphism and Lorentz invariance,and it is also true for the Coulomb gauge we are using. An E -invariant gauge choicefor local supersymmetry can be achieved in terms of the SU (8)-covariant derivative D µ ψ iν = ∂ µ ψ i − (cid:16) u jkIJ ∂ µ u ikIJ − v jkIJ ∂ µ v ikIJ (cid:17) ψ jν . (5.1)(for instance by setting D µ ψ iµ = 0), with the extra proviso that the supersymmetryantighost and the Nielsen–Kallosh field transform in the non-linear representation of E conjugate to the one of ψ i and ǫ i .In the conventional formulation of the theory, the supersymmetry algebra closes onthe fermionic fields only modulo terms linear in the fermionic equations of motion. Withinthe Batalin–Vilkovisky approach, this problem is cured by introducing terms quadraticin the fermion antifields in the action. The functional form of the BRST operator then59ncludes these terms in the BRST transformation of the fermions. Let us briefly recallhow this works. Collectively designating the fields and ghosts as ϕ a , and their Grassmannparity as ( − a , we have s ϕ a = X b K ( ϕ ) ab δ L Σ δϕ b . (5.2)This equation simply expresses the fact that algebra closes (that is, s ≈
0) only if theequations of motion are imposed. Introducing antifields ϕ ‡ a , the action Σ[ ϕ, ϕ ‡ ] readsΣ( ϕ a , ϕ ‡ a ) = 1 κ S [ ϕ ] − Z d x X a ( − a ϕ ‡ a sϕ a + κ X ab ϕ ‡ a K ab ( ϕ ) ϕ ‡ b ! , (5.3)The symmetry of the action and the closure of the algebra can then be combined into asingle BRST master equation (indexed by a ‡ to distinguish it from the E masterequation to be introduced below) (cid:0) Σ , Σ (cid:1) ‡ ≡ X a Z d x δ R Σ δϕ ‡ a δ L Σ δϕ a = 0 . (5.4)This equation requires in addition that sK ab + 12 X c (cid:18) K ac ∂ L sϕ b ∂ϕ c + ( − ( a +1)( b +1) K bc ∂ L sϕ a ∂ϕ c (cid:19) = 0 , (5.5) X d (cid:18) K cd ∂ L K ab ∂ϕ d + ( − c ( a + b ) K ad ∂ L K bc ∂ϕ d + ( − b ( c + a ) K bd ∂ L K ca ∂ϕ d (cid:19) = 0 . (5.6)These identities are automatically satisfied modulo the equations of motion by integrabil-ity of definition (5.2). For N = 8 supergravity in the conventional formulation, the termquadratic in the antifields in (5.3) only involves the fermionic antifields ψ ∗ iµ and χ ∗ ijk ina first approximation ( i.e. neglecting the antifield dependent terms in (5.2)). In super-gravity, such K ab components associated to the fermions are bilinear in the superghosts ǫ i (and depend as well on the vierbeine and the scalar fields), and the validity of the identity(5.5) is ensured by certain cubic Fierz identities in ǫ i . (5.6) is trivially satisfied because K ab only depends on fields on which the algebra is satisfied off-shell. Nevertheless, thismodification of the BRST transformation of the fermions also affects the closure of thealgebra on the bosons, such that the BRST transformation of the Lorentz ghost Ω ab mustinclude terms linear in the fermion antifields as well. This entails terms quadratic in the See e.g. [40] for further information. ‡ ab as well. Considering these terms inthe nilpotency of the linearised Slavnov–Taylor operator (Σ , · ) on the vierbeine, i.e. (cid:0) Σ , (cid:0) Σ , e aµ (cid:1) ‡ (cid:1) ‡ = κ X a (cid:16) ¯ ǫ i γ a K i a µ ( ϕ ) + ¯ ǫ i γ a K µi a ( ϕ ) + K ab a ( ϕ ) e bµ (cid:17) ϕ ‡ a , (5.7)where K i a µ , K µi a and K ab a are the components of K ab to be contracted with ¯ ψ ‡ µi , ¯ ψ ‡ µi and Ω ‡ ab , respectively, one observes that K ab a is determined in function of K i a µ , K νi a , suchthat the modification of the action to be carried out amounts to replacing the gravitinoantifields appearing in the term quadratic in the fermion antifields by ψ ‡ µi → ψ ‡ µi − e µa γ b Ω ‡ ab ǫ i . (5.8)In our manifestly E -invariant formulation, the situation is the same with regard tothe fermion fields, but now the vector fields are also governed by a first order Lagrangian(first order in the time derivative), and hence the algebra of gauge transformations onthe vectors likewise involves the equations of motion. It is important that the equationof motion of the vector fields here appears as (2.6), and not in its integrated form (2.8),as required for the consistency of the Batalin–Vilkovisky formalism. We have checkedthat the diffeomorphism transformations do close among themselves, and that local su-persymmetry closes on the vector fields. However, their commutator on the vector fieldsclose modulo the equations of motion of the fermion fields, viz. s A IJ i = N ξ √ h e a e b i (cid:20) u ijIJ (cid:18) ¯ ǫ i γ µ γ ab δ L Sδ ¯ ψ µ j + 12¯ ǫ k γ ab δ L Sδ ¯ χ ijk (cid:19) − v ijIJ (cid:18) ¯ ǫ i γ µ γ ab δ L Sδ ¯ ψ jµ + 12¯ ǫ k γ ab δ L Sδ ¯ χ ijk (cid:19)(cid:21) . (5.9)To remain consistent with the basic symmetry property K ab = − ( − ab K ba , the closureof diffeomorphisms with local supersymmetry on the fermion fields then requires corre-spondingly the equations of motion of the vector fields. We checked that this is indeedthe case, and that the fermion equations of motion are not involved (as follows triviallyfrom Lorentz invariance). The quadratic terms in the fermion antifields are also modifiedby non-manifestly diffeomorphism invariant terms, such that they are manifestly dualityinvariant (and so do not depend on the scalar fields).The quadratic terms in the antifields of the gauge fields are responsible for the quarticterms in the ghosts that appear in supergravity [42, 43]. It follows that in the dualityinvariant formulation, we will also have quartic terms depending on the diffeomorphism61host ξ , the supersymmetry ghost ǫ i , the abelian antighost ¯ c m and the supersymmetryantighost η i , which in a flat Landau-type gauge for local supersymmetry like D µ ψ iµ ≈ N ξ √ h e a e b i ∂ i ¯ c IJ (cid:16) u ijIJ ¯ ǫ i γ µ γ ab D µ η j − v ijIJ ¯ ǫ i γ µ γ ab D µ η j (cid:17) + c.c. . (5.10)In general, such vertices do not contribute to amplitudes of physical fields. However,the renormalisation of the theory in the absence of regularisation preserving all gaugesymmetries requires the renormalisation of the composite BRST transformations. Inconsequence, the correlation functions involving the insertion of the BRST transformationof the vector fields do involve such vertices.Note that one can obtain the solution Σ of the master equation in the covariantformulation form the duality invariant one, by carrying out the Gaussian integration ofthe momentum variable Π m i for the complete action Σ with antifields, similarly as inthe second section. Considering for example the terms S ghostvec = Z d x (cid:18) A ∗ i m (cid:16) − (cid:0) ξ j + N j ξ (cid:1) F m ij + N √ h ξ h ij ε j kl J mn F n kl (cid:17) − c ∗ m (cid:16) ξ i ξ j F m ij + ξ ξ i (cid:0) − N j F m ij + N √ h h ij ε j kl J mn F n kl (cid:1)(cid:17) + · · · (cid:19) (5.11)and the F m ij dependent terms that appear in the supersymmetry transformations of thefermions, as well as the gauge-fixing terms, one sees that the vector fields only appearthrough their field strength F m ij . It is therefore important (and true!) that the wholequantum action can be treated in the way described in the second section. The Re[ F IJ ij ]dependent terms in the supersymmetry variation of the fermions and of the sl ( C ) ghostΩ ab , as well as the ones in the diffeomorphism variation of the vector fields and theirghost c IJ , are replaced upon Gaussian integration of the momentum variables Π IJ i bythe solution of Π IJ i according to their equation of motion. This step will restore manifestdiffeomorphism invariance. All these terms will also produce quadratic terms in thesources, which define the required equations of motion in order to close the gauge algebrain the E invariant formulation of the theory. This way one obtains that the only termsquadratic in Im[ A ∗ i IJ ] involve the diffeomorphism ghost ξ µ , and that they vanish once oneputs the source Re[ A ∗ i IJ ] equal to zero, in agreement with the explicit computation in theformalism. 62 .2 BRST extended current Having set up the BRST transformations and the Batalin–Vilkovisky framework, thenext task is to define the e current Ward identities in such a way that their mutualconsistency is preserved also in perturbative quantisation. To this aim, one must inprinciple couple the whole chain of operators associated to the current via the BRSTdescent equations, that is, extend the current constructed in section 2.4 by appropriateghost and antifeld terms. Because the classical current defines a physical Noether chargewhich is BRST invariant, one has sJ (3 , (Λ) = − dJ (2 , (Λ) , (5.12)where we now write J (3 , ≡ J , indicating the form degree and ghost number. Consideringthe functional BRST operator s acting on both fields and antifields, the conservation ofthe current reads dJ (3 , (Λ) = − sJ (4 , − (Λ) , (5.13)where J (4 , − is the composite operator linear in the antifields J (4 , − (Λ) ≡ X a ϕ ‡ a δ e (Λ) ϕ a . (5.14)Then dJ (3 , (Λ) = X a (cid:18) δ e (Λ) ϕ a δ L Σ δϕ a + δ e (Λ) ϕ ‡ a δ L Σ δϕ ‡ a (cid:19) , (5.15)as defined by the Noether procedure on the complete gauge fixed action Σ[ ϕ, ϕ ‡ ], andwhere ϕ ‡ a transforms with respect to E in the representation conjugate to ϕ a .The whole chain of operators appearing in the descent equations defines an extendedform ˜ J which is a cocycle of the extended differential d + s [35],( d + s ) (cid:0) J (4 , − + J (3 , + J (2 , + J (1 , + J (0 , (cid:1) = 0 . (5.16)The complete form of the extended current ˜ J which now also depends on the ghosts andantifields is again very complicated, and its explicit form would not be very illuminating.Let us nevertheless discuss some salient features of this extended current, neglectingterms depending on the antifields and terms linear in the equations of motion. Withthese assumptions we can take J (4 , − to vanish, and J (3 , can be identified with theGaillard–Zumino current constructed in section 2.4, where we also disregard the ‘curl63omponent’ leading to a trivial cocycle. Let us first rewrite the components of J (3 , interms of differential forms, cf. (2.51, 2.53) R ij = − ie a ∧ (cid:16) ¯ ψ i ∧ γ a ψ j − δ ij ¯ ψ k ∧ γ a ψ k (cid:17) − ε abcd e b ∧ e c ∧ e d (cid:16) ¯ χ ikl γ a χ jkl − δ ij ¯ χ klp γ a χ klp (cid:17) ,R ijkl = − ⋆ ˆ A ijkl + i e a ∧ e b ∧ (cid:16) ¯ χ [ ijk γ ab ψ l ] + 14! ε ijklpmnpq ¯ χ mnp γ ab ψ q (cid:17) , (5.17)and J (2) GZ (Λ) = − A m ∧ F n Λ mp Ω pn . (5.18)Conveniently, the extended current ˜ J takes a form similar to the current constructed insection 2.4 ˜ J (Λ) = − e i ξ tr (cid:0) V − ˜ R V Λ (cid:1) + ˜ J (2) GZ (Λ) . (5.19)so we only need to explain how to obtain the ‘tilded’ version of the above currents. Theoperator i ξ is the (commuting) Cartan contraction with respect to the anti-commutingvector ξ µ ; its exponentiated action takes care automatically of all modifications involv-ing the diffeomorphism ghost fields ξ µ [44]. In order to understand how to extend theremaining piece J (2) GZ (Λ) to ˜ J (2) GZ (Λ), it is again convenient to write a Russian formula( d + s ) (cid:0) A m + c m (cid:1) = e i ξ ˜ F m , (5.20)where the extended curvature ˜ F is defined as˜ F IJ ≡ F IJ + u ijIJ (cid:16)
14 ¯ ǫ k e a γ a χ ijk + 2¯ ǫ i ψ j + ¯ ǫ i ǫ j (cid:17) − v ijIJ (cid:16)
14 ¯ ǫ k e a γ a χ ijk + 2¯ ǫ i ψ j + ¯ ǫ i ǫ j (cid:17) = ˆ F IJ + u ijIJ (cid:16)
14 [ ψ + ǫ ] k e a γ a χ ijk + [ ψ + ǫ ] i [ ψ + ǫ ] j (cid:17) − v ijIJ (cid:16)
14 [ ψ + ǫ ] k e a γ a χ ijk + [ ψ + ǫ ] i [ ψ + ǫ ] j (cid:17) . (5.21)The gravitinos here appear only in the supercovariantisation of F IJ or through the com-bination ψ i + ǫ i . In addition we need the nilpotent extended differential [44]˜ d ≡ e − i ξ ( d + s ) e i ξ = d + s − L ξ + i (¯ ǫγǫ ) , (5.22)where i (¯ ǫγǫ ) is the Cartan contraction with respect to the vector ¯ ǫ i γ µ ǫ i . Defining˜ A m ≡ A m + c m − i ξ A m , (5.23)it is obvious that ( d + s ) e i ξ (cid:16) ˜ A m ∧ ˜ F n Λ mp Ω pn (cid:17) = e i ξ (cid:16) ˜ F m ∧ ˜ F n Λ mp Ω pn (cid:17) . (5.24)64he right-hand-side being gauge-invariant, the extended form ˜ R can be obtained fromthe equation ˜ d (cid:18)
124 tr (cid:0) V − ˜ R V Λ (cid:1)(cid:19) = 14 ˜ F m ∧ ˜ F n Λ mp Ω pn . (5.25)which is an extended version of the Gaillard-Zumino construction. For any gauge in-variant extended form, such as ˜ R or ˜ F m , supersymmetry covariance implies that thegravitino field ψ i only appears in supercovariant forms, or ‘naked’, through the wedgeproduct of ψ i + ǫ i with supercovariant forms. It follows that ˜ R is simply obtained from R by performing the replacement ψ i → ψ i + ǫ i everywhere inside (5.17).The (4 ,
0) component of (5.25) is simply the current conservation. To see that (5.25)is indeed satisfied for the other components, let us consider the (0 ,
4) component of thisequation. From (5.21) we see that the right hand side is the e component of thesquare of u ijIJ ¯ ǫ i ǫ j − v ijIJ ¯ ǫ i ǫ j . By E covariance, the scalar fields dependence thenreduces to a similarity transformation with respect to V (as the left hand side), and onecan concentrate on the e element quadratic in ¯ ǫ i ǫ j . Because (for commuting spinors)¯ ǫ [ i ǫ j ¯ ǫ k ǫ l ] = 0 , (5.26)this term only contributes in the su (8) component i (¯ ǫ i ǫ k )(¯ ǫ j ǫ k ) − i δ ij (¯ ǫ k ǫ l )(¯ ǫ k ǫ l ). Because˜ R has a vanishing (0 ,
3) component, the left hand side is the Cartan contraction of its(1 ,
2) component − ie a (cid:0) ¯ ǫ i γ a ǫ j − δ ij ¯ ǫ l γ a ǫ l (cid:1) with the vector ǫ i γ µ ǫ i . Using the Fierz identity(¯ ǫ i ǫ k )(¯ ǫ j ǫ k ) − δ ij (¯ ǫ k ǫ l )(¯ ǫ k ǫ l ) = −
12 (¯ ǫ k γ a ǫ k ) (cid:16) ¯ ǫ i γ a ǫ j − δ ij ¯ ǫ l γ a ǫ l (cid:17) , (5.27)one obtains the validity of the (0 ,
4) component of (5.17).Considering the complete antifield dependent extended current ˜ J , one can couplethe E current to the action in a way fully consistent with BRST invariance. Indeed,considering sources B ( p, − p ) for each component of the current ˜ J , one obtains thatΣ[ B ] = Σ + Z ˜ B ∧ ˜ J , (5.28)where we use the Berezin notation Z Tr ˜ B ∧ ˜ J = Z Tr (cid:16) B (0 , J (4 , − + B (1 , ∧ J (3 , + B (2 , − ∧ J (2 , + B (3 , − ∧ J (1 , + B (4 , − J (0 , (cid:17) , (5.29) We have computed the complete ξ µ dependent part of ˜ J including the antifields to check that thenon-manifest Lorentz invariance does not give rise to extra difficulties. Nevertheless, its exhibition wouldnot shed much light in this discussion. However we have not computed explicitly the ǫ i dependent termsthat would involve the quadratic terms in the antifields of the solution Σ of the master equation. (cid:0) Σ , Σ (cid:1) ‡ − Z d ˜ B ∧ δ Σ δ ˜ B = 0 . (5.30)This formal notation means that ( d + s ) ˜ B = 0 . (5.31)This would be enough for insertions of one single current in a BRST invariant way, butconsistency with E will require the consideration of higher order terms in ˜ B in Σ, suchthat these equations are then only valid up to quadratic terms in the sources B ( p, − p ) .Introducing a source for the E current, the rigid e Ward identity is promotedto a local e Ward identity expressing the conservation of the E current, such that δ e ˜ B = − dC − { ˜ B, C } . (5.32)All the components of ˜ B thus transform in the adjoint representation, and B (1 , trans-forms as an e gauge field. In order for the current Ward identity to be satisfied, eachderivative in the action must be replaced by an e covariant derivative with respect tothe gauge field B (1 , . It follows that the linear component is defined as R Tr B (1 , ∧ J (3 , ,by definition of the Noether current. The kinetic terms of the scalar fields, the Maxwellfields, their ghosts, and the supersymmetry ghost being quadratic in derivatives, they giverise to bilinear terms in B (1 , in the action. The compatibility with BRST invariancetherefore requires to also add quadratic terms in the other sources defining ˜ B .In order to ensure that δ g anticommutes with s , one must then define the BRSTtransformation of ˜ B such that ( d + s ) ˜ B + ˜ B = 0 . (5.33)In this way one has the consistent ‘very extended’ Russian formula( d + s + δ g ) (cid:0) ˜ B + C (cid:1) + (cid:0) ˜ B + C (cid:1) = 0 , (5.34)and sB (0 , = − B (0 , sB (1 , = − dB (0 , − [ B (1 , , B (0 , ] . (5.35)The master equation for the completed Σ[ ˜ B ] (including quadratic couplings in ˜ B istherefore (cid:0) Σ , Σ (cid:1) ‡ − Z ( d ˜ B + ˜ B ) ∧ · δ Σ δ ˜ B = 0 , (5.36)66t is straightforward to compute the solution Σ[ ˜ B ] for a non-linear sigma model coupledto gravity, but the derivation of the complete solution in the case of N = 8 supergravityis beyond the scope of this paper. Nevertheless, one can say that this solution can bewritten asΣ[ ˜ B ] = 1 κ S [ ϕ, ˜ B ] − Z d x X a ( − a ϕ ‡ a s ˜ B ϕ a + κ X ab ϕ ‡ a K ab ( ϕ ) ϕ ‡ b ! , (5.37)such that s ˜ B defines a differential operator which is nilpotent modulo the equations ofmotion of S [ ϕ, ˜ B ] satisfying s ˜ B S [ ϕ, ˜ B ] = 0 , (5.38)and which anti-commutes with δ g ( C ) for a x dependent parameter C . We emphasisethat this is not equivalent to gauging the theory with respect to a local E symme-try, because the components of ˜ B are classical sources and do not constitute part of asupermultiplet in the conventional sense.In order to arrive at a consistent definition of the BRST master equation (5.4) and the e master equation (4.14), one has to introduce sources ϕ g a for the non-linear symmetry,sources (or antifields) for the BRST transformations, as well as sources ϕ ‡ g a for the non-linear transformations of the BRST transformations [45], which all transform with respectto E in the representation conjugate to the one of the corresponding fields. Given the E invariant solution (5.37) to the BRST master equation, one computes that thecomplete action Σ = 1 κ S [ ϕ, ˜ B ] − Z d x X a ( − a (cid:16) ϕ ‡ a s ˜ B ϕ a + ϕ g a δ p ( C p ) ϕ a + ϕ ‡ g a δ p ( C p ) s ˜ B ϕ a (cid:17) − κ Z d x X ab (cid:16) ϕ ‡ a − ( − a δ p ( C p ) ϕ ∗ g a (cid:17) K ab ( ϕ ) (cid:16) ϕ ‡ a − ( − b δ p ( C p ) ϕ ‡ g b (cid:17) , (5.39)yields a consistent solution of the BRST master equation Z d x X a (cid:18) δ R Σ δϕ ‡ a δ L Σ δϕ a − ( − a ϕ g a δ L Σ δϕ ‡ g a (cid:19) − Z ( d ˜ B + ˜ B ) ∧ · δ Σ δ ˜ B = 0 , (5.40) Whereas the BRST operator s anti-commutes with δ g ( C ) only for constant parameter C . The only sources ϕ ‡ g a that are involved quadratically in the action are ψ ‡ g µi , χ ‡ g ijk , A ‡ g i m , and − δ p ( C p ) is defined as a linear e transformation on A ‡ g i m , and as an su (8) transformation of parametertanh(Φ / ∗ C p on ψ ‡ g µi and χ ‡ g ijk . su (8) Ward identity Z d x X a (cid:18) δ k ( C k ) ϕ a δ L Σ δϕ a + δ k ( C k ) ϕ g a δ L Σ δϕ g a + δ k ( C k ) ϕ ‡ a δ L Σ δϕ ‡ a + δ k ( C k ) ϕ ‡ g a δ L Σ δϕ ∗ g a (cid:19) − Z (cid:0) dC k + { ˜ B k , C k } (cid:1) ∧ · δ L Σ δ ˜ B k + { C k , ˜ B p } ∧ · δ L Σ δ ˜ B p + { C k , C p } · δ L Σ δC p ! = 0 , (5.41)and the E master equation Z d x X a (cid:18) δ R Σ δϕ g a δ L Σ δϕ a + ( − a ϕ ‡ a δ L Σ δϕ ‡ g a + ( − a δ k ( C p ) ϕ ‡ g a δ L Σ δϕ ‡ a − ϕ g a δ k ( C p ) ϕ a (cid:19) − Z (cid:0) dC p + { ˜ B k , C p } (cid:1) ∧ · δ L Σ δ ˜ B p + { C p , ˜ B p } ∧ · δ L Σ δ ˜ B k ! = 0 . (5.42)According to the quantum action principle [14], these functional identities are satisfiedby the n -loop 1PI generating functional Γ n , modulo possible anomalies defined by localfunctionals A g n and A ‡ n . We have established in this paper that there is no non-trivialanomaly for the non-linear E master equation. It is commonly admitted (although nogeneral proof exists to our knowledge) that there is no non-trivial anomaly to the BRSTmaster equation in four dimensions (that is, diffeomorphisms and local supersymmetryare non-anomalous in four space-time dimensions). Once one has enforced the E master equation, the cohomology of the BRST operator of ghost number one associatedto the possible anomalies to the BRST symmetry must be defined on the complex of E invariant functionals. Nevertheless, it rather obvious that the a BRST antecedent of an E invariant solution to the BRST Wess–Zumino consistency condition can always bechosen to be E invariant. We therefore conclude that there exists a renormalisationscheme such that these three functional identities are satisfied by the 1PI generatingfunctional Γ to all orders in perturbation theory.The Pauli–Villars regularisation employed in this paper breaks all these Ward iden-tities, and so the determination of the non-invariant finite counterterms would requirechecking their validity in each order of perturbation theory. In principle, preserving E invariance requires testing the e Ward identities separately, and local supersymmetrywill not be enough. As an example, the three-loop supersymmetry invariant startingas the square of the Bel–Robinson tensor does not preserve E invariance [22, 23].Therefore, the supersymmetry master equation does not determine its coefficient in thebare action, independently of the property that there is no logarithmic divergence atthis order, and one must use the E master equation to determine its value. L = 3 is68herefore the first loop order at which a renormalisation prescription may fail to preserve E invariance. One would expect that the prescription used in [5, 6] to compute N = 8on-shell amplitudes should satisfy the e Slavnov–Taylor identities, but this needs tobe checked.The BRST master equation and the E master equation are more constrainingthan the requirement of local supersymmetry and rigid E invariance. For this reasonit would be interesting to see if the prospective divergent counterterms at 7 and 8 loopcould possibly be ruled out by these master equations. There is still one subtlety concerning the Coulomb gauge which we have not yet addressed.It is well known that the Coulomb gauge in non-abelian gauge theories gives rise toenergy divergences which are not easily dealt with in the renormalisation program [46,47]. Because the ghost ‘kinetic’ term does not involve a time derivative, any ghost loopcontribution is the energy integral of a function independent of the energy k , whichdiverges linearly. However, in the flat Coulomb gauge we use the ghost field c m onlyappear in its free ‘kinetic’ term − ¯ c m ∂ i ∂ i c m . (5.43)Therefore, although the antighost ¯ c m couples to the other fields via the diffeomorphismghosts ξ µ and the supersymmetry ghosts ǫ i , and so ‘ghost particles’ can decay, theycannot be created, and there is no closed loops involving the ghost c m . It follows thatthe Coulomb energy divergences do not appear in the loop corrections to amplitudes. Itis in fact very important that the Coulomb gauge we use is field independent for thisproperty to be true. For instance, a metric dependent gauge such as ∂ i ( √ hh ij A j ) wouldgive rise to the ghost Lagrangian − ¯ c m ∂ i (cid:0) √ hh ij ∂ j c m (cid:1) , (5.44)whence perturbation theory would involve energy divergences through the couplings tothe metric. Although BRST invariance in principle guarantees that these energy diver-gences should cancel with the energy divergences involving vector fields, the compensatingprocess might be difficult to exhibit.Even within the ‘free Coulomb gauge’, the energy divergences do not disappear whenone considers insertions of non-gauge-invariant composite operators, and in particularwhen one considers insertions of the E current, since the latter couples to the ghosts69n a way very similar as in non-abelian gauge theory, in such a way that (5.43) is replacedby − ¯ c m D i D i c m (5.45)with the E covariant derivative D i c m ≡ ∂ i c m + B m i n c n . For all (and only for) thecorrelation functions of N E currents, there is one one-loop diagram associated to a‘ghost particle’ interacting with each of the currents for each ordering of the currents,which gives an integral of the form D N Y a =1 J i a ( X a , p a ) E ghost = − X ς Tr N Y ς ( a )=1 X ς ( a ) × Z d k (2 π ) (cid:16) k i N − P N − c =1 p i N c (cid:17) Q N − a =1 (cid:16) (cid:0) k i a + P a − b =1 p i a b (cid:1) + p i a a (cid:17)Q Na =1 (cid:0) k + P a − b =1 p b (cid:1) + C.T. , (5.46)where the sum over ς is the sum over non-cyclic permutations, ( i.e. the permutationsidentified modulo cyclic ones), and C.T. correspond to the diagrams involving contactterms.The contributions of the vector fields to such insertion is given at one-loop by D N Y a =1 J i a ( X a , p a ) E vec = ( − i ) N X ς Tr N Y a =1 X ς ( a ) Z d k (2 π ) N Y b =1 (cid:16) ∆( k ς,b )Υ i ς ( b ) ( k ς,b , k ς,b + p ς ( b ) ) (cid:17) + C.T. , (5.47)where k ς,a = k + P ς ( a ) − ς ( b )=1 p ς ( b ) and the sum over ς is the sum over non-cyclic permutations.The leading order in k in the limit k → + ∞ of the product∆( k )Υ k ( k, k + p ) =1 k iδ mn (cid:0) δ ji k k − δ k i k j + k i δ k j (cid:1) + O ( k − ) Ω mn ε k i l k l k − + O ( k − )Ω mn ε j kl k l k + O (1) ik k δ mn (5.48)is such that D N Y a =1 J i a ( X a , p a ) E vec = X ς tr N Y a =1 X ς ( a ) Z d k (2 π ) tr N Y b =1 K i ς ( b ) ( k ς,b ) + O ( k − ) ! , (5.49)with K k ( k ) = 1 k δ ji k k − δ k i k j + k i δ k j ε k i l k l ε j kl k l k k , (5.50)70here we used the property that the trace is invariant with respect to inverse rescal-ings of the two off-diagonal components, and the property that the contact terms aresubleading in k because ∆( k ) R ij = O ( k − ) 0 O (1) 0 . (5.51)We observe that this matrix can be written K k ( k ) = k i σ i k σ k , (5.52)where the σ i are the 4 × σ k ≡ i ε ij k δ k i − δ k j , (5.53)satisfying σ i σ j = δ ij − iε ij k σ k . (5.54)Rewriting the ‘leading’ vector field contribution to the N su (8) currents insertion in thisway, D N Y a =1 J i a ( X a , p a ) E vec = Z dk π X ς tr N Y a =1 X ς ( a ) Z d k (2 π ) tr N Y b =1 /k ς,b σ i ς ( b ) + O ( k − ) ! , (5.55)one recognises that the integrand X ς tr N Y a =1 X ς ( a ) Z d k (2 π ) tr N Y b =1 /k ς,b σ i ς ( b ) , (5.56)is the one-loop N su (8)-currents insertion in a three-dimensional theory of free bosonicspinor fields.It follows that the contribution to the N su (8)-current insertions responsible for energydivergences can be computed in an Euclidean three-dimensional effective theory, with 56 This can easily be proved using a similarity transformation of the form K → S − KS with S = δ ji k k − c m , c m and 56 Dirac spinor fields λ m , understoodas SU (2) ⊕ ¯ real spinors with ¯ λ m = λ n T G nm , (5.57)coupled to an external su (8)-current as S = Z d x (cid:16)
12 ¯ λ m /Dλ m − ¯ c m D i D i c m (cid:17) . (5.58)The corresponding contributions to the N su (8)-currents insertions areexp( − Γ[ B ]) = Det[ D i D i ]Det[ /D ] . (5.59)and therefore do not vanish. Nevertheless, they can be compensated by the contribu-tion of a trivial free-theory. Consider the fermionic fields θ m i , ¯ θ m and the bosonic fields L m , ¯ L m , with BRST transformations sθ m i = ∂ i L m , sL m = 0 , s ¯ L m = ¯ θ m , s ¯ θ m = 0 . (5.60)The BRST invariant Lagrangian12 Ω mn ε ij k θ m i ∂ j θ n k + s (cid:16) ¯ L m ∂ i θ m i (cid:17) = 12 Ω mn ε ij k θ m i ∂ j θ n k + ¯ θ m ∂ i θ m i + ¯ L m ∂ i ∂ i L m , (5.61)is a fermionic equivalent of the abelian Chern–Simons Lagrangian. The coupling ofthis theory to the current gives rise to a contribution to the N su (8)-current insertionswhich cancels the ratio of determinants (5.59). One can therefore disregard the energydivergences without affecting the BRST symmetry, although the extended current (5.19)is modified by a non-trivial BRST cocycle˜ J (Λ) C ≈ dt ∧ (cid:0) dx i θ m i + L m (cid:1) ∧ (cid:0) dx j θ n j + L n (cid:1) Ω np Λ mp . (5.62)Nevertheless, this term vanishes when the equations of motion are imposed with theappropriate boundary conditions, ∂ [ i θ m j ] = 0 , ∂ i θ m i = 0 ⇒ θ m i = 0 . (5.63)This contribution to the energy divergences is reproduced by the Pauli–Villars fields,within the prescription for the vector fields defined in section 3.2, and the prescription forthe ghosts that their Pauli–Villars Lagrangian is mass-independent. For the ghosts, this72mplies that their contribution is entirely eliminated by their Pauli–Villars ‘partners’,and one simply omits them at one-loop. This prescription is rather natural, since itpreserves the BRST symmetry associated to the abelian gauge invariance of the Pauli–Villars vector fields (the mass term in (3.31) being M Γ mn ε ij k A m i F n j k ). The leading k independent integrand in (5.47) is mass-independent for the Pauli–Villars vector fieldFeynman rules as well, and that is why their contribution cancel precisely the vectorfields energy Coulomb divergences.By property of the Pauli–Villars regularisation, the regularised divergences in M can be computed by expending the integrant in powers of the external momenta (since p ≪ M and p ≪ M ), and no non-local divergent contribution can be produced.The energy divergences are therefore consistently eliminated within the Pauli–Villarsregularisation. One computes indeed that the divergent contribution to the regularisedtwo-points function is D J i ( X , p ) J j ( X , − p ) E vec+PV ∼ i π Tr (cid:0) X X (cid:1) (cid:18) aM − (cid:16) δ ij (cid:0) p − p (cid:1) − p i p j (cid:17) ln M (cid:19) , (5.64)similarly as for the Dirac fermion contribution. In particular, we see that the Coulombenergy divergence D J i ( X , p ) J j ( X , − p ) E ghost+ λ ¯ λ = Z dk π Tr (cid:0) X X (cid:1) | p | (cid:0) δ ij p − p i p j (cid:1) , (5.65) does not require a ‘catastrophic’ non-local renormalisation ∝ Z d p (2 π ) M | p | (cid:0) δ ij p − p i p j (cid:1) Tr B i ( p ) B j ( − p ) , (5.66)within the prescription. The coefficient a depends on the axial / vector character ofthe elements X and X , and is not unambiguously determined within the prescription,because it diverges logarithmically in the UV ( i.e. at α → a A = Z ∞ dα (cid:18) M − α − (cid:16) e − αM − (cid:17) + (cid:16) α − + 2 M (cid:17) e − αM (cid:19) a V = Z ∞ dα (cid:18) M − α − (cid:16) e − αM − (cid:17) + 13 α − e − αM (cid:19) . (5.67)This difficulty is not associated to the Coulomb divergences, but to the general propertythat the Pauli–Villars regularisation does not permit to regularise divergences behavinglike ∼ M ln M . For example, the same problem appears in the Dirac fermion contribu-tion to the two-point function when X and X are axial. These divergences are irrelevantanyway, since they do not affect the renormalised correlation functions at higher orders.73 Conclusions
We have exhibited in this paper the consistency of the duality invariant formulation of N = 8 supergravity in perturbation theory. The non-standard non-manifestly Lorentzinvariant Feynman rules turn out to satisfy the quantum action principle, and diffeo-morphism invariance can therefore be maintained through appropriate renormalisations.The theory can be gauge-fixed within the Batalin–Vilkovisky formalism, and althoughthe abelian ghosts exhibit Coulomb energy divergences in insertions of the E current,these divergences are consistently removed within the Pauli–Villars regularisation.Furthermore, we have solved the Wess–Zumino consistency conditions for the anomalyassociated to the non-linear e -current Ward idendities, and shown that these solutionsare uniquely determined in terms of the corresponding solutions to the Wess–Zuminoconsistency condition associated to the linear su (8)-current Ward identity. It follows thatany non-linear E anomaly in perturbation theory is entirely determined by the one-loop coefficient of the linear su (8) anomaly. In particular, we have explicitly computedthe one-loop contribution of the vector fields to the anomaly, establishing the validity ofthe family’s index prediction, and therefore the vanishing of the anomaly at one-loop.The main result of the paper is that the non-linear Slavnov–Taylor identities associ-ated to the e Ward identities are maintained at all orders in perturbation theory, if onerenormalises the theory appropriately. Although we proved this theorem within the sym-metric gauge, it remains in principle valid within the SU (8) gauge invariant formulation[16].What are the implications of the non-linear E symmetry for possible logarithmicdivergences of the theory? Regarding the definition of BPS supersymmetric invariantswhich cannot be written as full superspace integrals (but as integrals over subspacesof superspace classified by their BPS degree), the linear approximation suggests thatthey cannot be duality invariant. Indeed, the BPS invariants are defined in the lin-earised approximation as partial superspace integrals of functions of the scalar superfield W ijkl ( x, θ ) = φ ijkl + O ( θ ), but there is no E invariant function that can be built outof these scalar fields in any SU (8) representation. It is therefore hard to see how suchsupersymmetric invariants ( i.e. the supersymmetrisations of the Bel–Robinson square R , ∂ R and ∂ R ) could be made invariant under the full non-linear duality symmetry.Nevertheless, this argument may not be entirely ‘watertight’, as a similar argument ap-pears to fail in higher dimensions, where, however, the duality groups are non-exceptional.For instance, the logarithmic divergences found in dimensions ≥ SO (5 ,
5) invariant 1/8 BPS counterterm in six dimensions, an SL (5 , R ) invari-ant 1/4 BPS counterterm in seven dimensions, and an SL (2 , R ) × SL (3 , R ) invariant 1/2BPS counterterm in eight dimensions. Nevertheless, [22, 23] exhibited that the 1/2 BPSinvariant is not E invariant, which implies that the absence of logarithmic divergenceat 3-loop is a consequence of the e Ward identities.The duality invariance may therefore entail various non-renormalisation theorems,which might explain the absence of logarithmic divergences in maximal supergravityin five dimensions at four loops [6], and in maximal supergravity in four dimensionsat three, five and six loops. A similar argument would lead to the conclusion that N = 6 supergravity admits its first logarithmic divergence at five loops, and N = 5supergravity at four loops. However, establishing such non-renormalisation theoremswill require further investigation of BPS invariants in supergravity.As another application, the e Slavnov–Taylor identities such as (4.16) may im-ply special identities among the on-shell amplitudes in the ‘multi-soft-momenta limit’,generalising the ones derived in [48] at all orders in perturbation theory.As shown by several examples (see e.g. [7]), the study of supersymmetric countertermsis not enough to reach definite conclusions regarding the appearance of certain logarith-mic divergences in supersymmetric theories. The non-linear e Ward identities maytherefore imply more stringent restrictions than one would deduce from the existence of E invariant supersymmetric counterterms. Acknowledgements
We would like to thank Ido Adam, Thibault Damour, Paul Howe, Ilarion Melnikov,Pierre Ramond, Hidehiko Shimada, Kelly Stelle, Pierre Vanhove and Bernard de Witfor discussions related to this work. We are grateful to the referee for suggesting severalimprovements in the original version of this article.
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