Invariants and divergences in half-maximal supergravity theories

Journal of High Energy Physics, Jul 2013

The invariants in half-maximal supergravity theories in D = 4, 5 are discussed in detail up to dimension eight (e.g. R 4). In D = 4, owing to the anomaly in the rigid SL(2, \( \mathbb{R} \)) duality symmetry, the restrictions on divergences need careful treatment. In pure \( \mathcal{N}=4 \) supergravity, this anomalous symmetry still implies duality invariance of candidate counterterms at three loops. Provided one makes the additional assumption that there exists a full 16-supercharge off-shell formulation of the theory, counterterms at L ≥ 2 loops would also have to be writable as full-superspace integrals. At the three-loop order such a duality-invariant full-superspace integral candidate counterterm exists, but its duality invariance is marginal in the sense that the full-superspace counter-Lagrangian is not itself duality-invariant. We show that such marginal invariants are not allowable as counterterms in a 16-supercharge off-shell formalism. It is not possible to draw the same conclusion when vector multiplets are present because of the appearance of F 4 terms in the SL(2, \( \mathbb{R} \)) anomaly. In D = 5 there is no one-loop anomaly in the shift invariance of the dilaton, and we argue that this implies finiteness at two loops, again subject to the assumption that 16 supercharges can be preserved off-shell.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

Invariants and divergences in half-maximal supergravity theories

G. Bossard 2 P.S. Howe 0 K.S. Stelle 1 Open Access 0 Department of Mathematics, King's College, University of London , Strand, London WC2R 2LS, U.K 1 Theoretical Physics Group, Imperial College London , Prince Consort Road, London SW7 2AZ, U.K 2 Centre de Physique Theorique , Ecole Polytechnique, CNRS, 91128 Palaiseau Cedex, France The invariants in half-maximal supergravity theories in D = 4, 5 are discussed in detail up to dimension eight (e.g. R4). In D = 4, owing to the anomaly in the rigid SL(2, R) duality symmetry, the restrictions on divergences need careful treatment. In pure N = 4 supergravity, this anomalous symmetry still implies duality invariance of candidate counterterms at three loops. Provided one makes the additional assumption that there exists a full 16-supercharge off-shell formulation of the theory, counterterms at L 2 loops would also have to be writable as full-superspace integrals. At the three-loop order such a duality-invariant full-superspace integral candidate counterterm exists, but its duality invariance is marginal in the sense that the full-superspace counter-Lagrangian is not itself duality-invariant. We show that such marginal invariants are not allowable as counterterms in a 16-supercharge off-shell formalism. It is not possible to draw the same conclusion when vector multiplets are present because of the appearance of F 4 terms in the SL(2, R) anomaly. In D = 5 there is no one-loop anomaly in the shift invariance of the dilaton, and we argue that this implies finiteness at two loops, again subject to the assumption that 16 supercharges can be preserved off-shell. 1 Introduction 2 Supergeometry of N = 4 supergravity theories in four dimensions 2.1 Field-strength tensors 2.2 Scalar fields and supersymmetry relations 2.3 Harmonic superspaces 3 Invariants in N = 4 supergravity theories 3.1 Linearised invariants 3.2 Dimension-four invariants: R2 and F 4 3.2.1 R2 invariants in pure supergravity 3.2.2 The F 4 invariant 3.2.3 R2 type invariants with vector multiplets 3.3 Dimension-six invariants: 2F 4 and R2F 2 3.4 Dimension-eight invariants 3.4.1 R4 type invariants 3.4.2 4F 4, (F )2R2 and (2T )2R2 type invariants 3.4.3 Protected R4 type invariants 4.1 The one-loop sl2R anomaly 4.2 Renormalisation of the anomaly at higher orders 4.3 Recovering SL(2, Z) symmetry 5 Implications for perturbative quantum field theory 5.1 Consequences of the anomaly 5.2 Superspace non-renormalisation theorems 5.3 Descent equations for co-forms 5.4 The N = (2, 2) non-linear sigma model 5.5 Non-renormalisation in N = 4 supergravity 6 Supergeometry in five dimensions 6.1 Maximal (N = 4) supergravity 6.2 Pure N = 2 supergravity 6.3 Harmonic superspace 7 Invariants in five dimensions 7.1 Normal-coordinate expansion of E 7.2 R4 type invariants 7.3 Protected invariants 7.4 Consequences for logarithmic divergences i 1 5 Developments in the evaluation of scattering amplitudes using unitarity methods over the past decade or so have made it possible to push the investigation of the onset of ultraviolet divergences in maximal supergravity theories to higher loop orders than would have been possible using conventional Feynman diagram techniques. In particular, it has been shown that D = 4, N = 8 supergravity is finite at three loops (R4) [1], and that D = 5 maximal supergravity is finite at four loops (6R4) [2], despite the existence of corresponding counterterms, at least at the linearised level [35]. Since these invariants are of F-type, i.e. correspond to integrals over fewer than the maximal number of odd superspace coordinates, it might have been thought that they should be protected by superspace nonrenormalisation theorems [5], but it is difficult to justify this argument because there are no known off-shell versions of maximal supergravity that realise all of the supersymmetries linearly. Indeed, such off-shell versions cannot exist in every dimension because it is known that divergences do occur for F-type counterterms in D = 6 and D = 7 above one loop [6]. However, these finiteness results can be explained instead by duality-based arguments. E7(7) Ward identities can be defined at the cost of manifest Lorentz covariance [7, 8], and can be shown to be non-anomalous.1 These Ward identities imply that the counterterms associated to logarithmic divergences must be E7(7) invariant. The unique SU(8) invariant R4 candidate counterterm can be proved to violate E7(7) symmetry from a perturbative scattering amplitude approach [9] and from a direct field-theoretic argument [10] that makes use of dimensional reduction and of the uniqueness of the D = 4 counterterms at the linearised level [11]. In addition, there is no superspace measure for the R4 invariant at the full non-linear level, while an analysis of the closed super-four-form that does define this supersymmetric invariant leads to the same conclusion: there is no three-loop acceptable counterterm that is both N = 8 supersymmetric and E7(7) duality invariant [10]. Furthermore, these arguments can be extended to the other two F-term invariants in D = 4 arising at the five and six-loop orders [10, 13], there being no four-loop invariant [11]. One can then use dimensional reduction and the known divergences at one, two and three loops in D = 8, 7 and 6, respectively, to show that these are the only F-term divergences that can arise in maximal supergravity in any dimension. This result can also be seen from an analysis of the conjectured duality properties of superstring theory [14, 15]. It therefore seems that maximal supergravity must be ultra-violet finite up through at least six loops in D = 4, and that there are no divergences that correspond to the known 1The absence of a supersymmetric anomaly for the E7(7) Ward identities that cannot be removed by supersymmetric non-invariant counterterms has not been rigourously established at all orders in perturbations theory. Nonetheless, the complete characterisation of the supersymmetry invariants of type R4, 4R4, 6R4, 8R4 [9, 11, 12] allows one to prove that such an anomaly cannot appear before eight loops. linearised BPS counterterms (F-terms) [3, 4, 11, 16, 17]. At the seven-loop order, we reach the borderline between F-term and D-term invariants. At this order there would seem to be a candidate D-term invariant, the volume of superspace, which is manifestly symmetric with respect to all symmetries and which would be difficult to protect by conventional fieldtheoretic arguments. However, it is now known that the volume of superspace vanishes on-shell for any N in D = 4, although there is still an N = 8 seven-loop invariant that can be written as a manifestly duality-invariant harmonic-superspacee integral over 28 odd coordinates [12]. The situation at this order is therefore somewhat ambiguous, although it is unlikely that there is an off-shell formulation of the maximal supergravity theory preserving all the supersymmetries linearly which could be used to try to justify the absence of a sevenloop divergence. A direct computational resolution of this ambiguity would seem to be a tall order, at least in the near future, but a similar situation arises in the half-maximal case which is more tractable from both the computational and formal points of view. In D = 4, N = 4 supergravity the F/D borderline occurs at the three-loop level, i.e. for R4 type counterterms. It has recently been shown that half-maximal supergravity is finite at this order [1820] and that this state of affairs persists in D = 5 [21] (where the relevant loop order is two) and in the presence of vector multiplets [19, 20]. These finiteness results have been obtained from scattering amplitude computations [18, 21] in pure supergravity and from string theory [19, 20] in supergravity coupled to vector multiplets. Field-theoretic arguments in support of these results have been given using duality arguments2 [25] and conformal symmetry [26]. From the counterterm point of view, the situation resembles seven loops in N = 8 because the natural candidate for the R4 invariant would be the volume of superspace. As in N = 8, this turns out to vanish in both D = 4 and 5, but in both cases one can also construct R4 invariants as harmonic-superspacee integrals over twelve odd coordinates instead of the full sixteen. As we shall show, duality-invariant counterterms of this type can be re-expressed as full-superspace integrals with integrands that are not themselves duality-invariant. The issue is therefore to understand if this property is enough to rule out these counterterms as possible divergences. In order to do so, one has to assume that there are off-shell formulations of the theories under consideration that preserve all the supersymmetries as well as the duality symmetries, or at least a large enough subgroup thereof. Perhaps the simplest case to consider is half-maximal D = 5 supergravity which has only one scalar, the dilaton , and for which the duality symmetry is simply a shift of this field. As mentioned above, the superspace volume vanishes, but the full-superspace integral of any function of will give rise to a supersymmetric R4 invariant. The only choice of function for which this integral is shift-invariant is a linear one, because under a shift this would give rise to the volume, which vanishes. Moreover, one can rewrite this invariant as a twelve-theta harmonic-superspacee integral of a Grassmann-analytic function that is manifestly shift-invariant. In a sense, one can regard as a zero-form potential for a gauge transformation with a closed zero-form parameter, i.e. a constant, and so the integral could 2A similar controversial argument, that E7(7) symmetry could be much more restrictive and that N = 8 supergravity might in consequence be finite at all orders, was made in [22, 23], despite the existence of an infinite number of duality-invariant full-superspace counterterms starting at eight loops [3, 24]. be thought of as the simplest type of Chern-Simons invariant. One could therefore rule this out as a divergence if one could show that divergences have to correspond to full-superspace integrals of integrands that are manifestly invariant under duality transformations. For D = 4, N = 4 supergravity, the duality group is SL(2, R). This symmetry is anomalous [27], but we shall show that the anomalous Ward identities still require the three-loop counterterm to be duality-invariant. It turns out that this counterterm can be expressed either as a twelve-theta harmonic-superspacee integral with a manifestly dualityinvariant integrand or as a full-superspace integral of the Kahler potential on the scalar manifold. This potential can be regarded as a local function depending on the complex scalar field T , the unique independent chiral superfield in N = 4 supergravity, and on its complex conjugate. One would therefore need to show that the only allowable counterterms should be full-superspace integrals of integrands that are manifestly symmetric under the isometries of this space in order to rule out a possible divergence at three loops. This case has some features in common with (2,2) non-linear sigma models in D = 2, which also have Kahler target spaces. In [28] it was shown, using the background field method in superspace, that all possible counterterms beyond one loop have to be full-superspace integrals of tensorial functions of the background field, i.e. constructed from the Riemann tensor and its target-space derivatives. So in the case of a symmetric target space these functions would be constant, i.e. manifestly invariant under the isometries. If we could prove a similar result for N = 4 supergravity then three-loop finiteness would be a consequence. As well as studying half-maximal supergravity theories we shall also consider the inclusion of n extra vector multiplets. In D = 4 this leads to an additional SO(6, n) duality symmetry that acts on the manifold of the extra scalars, while in D = 5 we have SO(5, n). In five dimensions there are no one-loop divergences to complicate matters and we shall argue that the extra multiplets do not affect the counterterm analysis at two loops. In four dimensions, on the other hand, there is a one-loop divergence corresponding to an F 4 counterterm [29]. We shall show that the full super-invariant corresponding to this is also invariant under SL(2, R) SO(6, n), a fact that is non-trivial to prove. This invariant turns out to affect the analysis of divergences at three loops because the one-loop anomaly functional, in the presence of vector multiplets, necessarily includes an F 4 term. The presence of this term implies that the theory is no longer invariant at one loop under shifts of the dilaton, which it is in the absence of vector multiplets. In order to argue our case, we shall have to assume the existence of a suitable offshell formalism that preserves all of the supersymmetries linearly. It was shown some time ago by a counting argument [30, 31] that, in ordinary superspace with a finite number of component fields, this is only possible for N = 4 supergravity coupled to 6 + 8k vector multiplets. The case k = 0 corresponds to dimensional reduction of the off-shell D = 10, N = 1 construction given in [32] which makes use of a six-form potential rather than the usual two-form. However, even if one were able to construct these multiplets for arbitrary k, there would still be a difficulty with duality symmetry because some of the scalars appear as antisymmetric two-form potentials off-shell. For example, for k = 0, i.e. with six vector multiplets, one finds that the 36 scalars appear as 20 + 1 scalars and 15 two-forms. For this reason, we shall instead assume that a suitable off-shell theory can be constructed in harmonic superspace even though no one has succeeded in doing this for a single N = 4 vector multiplet, owing in part to the vector multiplets self-conjugate nature.3 This difficulty should be contrasted with the successful construction of off-shell harmonic-superspacee formulations of the 8-supercharge hypermultiplet in D = 4, 5 & 6. Similarly to the finite-field N = 4 cases, there is a no-go theorem for finite-field off-shell formulations of hypermultiplets where the physical modes involve only scalar fields [34], but harmonic-superspacee constructions, with infinite numbers of component fields, nonetheless exist [35, 36]. So the question of whether an off-shell formulation exists for half-maximal supergravity and its vector-multiplet couplings still remains open. To summarise, the dimension-eight (R4) invariants in pure half-maximal supergravity theories in D = 4, 5 can be considered to be on the F/D borderline because they can be expressed either as integrals of duality-invariant integrands over twelve odd coordinates or as full-superspace integrals whose integrands are not themselves invariant even though the integrals are. If we make the assumption that there exist off-shell versions of these theories that preserve all of the supersymmetries linearly as well as the duality symmetries, then the divergences would have to correspond to full-superspace integrals with integrands that are duality-invariant and hence would be absent at the corresponding (3,2) loop orders. When vector multiplets are present, this result still holds in D = 5 but is vitiated in D = 4 owing to the fact that the anomaly is altered in the presence of the vector multiplets. In D = 4, the requirement of duality symmetry is not compatible with manifest Lorentz invariance, so this would presumably have to be given up in superspace, as it must in the component formalism [8, 37]. In superspace, it might be possible to make use of light-cone harmonic-superspacee techniques to help out in this context [3840]. In D = 5, on the other hand, duality is fully compatible with manifest Lorentz symmetry. Although our main motivation is the understanding of the ultra-violet behaviour of half-maximal supergravity theories, our classification of supersymmetry invariants is also of interest for understanding string-theory effective actions with half-maximal supersymmetry. In particular, the non-linear structure of the R2 type supersymmetry invariant is shown to be such that the threshold function multiplying the F 4 type structure is not independent of the R2 threshold. Our results are less conclusive for higher-derivative invariants like 2F 4, since we do not discuss in this paper the higher-order corrections required to promote a supersymmetry invariant considered modulo the equations of motion to an actual supersymmetric effective action. Nonetheless, the structure of R4 type invariants is clarified, shedding light on the property that the tree-level R4 invariant does not get corrected at one- and two-loops in heterotic string theory. In the next section we give a brief summary of D = 4 supergeometry, including vector multiplets, and then continue in section 3 with a comprehensive study of invariants up to dimension eight. In order to do this, we make extensive use of harmonic-superspacee methods and the ectoplasm formalism. At dimension four there are R2 and F 4 invariants, both of which appear in the cocycle (i.e. closed super-D-form) that determines the SL(2, R) anomaly in the presence of vector multiplets. The F 4 invariant can be written as a harmonic-superspacee integral that is not manifestly SO(6, n) invariant, but that can also be expressed as a cocycle that is so invariant. At dimension six, the 2F 4 one-quarter BPS invariant is also writable as a harmonic-superspacee integral, but in this case we have not been able to establish SO(6, n) invariance. In addition, there are invariants involving R2F 2 terms. At dimension eight, we show that the volume of superspace vanishes even in the presence of vector multiplets. As a consequence, the R4 type duality invariant can be written either as a sixteen-theta full-superspace integral of a non-duality-invariant function of the N = 4 supergravity complex scalar field T or as a manifestly invariant twelve-theta harmonic-superspacee integral. There is also a 4F 4 type duality invariant that can only be written as a twelve-theta harmonic-superspacee integral. In section 4, we discuss the SL(2, R) anomaly. It is given by a dimension-four (R R)-type invariant which involves F 4 in the presence of vector multiplets, and in the following section we discuss the implications of the anomaly in perturbation theory. This section includes a brief recap of superspace non-renormalisation theorems as well as a discussion of the notion of superspace co-forms that is needed in order to justify non-renormalisation of marginal counterterms. The nonrenormalisation theorem is illustrated by the example of (2, 2) non-linear sigma models in D = 2. A brief discussion of pure N = 4, D = 4 supergravity was given in reference [41]. In sections 6 and 7, we discuss the situation in D = 5. We give some details of D = 5 supergeometry, starting with the maximal N = 4 case, and then go on to discuss the invariants for the half-maximal N = 2 case, up to dimension eight. We show that the volume of D = 5 half-maximal superspace vanishes (although it does not in the D = 5 maximal case), and analyse the effect that this has on dimension-eight invariants. Since there is no one-loop anomaly for the D = 5 supergravity duality symmetry, which is simply invariance under shifts of the dilaton, it is more straightforward to argue that halfmaximal theories should be two-loop finite in D = 5, given the assumption of a suitable off-shell formulation. We also show that certain R4-type invariants cannot be written as full-superspace integrals, depending on the function of the dilaton that multiplies R4 in the spacetime invariant. In the context of the heterotic string, we argue that this suggests that R4 is only protected, i.e. of BPS type, for loop-orders one to four, but not beyond. We state our conclusions in section 8. The borderline F/D problem is difficult to analyse from our field-theoretic point of view, owing to the need to assume the existence of a full off-shell formalism, but for higher loops there will certainly be candidate counterterms that are purely D-type and whose integrands are invariant with respect to all known symmetries. In this sense, an unambiguous test of miraculous ultra-violet cancellations in half-maximal supergravity will require calculations at one loop higher than those that have been carried out to date. Supergeometry of N In the superspace formulation of D = 4, N -extended supergravity there are preferred basis one-forms EA = (Ea, Ei, E i) related to the coordinate basis forms by the supervielbein matrix EA = dzM EM A, where a, , are respectively Lorentz vector and two-component spinor indices and i = 1, . . . N is an internal index for the fundamental representation of U(N ) (SU(8) for N = 8). The structure group is SL(2, C) (S)U(N ). The connection AB and the curvature RAB are respectively one- and two-forms that take their values in the Lie algebra of this structure group, so that there are no mixed even-odd components. For general N , one can impose a conformal constraint on the dimension-zero torsion requiring that it be flat, i.e. that the only non-vanishing part is [42] This constraint, together with the imposition of conventional ones, implies that the only non-vanishing component of the dimension-one-half component of the torsion is together with its complex conjugate. For N 4 this leads to an off-shell conformal supergravity multiplet [43], while for N > 4 one finds a partially off-shell system [42, 44]. To go on-shell in Poincare supergravity, one needs to specify in addition a number of dimension-one superfields in terms of the physical fields. One can show that there are additional field strengths corresponding to the vector and scalar fields in the theory, as discussed in detail in [42, 45]. The geometry relevant for N = 4 supergravity coupled to n vector multiplets can be obtained by truncating N = 8 supergravity to N = 4 supergravity coupled to six vector multiplets and then extending to n using SO(n) symmetry. Field-strength tensors In this subsection we give the non-vanishing components of the superspace field-strength tensors up to dimension one. As well as the torsion and the curvature we also have the field-strengths for the vectors and the scalars. In N = 4 we can replace the three-index spinor field by a one-index one, ijk = ijkll, so that the dimension-one-half torsion can be rewritten as The non-vanishing components of the torsion at dimension one are T ,j,k = ij k + 8i kj 2(l) l + 5l l where Mij is the field-strength for the six vectors in the supergravity multiplet and iA, A = 1, . . . n, denote the n vector-multiplet spinor fields.4 The non-vanishing dimension-one 4The index A is also used as a tangent space super-index but it should be clear from the context which is meant. components of the Lorentz curvature are Rij = 14 ijklkAlA , together with their complex conjugates. The dimension-one components of the U(4) curvature, Rkl, are Rijkl = 41 ijkppAlA 12 l(ij)kpqpAqA , its complex conjugate, and Rj kl = 21 lkij 2 l i 1 ikAjA 12 jk iAlA + + 41 lk iAjA + 21 ji kAlA + 4 (2lijk ji lk)pApA . 1 In the absence of the vector multiplets, the U(4) curvature collapses to U(1). dimension-one SO(n) curvatures are As well as the geometrical tensors there are (complex) spin-one field strengths for the supergravity vectors, Fij , and for the n vector multiplets FA. For the scalars there are one-form field strengths P and PijA. These fields obey the Bianchi identities DFij = P Fij + PijA FA DFA = PAij Fij + P FA , where antisymmetric pairs of SU(4) indices are raised or lowered with 21 ijkl. The oneforms are covariantly constant, DP = DPijA = 0, where D is covariant with respect to the local U(4) SO(n). These forms are given by PijA = E[i j]A 21 E kijkl l A + EaPaijA , Scalar fields and supersymmetry relations The scalar fields in the supergravity sector take their values in the coset SO(2)\SL(2, R) and can be described by an SU(1, 1) (= SL(2, R)) matrix V. The Maurer-Cartan form dVV1 can be written where Q is the composite U(1) connection. The scalar matrix V can be parametrised in the form Rij = 12 ji P P 21 ikmnPmAn PjkA , dVV1 = V = U U (1 T T) = 1 . The U(4) and SO(n) curvatures can be expressed in terms of the one-forms by The U(1) gauge-invariant complex scalar superfield T can be considered to be a coordinate of the SO(2)\SL(2, R) coset on the unit disc. T is chiral while U = The scalar fields in the vector multiplets parametrise the coset (SO(6) SO(n))\SO(6, n). They can be represented by an SO(6, n) matrix U obeying the constraint U U T = , where is the SO(6, n)-invariant metric in R6,n. If we set where the pair ij represents an SO(6) vector index, A is acted on by SO(n), and I run from 1 to 6 and I from 7 to n + 6 and together make up an SO(6, n) vector index, and where then the SO(6, n) constraint can be written Here, and elsewhere in the text, the indices A, I, I and ij are raised and lowered with the appropriate Euclidean metrics (and 12 ijkl for the ij). We have DU U 1 = where P = PijA is the one-form given in (2.10). In components, these Maurer-Cartan equations read The supersymmetry variations of the dimension-one-half fermions are given by DijA = ji MA + i Aj 21 ji k Ak In this paper we shall also need the spinorial derivatives of the some of the dimension-one fields. These are 12 k [iAj]A + 2i ijpq ( P)pqAkA 12 ijpqM pq Ak + 2[ij]Ak kBB[ij]A where the dots state for terms involving the matter fields that will not be relevant for our computations and i is the gravitino field strength. To compute these derivatives, one uses the supersymmetry algebra as defined in [42] for N = 4 supergravity coupled to six vector multiplets, and one extends the corresponding expressions to n vector multiplets using the SO(n) symmetry. It turns out to be helpful to use such expressions to compute first the torsion [42], and then to make use of the U(4) SO(n) covariant supersymmetry algebra in order to obtain the expressions for n vector multiplets. Harmonic superspaces Since we shall be interested in investigating F-term invariants in the full non-linear theory, we shall need to know which sub-superspace measures are available. The simplest possibility is chiral superspace, and we know that the supergravity scalar fields T, U are indeed chiral. However, these are in fact the only independent chiral superfields due to the the fact that the dimension-one-half torsion is non-zero. This is similar to IIB supergravity, where it is known that there is no chiral measure for similar reasons. The other possible measures can be investigated using harmonic-superspacee methods. In flat superspace, we recall that a G-analytic (G for Grassmann) structure of type (p, q) consists of a set of p Ds and q D s that mutually anti-commute, and that such sets can be parametrised by the coset spaces (U(p) U(N (p + q)) U(q))\U(N ), which are compact, complex manifolds (flag manifolds) [35, 46, 47]. Harmonic superspaces of type (N , p, q) consist of ordinary superspaces augmented by the above cosets. Analytic superspaces, which have 2(N p) + 2(N q) odd coordinates as well as the harmonic cosets, are generalisations of chiral superspace on which (p, q) G-analytic fields can be defined in a natural way.5 In curved superspace one has to check that the corresponding sets of odd derivatives, suitably extended to include the harmonic directions, remain involutive in the presence of the non-trivial geometry. It turns out that this is only possible when both p, q 1 for N > 4, but for N = 4 one can also have (p, q) = (1, 1), (2, 2) or (1, 0), (0, 1) [50]. Let us consider first (4, 1, 1) harmonic superspace. The harmonic variables, u1i, uri, u4i (and their inverses ui1, uir, ui4), where r = 2, 3, can be used to parametrise the coset of U(4) with isotropy group (U(1) U(2) U(1)) in an equivariant fashion. The first obstruction to involutivity vanishes because T11 k = 0 (where quantities with indices (1, r, 4) are projected by means of the appropriate harmonic matrices), but we still need to check that the dimension-one curvatures do not cause problems. It is straightforward to show, using (2.6), (2.7), that no further obstructions to involutivity arise because and similarly for R14,kl and R 44,kl , for the same projections of the Lie algebra indices. The (4, 1, 0) (and (4, 0, 1)) cases are also fine. For the former, one splits the index into (1, r), r = 2, 3, 4, and the only possible curvature obstruction involves R11,1r, which clearly vanishes. For (4, 2, 2) harmonic superspace, the relevant harmonic coset is (U(2) U(2)) \ U(4) for which we define harmonic variables (uri, uri) and their inverses (uir, uir) where r = 1, 2 and r = 3, 4, with uriuis = 0 , where unitarity implies that, e.g. , uir = uri. G-analyticity of type (2, 2) is allowed [50] because the appropriate projections of the dimension one-half component of the torsion are 5Further discussions of harmonic or projective superspace techniques in N = 2 supergravity theories can be found e.g. in [48, 49]. while the dimension-one components of the Riemann tensor satisfy Although the torsion and curvature tensors are compatible with involutivity for each of the above cases, harmonic-analytic fields will transform under non-trivial representations of the isotropy groups and there are constraints on the allowed representations that arise because of the non-vanishing components of the dimension-one curvature components restricted to the isotropy sub-algebra. It is straightforward to check that the representations under which putative integrands must transform are allowed. For example, in the (4, 1, 1) case, a G-analytic integrand should be of the form O4141. Such fields are allowed because R1411 = R1444. In the (4, 2, 2) case there is again no obstruction to the existence of a G-analytic superfield of the correct U(1) weight, because In other words, the su(4) curvature decomposes as the sum of two su(2) curvatures. In all of the above cases one can define measures for G-analytic fields by means of expansions of the Berezinian of the supervielbein with respect to 2p+2q normal coordinates. This will be done explicitly for (4, 1, 1) case later on; it is more difficult for (4, 2, 2) but it is clear from the construction that such a measure does exist. These integrals are the curved-space versions of integrals over analytic superspaces in the flat case. Invariants in N = 4 supergravity theories Linearised invariants We begin with a brief survey of invariants constructed from the on-shell linearised fieldstrength superfields that transform as primary superconformal fields. For N = 4 the superconformal algebra can be taken to be su(2, 2|4) but the abelian subalgebra generated by the unit matrix, i.e. R-symmetry, does not act on superspace only the quotient algebra psu(2, 2|4) does. The quantum numbers associated with the latter are the dilation weight, L, the two spin quantum numbers, J1, J2, and three Dynkin labels (r1, r2, r3) specifying an su(4) representation.6 The chiral supergravity field strength superfield W has L = 1 and will also be assigned R = 1. (W is the linearised limit of T and we use the U(1) gauge in which U = 1 1T T = 1 + O(T T).) The vector-multiplet fields strengths WA have L = 1, R = 0 and transform in the [0, 1, 0] of su(4). Note that the U(1) symmetry of supergravity is not part of the superconformal symmetry for N = 4; it has been referred to as U(1)Y in the literature [51]. It does act on superspace, however, since it can be considered as a subgroup of U(2, 2|4) [52]. The invariants are integrals of monomials in the field strengths and their derivatives. These can be put into representations of the superconformal group, either primaries or 6The superconformal dimension L of a field is its canonical one, while the dimensions used in the full theory are geometrical, e.g. scalars have L = 1 but geometrical dimension zero. descendants. Because we are integrating, we need only consider the primaries, and since primaries with non-zero spin quantum numbers do not have Lorentz scalar top components, we can restrict our attention to primaries with J1 = J2 = 0 [17]. An important class of invariants consists of those that are short, or BPS. These correspond to the unitary series B and C representations with J1 = J2 = 0. The series B unitarity bounds are L + R = 21 m, or L R = 2m1 12 m, together with L 1 + m1, while for series C we have L = m1, R = 12 m m1, where m1 = Pk rk is the number of boxes in the first row of the su(4) Young tableau and m = Pk krk is the total number of boxes [53]. In fact, the series C representations do not involve the supergravity field strength and thus have R = 0. The simplest pure supergravity invariants are the chiral ones, W p, and of these only p = 2 gives rise to a U(1) invariant integral. This is the linearised R2 invariant and is in fact a total derivative on-shell. We can also construct (4, 0, 2) invariants with integrands of the form M12M12W p+2, for p 0, although only the one with p = 2 would be U(1) invariant (and is also a total derivative). The corresponding chiral primaries are of the form ui1uj 2uk1ul2 Mij Mkl W p+2 ijmnmnMkl W p+1 + 41 ijmnklrs(mn)rs W p , (3.1) where the three terms define integrands that are equivalent up to total derivatives. These are series B with L + R = 2 and su(4) representation [0, 2, 0]. The pure vector-multiplet invariants were considered in [11]. Here, we must insist on su(4) symmetry (and so(n)), and there is only one one-half BPS invariant WAW AWBW B in the [0, 4, 0] representation of su(4). It can be integrated over an eight-theta measure, either as a super-invariant, or in (N , p, q) = (4, 2, 2) analytic superspace. There is also only one one-quarter BPS invariant. It is W[AWB]W AW B in the representation [2, 0, 2]. It can be integrated with respect to a twelve-theta measure as a superaction or as an integral over (4, 1, 1) analytic superspace. We can also have mixed supergravity-vector invariants. Since W is chiral the short ones in this series will be chiral with respect to fewer than four dotted derivatives. The simplest case is (4, 0, 2). Invariance with respect to su(4) means that we should have WAW A in the [0, 2, 0] representation. This superfield is the energy-momentum tensor for the vector multiplet. It can be defined on (4,2,2) superspace, but if we multiply it by some power p of W , then we get superfields of the required (4, 0, 2) type. This series of superfields has L = p + 2, R = p, m = 4, m1 = 2 and saturates the series B bound. Note that one must have p 2 since otherwise the integral would give zero due to the second-order constraint on W . The final possibility for BPS invariants is (4, 0, 1). In order to ensure su(4) symmetry we require two powers of WA, but also two undotted spinorial derivatives. The representation has L = p + 3, for a factor of W p, p 2, and must be in the [0, 0, 2] representation which has m1 = 2, m = 6. These fields again satisfy the series B bound. If we define the energy-momentum tensor superfield by Tij,kl = WkA(iWj)lA then the primary linear combination is 1 Oij := 2Tij,kl DklW p + (p 1)DkTij,kl DlW p + 4 where Dij := DiDj . The analytic superspace integrand is O44 = Oij ui4uj 4. p(p 1)DklTij,kl W p , Note that the (4, 0, 1) and (4, 0, 2) chiral primaries give rise to invariants that reproduce (4, 2, 2) harmonic integrals discussed in [54]. Dimension-four invariants: R2 and F 4 We shall now analyse the various nonlinear invariants in N = 4 supergravity starting at dimension four. R2 invariants in pure supergravity The first possible on-shell invariants in pure N = 4 supergravity are all of generic R2 structure. Although the relevant SL(2, R) invariants vanish on-shell for trivial spacetime topology [74], there do exist non-vanishing invariants with a non-trivial dependence on the complex scalar field. Because both U and T are antichiral, one might think that one could define R2 type invariants from an anti-chiral superspace integral of the kind for some antichiral measure d8E . However, owing to the presence of the dimension onehalf torsion component, U and T are the only fields satisfying the antichirality constraint, and such a nave measure does not exist. This should be compared to the attempt to construct a superspace-integral R4 invariant in type IIB supergravity [55], a construction that similarly failed owing to the lack of a chiral measure, again due to the presence of a non-trivial dimension one-half superspace torsion component [56, 57]. Nevertheless, in D = 4, N = 4 one can define invariants involving U and T from closed super-four-forms using the ectoplasm formalism [5860]. These depend on an arbitrary holomorphic function F (0) which one can intuitively think of as being the second derivative of the function F (2). In pure supergravity, super-four-forms can be expanded in terms of the supervielbeins as follows representation, irrespectively of whether they can be defined as superspace integrals (also without involving prepotentials in a hypothetical off-shell formulation). Moreover, in order to prove the existence of such an invariant, one simply needs to check the lowest-dimension component of its d-closure, as we shall now demonstrate. The differential decomposes into four components [61] d = d0 + d1 + t0 + t1 of bi-degrees (1, 0), (0, 1), (1, 2) and (2, 1) respectively (where the first degree refers to the degree in the even basis forms Ea, and the second to the degree in the odd basis forms (Ei, E i)). The first two components of d are respectively even and odd differential operators while the other two are algebraic, involving the dimension-zero and dimensionthree-halves torsions. The latter is not relevant to the present discussion, while the former can be defined in terms of the contraction operator cEa ca and the dimension-zero torsion, given in (2.1), by t0Lp,q = Ej Ei Ti j c cLp,q . Since d2 = 0 it follows that t0 is nilpotent so that we can define cohomology groups Htp,q [61]. This means that we can analyse superspace cohomology in terms of elements of this group, in any spacetime dimension; see, for example, [6165].7 A key feature of D = 4 is that the only non-trivial t0-cohomology groups have p = 0 for N > 2. Since t0 maps onto lower-dimensional superform components, the lowest-dimensional component of a cocycle must itself necessarily be t0-closed, and one can remove any t0-exact component from it by redefining the cocycle by the addition of a trivial cocycle, L L + d. In N = 4 supergravity in four dimensions the only non-trivial t0 cohomology classes have bi-degree (0, 4) and correspond to irreducible representations of SU(4) for which the Dynkin labels [r1, r2, r3] satisfy r1 + 2r2 + r3 = 4. It is straightforward to check that all the irreducible representations of SU(4) for which the Dynkin labels [r1, r2, r3] satisfy r1 + 2r2 + r3 2 are t0 trivial, because they necessarily contain an SU(4) singlet that can be identified with Ei Ei.8 The first non-trivial cocycle equation therefore states that Using d2 = 0 together with t0L0,4[F ] = 0 one then has d1L0,4[F ] + t0L1,3[F ] = 0 . t0 (d0L0,4[F ] + d1L1,3[F ]) = 0 , 7This approach to superspace cohomology is related to pure spinors [6668] and also to the cohomology of supersymmetry algebras [6971]. 8In general, one can decompose a component of bi-degree (0, n) into the corresponding (0, p, q) components of degree p in Ei and q in Ej. For the components of degree (0, n, 0) and (0, 0, n), the symmetric tensor product of the associated representation decomposes into irreducible representations of SU(4) satisfying r1 + 2r2 + r3 = n because the antisymmetrisation of three SL(2, C) indices vanishes. For any other degree, the property that r1 + 2r2 + r3 < n implies, for the same reason, that the corresponding form includes a factor Ei Ei, and therefore that it is t0-trivial. Indeed, the only way to have lower values of r1 + 2r2 + r3 is either to have such a contraction, or to have three SU(4) indices antisymmetrised. and because there is no t0 cohomology class at bi-degree (1, 4), one concludes that there exists an L2,2[F ] such that d0L0,4[F ] + d1L1,3[F ] + t0L2,2[F ] = 0 . Similarly, for the other components one obtains t1L0,4[F ] + d0L1,3[F ] + d1L2,2[F ] + t0L3,1[F ] = 0 t1L1,3[F ] + d0L2,2[F ] + d1L3,1[F ] + t0L4,0[F ] = 0 t1L2,2[F ] + d0L3,1[F ] + d1L4,0[F ] = 0 . 0 0 0 0 where means equal in t0-cohomology, and + denotes the appropriate sum over symmetric permutations of the sets of indices {i,j,k,l,} and { k,l,p}. At bi-degree (0, 5), the t0-cohomology consists of all irreducible representations of SU(4) for which the Dynkin labels [r1, r2, r3] satisfy r1 + 2r2 + r3 = 5. After some work, one verifies that such a cocycle exists for any holomorphic function, F (0), with lowest components M ijkl = 6 F (0)(T)M ij M kl U 2F (0)(T)ijpq pqM kl F (0)(T)ijpqklrs pqr s !+ M ijkl = 6 kl F (0)(T)M ij 31 U 2F (0)(T)ijpqpq The computation goes as follows: (3.16) requires that 9Note that L0.4 determines an element of the spinorial cohomology group Hs0,4 which consists of nontrivial elements of Ht0,4 satisfying d1L0,4[F] 0 in t0-cohomology. 10A true anti-chiral cocycle in flat superspace would have only M0,4,0 non-zero. of SU(4). Then (3.15) requires that are G-analytic. Because of (2.20), one has for some N ij kl in the of SU(4). Because of the dimension-one-half component of the torsion, this equation is very constraining; the general solution depends on one antiholomorphic function M ij k = l F (0)(T)M ij 31 U 2F (0)(T)ijpqpq L[F (0)] = F (0)(T)R R + . . . . This invariant is of course complex, and the associated real invariants will be obtained from its real and imaginary parts, which are respectively even and odd with respect to parity. This may seem very different from the example of the type IIB superstring E (0)(T, T)R4 invariant, for which the function E (0) is also known to satisfy a Poisson equation [72]. However, note that any holomorphic function of this kind necessarily satisfies a Laplace equation The F 4 invariant The F 4 invariant in flat (4, 2, 2) analytic superspace was given in [11]. Here we show how to extend this result to the curved case. Because this superspace exists in the curved case there is no problem writing down this invariant in a similar way. However, it is not manifestly invariant under the SO(6, n) duality symmetry which is rather difficult to show directly. Nevertheless, we can show that this symmetry holds, as it must (because there is a known one-loop divergence), by use of the ectoplasm formalism to deduce that the analytic superspace integral is indeed duality-invariant by uniqueness. One can easily check that both and we obtain the unique U(4) SO(n) invariant G-analytic integrand with the correct U(1) weight: S(1) = Z The associated top component then clearly contains a term in F 4 and yields the one-loop counterterm for the matter vector fields. One proves, using superconformal representation theory, that this is the unique U(4) SO(n) invariant at this order [11]. It follows that this invariant must actually be invariant as well under SO(6, n), despite the fact that the integral formula (3.26) does not display this symmetry manifestly. To prove directly that this is indeed the case, one would need to show that d (4,2,2) K L V34K V34LU34I V34J = 0 . In the linearised approximation, the integral of (W34)4 over the (4, 2, 2) measure is indeed invariant under linearised duality transformations, i.e. constant shifts, since (W34)3 is ultrashort. If (3.28) were not zero, it would necessarily start at five points with the linearised invariant part of it defined as the integral of (W34)5 over the (4, 2, 2) measure, and there could also be a contribution from the integral of (W34)3 component over the non-linear (4, 2, 2) measure; in principle, these might cancel each other, and indeed they do. However, this would be very complicated to check in practice. Instead, we shall show that there is a duality-invariant cocycle associated to this invariant. Within the linearised approximation, one finds that the nave representative of this cocycle obtained from the one-half BPS supermultiplet in the of SU(4) [4] does depend explicitly on the scalar fields, and so looks non-shift-invariant. For example (with W ij A = V ij IUAI), one has M ijkl ABCD + 2AC BD W ij AW klBMC M D + W ij AM BklpqpC qD + ijpqklrspAqBrC sD projected into the of SU(4). However, one can trivialise the components depending explicitly on the scalar fields by adding the exterior derivative of a 3-form with (0, 3, 0) component ijk ABCD + 2AC BD projected into the . Here, formulae (3.30), (3.31) are only indicative, and the specific coefficients have not been precisely established. To prove that the complete non-linear cocycle can be chosen to be duality-invariant, we will compute its fermionic components directly. One finds that M ijkl = 4i ijpqklrs ABCD + 2AC BD pAqBrC sD+ i M ijkl = 2 ijpq ABCD + AC BD + ADBC pAqBlC kD+ together with their complex conjugates. The cocycle equations are then satisfied, i.e. 0 0 0 DpM ijkl + 2TpiqMq+ jkl DpM ijkl + 4DiMp + 6Ti jqMklpq+ jkl TplqM ijkq + 2DpM ijk l + 3DkM ijlp + 3Ti jqMklpq+ Equation (3.33) consists of one single term in AA3 in the [1; 0|1, 2, 0]. Equation (3.34) includes a term in PAA3 in the [0; 1|0, 2, 1] and a term in A2A2 in the [2; 1|2, 1, 1]. Equation (3.35) includes two terms in A2 APA in the [1; 0|1, 2, 0] and the [1; 2|1, 1, 2], one term in A4 in the [1; 0|1, 2, 0], and two terms in A A3 in the [1; 0|1, 2, 0] and the [1; 2|1, 1, 2]. Because the cocycle is unique in cohomology, the property that we can find a dualityinvariant representative is enough to prove that the integral (3.26) is duality-invariant. R2 type invariants with vector multiplets In this subsection we will generalise the cocycle derived in section 3.2.1 in the presence of vector multiplets. As we shall prove, the cocycle is still anti-chiral in the presence of vector multiplets, but in a weaker sense, i.e. only M0,1,3 = M0,0,4 = 0. The other lowestdimensional components satisfy TplqM ijkq + 2DpM ijk l + 3DkM ijlp+ 0 0 0 The last condition (3.40) is purely algebraic, and requires that M ij kl 4iN kl 2iN klj , j for some operator symmetric in exchange of the pairs of indices { k, l}. One computes that a (1, 3) component satisfying the corresponding (0, 5) constraint T1,2M1,3 0 is where the unhatted indices now define the N = 2 internal Sp(2) indices, whereas the hatted indices are for the complementary Sp(2) in Sp(4). The second term is determined as a function of the former by imposing symplectic tracelessness with respect to Sp(4). The vector field-strength tensor decomposes into the symplectic traceless component and the singlet according to F ij = F ij 14 ij F , F = 41 F (6.20) where F ij is symplectic traceless. The Sp(2) components of the scalar superfield reduce to Kij L 41 ij KL , V ij K L = 18 ij K L e where IJ and K L are respectively Sp(2) and Sp(2) rigid indices. It is consistent to identify rigid and local indices, because Rijkl = 0 by construction in the absence of matter. The bilinears in the fermion reduce to 18i)(kl)(j + 7i)(kl)(j pqpq It then follows from (6.16) that Dij = 2i ij aDa + 31 ab 2Maibj + ij Mab 2 1 where Maibj and Mab are defined as in (6.20). The scalar superfield is real, and satisfies the second-derivative constraints 6.3 Harmonic superspace Harmonic superspaces in five dimensions are of type (2N , p), meaning that a G-analytic field of this type will be annihilated by p four-component spinor derivatives that mutually anti-commute. We write 2N because a G-analytic field will depend on (2N p) fourcomponent odd coordinates, with N being the maximal possible value of p.18 In N = 4 supergravity, only the (8, 1) harmonic structure associated to the coset Sp(4)/(U(1)Sp(3)) is consistent with the dimension-one-half torsion component because where the harmonic variables are defined such that 18In other words, a G-analytic field of type (2N , p) is p/2N BPS. The Riemann tensor components are also consistent with the torsion because 95 i(kNal)bj + 772 i(kl)j Nab , It turns out that the Lorentz curvature u1iu1j Rijcd = 2i4 cdabeabu1iu1j Nei,j is expressible in terms of the G-analytic vector To show that Ba is indeed G-analytic, let us rewrite Ba in terms of Ba u1iu1j Nai,j . D11rs = 18 u1iu1j urkusl aNai,jkl 3N i,jkl D1Ba = 21 a = 12 a vwrtsu [1rv1sw1t]u Note that this is the unique G-analytic vector (up to an overall G-analytic function) since the obstruction only vanishes because the curvature is proportional to Ba itself. At the linearised level, one can define the 6R4 invariant starting with a G-analytic integrand quadratic in the linearised G-analytic superfield W 1rst V 1[rIJ V st]KLIK JL, but this does not extend to the non-linear level. Indeed, its nave generalisation V 1rIJ uiujrV ij IJ is not G-analytic since There is therefore no dimension-zero G-analytic superfield of the correct U(1) weight, and the 6R4 supersymmetry invariant cannot be defined as a harmonic-superspacee integral at the non-linear level. The N = 2 supergravity theory admits a (4, 1) harmonic-superspacee structure associated with the coset Sp(2)/(U(1) Sp(1)). The harmonic variables in this case satisfy where rs is the Sp(1) antisymmetric tensor and the other contractions are null. However, since the torsion supertrace does not vanish in this case, i.e. (where the index A runs over all bosonic and fermionic indices), the G-analyticity condition involves the flat U(1) connection u1ii We will prove this in the next section by consistency with the normal-coordinate expansion of a generic scalar superfield together with Stokes theorem in superspace. Invariants in five dimensions As in four dimensions, the first duality invariants will be of the same dimension as the full-superspace integral of a function of the D = 5 scalars. In N = 4 supergravity, this property was already discussed in [10]. In pure D = 5, N = 2 supergravity, the first Sp(2) invariant can be derived from the exact 6-superform L 12 abcdeeea Rbc Rde + A Rab Rab + . . . so it is clearly not invariant with respect to a shift of the dilaton superfield. In any case, it is not relevant in perturbation theory in five dimensions because of its mass dimension. In this section we will show that the integral of the Berezinian of the supervielbein does not vanish in maximal supergravity in five dimensions, but it does vanish in the halfmaximal theory. However, the volume of N = 2 superspace vanishes, so one can still write the shift-invariant R4-type invariant as a full-superspace integral of the dilaton superfield. In order to do this, we shall compute the normal-coordinate expansion of the supervielbein Berezinian. Normal-coordinate expansion of E One checks, in a similar fashion to [12], that all the requirements for the existence of complex normal coordinates A { ui1i , z1r, z11} , are satisfied, because the associated tangent vectors satisfy the involutive algebra with all other graded commutators vanishing. Here d1r and d11 are the vectors on the harmonic coset space that act on the harmonic variables by d11ui1 = 2u1i , and trivially on the others. The complex coordinates z1r, z11 are the complex normal coordinates on the coset space associated to these vectors. The vectors E1 are the horizontal lifts of the basis vectors E1 to the harmonic superspace (i.e. they contain connection terms). As in four dimensions, one can check that the normal-coordinate expansion of the supervielbein Berezinian multiplying any scalar superfield factorises into the normalcoordinate expansion of the harmonic measure and the normal-coordinate expansion of the supervielbein Berezinian together with the scalar superfield given in terms of the fermionic coordinates expansion alone. One can therefore forget about the complex normalcoordinate expansion of the harmonic measure, and simply consider the normal-coordinate expansion in terms of the fermionic coordinates as in [76]. Before discussing the specific examples of maximal and half-maximal supergravities in five dimensions, we shall rederive the formula for the normal-coordinate expansion of the supervielbein Berezinian in an alternative way. We will start quite generally and consider a supergravity theory that admits four fermionic normal coordinates (possibly together with harmonic ones which can be disregarded), with a possibly non-zero torsion supertrace Note that we use , . . . as indices for the 4 normal coordinates, which would be the Sp(1, 1) indices in five dimensions, and which would stand for both fundamental and complex conjugate SL(2, C) indices together with the associated U(1) weights in four dimensions. By Stokes theorem, the integral of a total derivative over superspace must vanish EEM = Z ddxd4k E (D + ) . If one assumes the existence of normal coordinates, with a general normal-coordinate expansion of the supervielbein Berezinian given by E = E 1 + e + e + e + e , then Equation (7.7) implies that = 214 (DDDD +4eDDD +12eDD +24eD +24e) (D +) = 0 for any tensor superfield , where the notation | to the right of a superfield indicates that the latter is evaluated at = 0, as in [76]. In more geometrical terms, X| is the pullback of the superfield X to the analytic superspace from which the normal-coordinate expansion is defined. Note that for a general vector field, A, this expression would only be required to be a total derivative in the analytic superspace, but for a vector, , normal to the analytic superspace, this expression must vanish. The various terms in D5n then determine uniquely the components en of the normal-coordinate expansion of E. To carry out this computation explicitly, we will consider theories for which one has e = e = 16 R + 21 D + 12 1 1 1 1 e = 6 [R|] + 6 D[D] + 2 [D] + 6 1 1 1 e = 180 R[ R|] + 72 R[R|] + 12 R[(D] + ]) + 214 D[DD] + 81 D[D] + 16 [DD] 1 1 + 4 [D] + 24 . One checks that these expressions are indeed compatible with the formula given in [76], reproducing the result Dln[ E] = + 31 R + 415 R R = (1)ATAA (E ) , where, by definition, = exp[D] 19 and where (7.6) and (7.10) have been used to show that the normal-coordinate expansions of and E take the same form as given in [76]. Let us now consider the full-superspace integral of a scalar superfield K. Using the normal-coordinate expansion, one computes that E = E 1 214 aBa 214 1144 abBaBb 19Or more explicitly = + (D) + 21 (DD) + 16 (DDD) . 20That would be (k, 1, 1) with k = N in four dimensions and (k, 1) with k = 2N in five dimensions. where d (k,1) is the measure over (k, 1) harmonic superspace20 d (k,1) = dxdd4(k1)du E with du being the Haar measure over the harmonic coset manifold, and E is the measure defined in (7.8), and where the function F [K] is equal to 1 (DDD D + 4eDD D + 12eD D + 24e D + 24e) K . (7.15) 24 Using this expression and using (7.10) and (7.11), one computes that F [K] satisfies i.e. F [K] is only G-analytic on a line bundle with a flat connection. Therefore we conclude in general that one can integrate any G-analytic section on this line bundle over (k, 1) harmonic superspace. R4 type invariants Before considering the full-superspace integral of a general function of the dilaton superfield in N = 2 supergravity in five dimensions, let us discuss the simpler N = 4 example. In this case, the torsion supertrace vanishes, and the only relevant component of the Riemann tensor is where Ba is the unique G-analytic vector (6.32). One straightforwardly computes that and so the volume is not zero. Note, however, that this result is irrelevant for the ultraviolet behaviour of the theory, because the 8R4 type invariant does not have the correct power counting to be a candidate counterterm in five dimensions. In N = 2 supergravity, the Riemann tensor satisfies the same equation, and is determined by the torsion supertrace u1ii as R = () () + 21 () . One straightforwardly computes that R = 3 41 , As a consistency check, one computes using this formula that indeed coincides with formula (7.13). This may be a surprise: the volume of superspace does not vanish on-shell in maximal supergravity, but it does in the half-maximal theory as a consequence of the contribution of the non-trivial torsion supertrace. Using this formula, one can now integrate an arbitrary function K of the dilaton superfield over full superspace to obtain d (4,1) ( 3) 23 + 23 (7.26) Since the integrand starts at 4-points, one can use the linearised analysis to compute that such an invariant contains a term in R4 of the form CCC C ( 3) 32 + 23 e3, e 32 or if it is a constant. Therefore the integral of the N = 2 supervielbein Berezinian vanishes. However, one can still write the only invariant that preserves the dilaton shift symmetry as the full-superspace integral Z d5xd16 E = d (4,1) . (7.28) 9 Z 32 This invariant satisfies all the required symmetries of the theory and is a full-superspace integral, so that there is no obvious non-renormalisation associated with it. However, we shall argue in the last section that the same reasoning that we sketched within supergravity in four dimensions would in principle imply that such an invariant would be forbidden by a non-renormalisation theorem within a hypothetical formulation of the theory in superspace with all 16 supercharges realised linearly. The situation remains very similar in the presence of vector multiplets. We shall not display the explicit components of the Riemann tensor and the torsion in the presence of vector multiplets in this paper. The latter can be straightforwardly extracted from the N = 4 ones for n = 5 vector multiplets, and, using the property that the vector multiplets only carry SO(5) vector indices contracted using the corresponding metric AB, one can then straightforwardly generalise all formulae to any number of vector multiplets. Here, we shall only explain how to extract the quantities that are important for the computation of the superspace volume. In this case the bilinears in the Dirac fermions also include the matter-field Dirac fermions A and one has N i,jkl = 3[kl[j]i] + [kl[Aj] i ]A . e = , 1 3 1 1 e = 4 + 8 12 AA 24 AA , 5 1 1 e = 12 6 [A]A + 24 []AA , The volume therefore still vanishes on-shell in the presence of vector multiplets. The full-superspace integral of a function of the dilaton K is expressible as a (4, 1) harmonicsuperspacee integral by 32 ( + 3) + 32 AA In particular, the integral of the dilaton is + 21 [AAB]B 13 AABB In the presence of vector multiplets, the most general SO(5, n) invariant G-analytic integrand of mass dimension two is 3 ( 3) 2 3 + 2 23 ( + 3) + 23 AA so that all such (4, 1) integrals can be expressed as full-superspace integrals of a primitive K() of G(). Note that the field Ba defining the component R of the Riemann tensor is the unique G-analytic vector, as in four dimensions, but its square is a G-analytic function whereas the integrand of the harmonic measure must be a G-analytic section (6.40) because of the non-vanishing torsion supertrace. Therefore there is no associated duality invariant in five dimensions. In fact one checks that for G() = e 23 one obtains an invariant that only depends on the matter fields in the quartic approximation. Through dimensional reduction, the latter gives rise to an SL(2, R) invariant in four dimensions that depends non-trivially on the matter scalar fields. We conclude that the (4, 1, 1) superspace integral (3.78) does not lift to five dimensions. In conclusion, N = 2 supergravity coupled to n vector multiplets in five dimensions admits only one invariant candidate that is invariant with respect to a shift of the dilaton superfield and that can be written as a (4, 1) harmonic-superspacee integral. It can also be written as the full-superspace integral of the dilaton superfield itself. Protected invariants We have discussed invariants that can be written as (4, 1) harmonic-superspacee integrals in the last section, but they do not exhaust all the possible R4 type invariants one can write in five dimensions. Similarly to four dimensions, some R4 invariants can only be written as (4, 2) harmonic-superspacee integrals. The main difference in five dimensions is that one can already distinguish these invariants by the structure of the four-graviton amplitude, because there are two distinguished Sp(1, 1) quartic invariants in the Weyl tensor that define supersymmetry invariants. The (4, 2) harmonic variables uri, uri parametrise the symmetric space Sp(2)/U(2), and satisfy and the reality condition uir = ij urj . As for the (4, 1) harmonic superspace, the nonvanishing supertrace of the torsion defines a flat connection (where we use (6.23)) and the appropriate G-analyticity condition for an integrand F(4,2) of the (4, 2) measure is that it satisfies In the linearised approximation, one can define the G-analytic superfield where we chose to define it in terms of the Sp(1, 1) spinor indices for convenience. Any quartic polynomial in M12 therefore defines a G-analytic integrand for the (4, 2) measure in the linearised approximation. One straightforwardly checks that Z d (4,2) M12M12M12M12 Z d5xd16 4 This class of invariant clearly corresponds to full-superspace integrals, and must admit a non-linear form for an arbitrary function of the dilaton that is not e 23 , e3 or 1. One has also the additional linearised invariant which cannot be written as a full-superspace integral, even at the linearised level. We will not prove the existence of such a G-analytic integrand in this paper, but one can infer, from the existence of the two ten-dimensional Chern-Simons type invariants associated to the gauge anomaly, that it exists at least for a power of the dilation of e 32 . However, it is not clear if one can define an independent duality invariant with this structure. Equivalently, one may wonder if one can define independent invariants involving the matter fields, which could only be written as (4, 2) superspace integrals. We will not answer this question in this paper. Nonetheless we note that such an invariant, even if it existed, would be more constrained by non-renormalisation theorems. Consequences for logarithmic divergences The situation for divergences in five dimensions is very similar to the one in four. There is only one available duality-invariant counterterm at two loops (7.35) which can be written as a (4, 1) harmonic-superspacee integral. It can also be written as a full-superspace integral, but not of a duality-invariant integrand. It has been recently computed [21] that the associated UV divergence is indeed absent in N = 2 supergravity, and there are hints from string theory suggesting that this result should apply independently of the number of vector multiplets.21 Before arguing that the result of the computation [21] may in principle be explained by a non-renormalisation theorem, let us point out that the uniqueness of the invariant (7.35) implies in principle that the finiteness of the four-graviton scattering amplitude in five dimensions extends to all scattering amplitudes at the two-loop order, including higher-point amplitudes and ones with external vector-multiplet states. Indeed, the only possible alternative invariants can be written only as (4, 2) harmonic-superspacee integrals, and there are several reasons to believe that they cannot support logarithmic divergences. First of all, there exists an N = 3 harmonic-superspacee formulation of Yang-Mills theory in four dimensions, and so extrapolating it to five-dimensional N = 2 supergravity, one expects only invariants that can be written at least as R d12 superspace integrals to contribute to logarithmic divergences. In components, one knows that genuine (4, 2) superspace integrals are associated to long cocycles, and therefore, applying the algebraic renormalisation arguments of [5], one would conclude as well that such invariants cannot be associated to logarithmic divergences. A similar argument to that given in [31] for the existence of auxiliary fields in fourdimensional theories implies that N = 2 supergravity in five dimensions can only admit an off-shell realisation with finitely many auxiliary fields when coupled to five modulo eight vector multiplets. In particular, the linearised theory with five vector multiplets can be obtained by dimensional reduction of the off-shell formulation of ten-dimensional supergravity [32]. Nonetheless, the argument of [31] does not rule out the existence of an off-shell formulation of the theory in harmonic superspace for an arbitrary number of vector multiplets. In this subsection we shall assume that such an off-shell formulation exists. One must note that the situation is much simpler in five dimensions, in the sense that realising the shift symmetry of the dilaton does not require us to consider a Lorentzharmonic formulation of the theory. Moreover, if we rely on the SO(n) symmetry to fix uniquely the allowed candidate counterterm, we do not require this symmetry at the level of the integrand in order to prove the non-renormalisation theorem. Therefore, the existence 21We are grateful to Pierre Vanhove for this comment. of an off-shell formulation of the theory for n = 5 + 8k vector multiplets would be enough to prove the non-renormalisation theorem in these special cases. In such a conventional superspace formulation of the theory, the non-renormalisation theorem sketched in section 4.4 would apply directly. Moreover, owing to the property that the coefficient for the logarithmic divergence is necessarily a polynomial in the number of vector multiplets, its vanishing for n = 5 modulo 8 then implies its vanishing for all n. Assuming the existence of such a formulation of the theory in superspace with all supercharges realised linearly, the beta function associated to a potential two-loop divergence will be, using the same argument as in section 5.1 [5], equal to the anomalous dimension of the classical Lagrange density in superspace for mixing under renormalisation with the density E. But the variation of this integrand with respect to a dilaton shift gives rise to an integrand with vanishing integral, i.e. the Berezinian of the supervielbeins. In an off-shell formulation, the latter will be a total derivative of a degree one co-form that will necessarily depend non-trivially on a prepotential. Assuming the existence of Feynman rules within the background field method that lead only to possible logarithmic divergences associated with functions of the potentials themselves (and not prepotentials), we conclude that the associated anomalous dimension must vanish. And in consequence, so must the beta function. Note that although the shift symmetry of the dilaton is subject to potential anomalies, there is no supersymmetry invariant with the appropriate power counting to define a oneloop anomaly for the shift symmetry. Although one expects this symmetry to become anomalous at two loops, this would not affect the potential logarithmic divergences at this order. Considering the full-superspace integral of a general function of both the dilaton and the scalar fields tm parametrising the SO(5, n)/(SO(5)SO(n)) symmetric space, one computes in the same way that the only contribution to the R4 coupling comes from the term In perturbative heterotic string theory, the -loop contribution to the effective action R4 coupling in five dimensions appears with a factor It follows that the one-, two-, three- and four-loop contributions to the effective action R4 coupling cannot be written as full-superspace integrals in five dimensions. In general, they cannot be written as (4, 1) harmonic-superspacee integrals either, except in the marginal case = 2 and K(tm) = 1. Therefore such couplings can only be defined as (4, 2) harmonicsuperspacee integrals, or as closed superforms, and can be considered as being one-half BPS protected. It is striking that we obtain precisely the same conclusion in four dimensions, for which the R4 coupling of the full-superspace integral of a function G(, tm) of the complex scalar and the scalar fields tm parametrising the SO(6, n)/(SO(6) SO(n)) symmetric space is multiplied by In perturbative heterotic string theory, the -loop contribution to the effective action R4 coupling in four dimensions appears with a factor with = a + ie2. It follows that the one-, two-, three- and four-loop contributions to the effective action R4 coupling cannot be written as full-superspace integrals in four dimensions either. The fact that the string-theory interpretation is the same in both four and five dimensions suggests that the same property should hold in ten dimensions, i.e. Z d10xg Since the -loop contribution gives rise to a factor it would then follow that the R4 coupling cannot appear as a full-superspace integral before five loops in ten dimensions. If it were possible to write e(2 23 )R4 couplings for = 1, 2, 3, 4 as full-superspace integrals in ten dimensions, then one would obtain by dimensional reduction that the corresponding dimensionally reduced invariants are also full-superspace integrals, which is in contradiction with our results. The only other invariants that include an R4 coupling are the Chern-Simons like invariants that are obtained from the d-exact 11-superforms H Rab Rbc Rcd Rda , H Rab Rab Rcd Rcd . These can only appear at one loop in string theory, because their R4 couplings come with a factor e 12 . This property is understood in string theory because these invariants are required as counterterms in order to cancel the gauge anomaly, which is itself subject to a non-renormalisation theorem [88]. The explicit computation in string theory indeed confirms that there is no R4 correction to the affective action at the 2-loop level [109], whereas only the Chern-Simons invariants appear at the 1-loop level. Nonetheless, it has been argued in [110] that this state of affairs cannot extend to all orders in perturbation theory, because it would be in contradiction with heterotic / type I duality. The structure of the supersymmetry invariants in four and five dimensions shows that R4 couplings can only be considered as being protected until four loops in string theory, and the above results suggest that the non-renormalisation theorem for the t8t8R4 81 1010R4 term in the effective action will apply until four loops, but not beyond. In this paper we have discussed the possible ultra-violet divergences that can arise in halfmaximal supergravity theories at three and two loops, in D = 4 and 5 respectively. We have shown, provided that some assumptions regarding off-shell formalisms are made, that the pure half-maximal supergravity theories should be finite in these cases, in agreement with the amplitude results and string theory. In the presence of vector multiplets, this conclusion remains unchanged in D = 5, but cannot be justified in D = 4 owing to appearance of an F 4 term in the SL(2, R) anomaly. The key observation is that, although the candidate counterterms seem superficially to be F-terms, they can be rewritten as D-terms, i.e. integrals over the full sixteen-theta superspaces. The fact that the volume of superspace vanishes, for both D = 4 and 5, implies that these full-superspace integrals are duality-invariant. There are, however, no candidate counterterms with manifestly duality-invariant full-superspace integrands. The relevant duality-invariant integrals can either be written as full-superspace integrals of integrands that are not themselves invariant, or they can be written as sub-superspace integrals of invariant integrands. We have called this situation the F/D borderline, since the status of these invariants is ambiguous. Given the existence of suitable off-shell versions of the theories preserving all of the supersymmetries linearly, as well as duality, we have argued that the F-term character wins out and that these invariants are therefore protected. This result is vitiated in the case of vector multiplets in D = 4 because the three-loop counterterm no longer needs to be fully duality-invariant. The recognition of F/D marginal structure further expands the class of special invariant structures that have a bearing on non-renormalisation properties of supersymmetric theories. By combining duality properties with supersymmetry structure, they expand the class of Chern-Simons-type invariants vulnerable to exclusion as candidate counterterms. Other examples of special structure that have been found in the now decade-long to- and fro- discussion of counterterm analysis versus unitarity-method loop calculations include special cohomology types in higher-dimensional super Yang-Mills theory [111]. For the future, it would clearly be of interest to construct the off-shell formalisms whose existence we have relied upon in our arguments. This is not an easy problem. We know that there is an off-shell version of N = 4 Yang-Mills theory in harmonic superspace, but that it only has linearly realised N = 3 supersymmetry. Moreover, this construction is rather special in that it relies upon the fact that the harmonic coset can be regarded as three-dimensional, so that the Chern-Simons action in this sector can be used to set the corresponding field strength to zero, thereby leading to the usual constraints in ordinary superspace, which are well-known to imply the equations of motion. So far, no other construction of this type has been made, except for the closely related D = 3, N = 6 Yang-Mills where such a construction leads to an off-shell version of non-abelian ChernSimons theory [112]. An additional complication is the requirement that duality symmetry be preserved. In four dimensions this is incompatible with manifest Lorentz symmetry and is therefore likely to require the use of Lorentz harmonics as well as those associated with R-symmetry. It may therefore be easier to try to tackle the five-dimensional case first where this last problem does not arise. Whether or not the above above programme can be implemented successfully, it seems difficult to imagine any purely field-theoretic argument that could protect yet higher-loop counterterms against ultra-violet divergences. This is because there are no further obstructions to the construction of counterterms that are manifestly invariant under all symmetries. This being the case, there is an obvious challenge on the computational side. If it turns out that, e.g., N = 4, D = 4 supergravity is finite at four loops, then all bets would be off regarding the perturbative finiteness of N = 8 supergravity. It is also important to stress the implications of the counterterm structures that we find, without relying on a hypothetical full-superspace off-shell formulation of the theory. In pure supergravity in four dimensions, we have shown that there is a unique dualityinvariant candidate counterterm at three loops. It therefore follows that the vanishing of the four-graviton amplitude at this order [18], implies the finiteness of all amplitudes at that order. In the presence of vector multiplets, in addition to the one-loop F 4 divergence, we find a unique potential two-loop candidate of generic form 2F 4 (whose invariance under SO(6, n) still remains to be established), and only three independent duality invariants at three loops, involving either R4 or 4F 4. Our analysis of the duality invariants that can be written as (4, 2, 2) harmonic superspace integrals is however not complete at this order, and there could be in principle more independent counterterms. Moreover, the one-loop SL(2, R) anomaly allows in principle for one additional non-duality-invariant R4 type counterterm. According to the string-theory arguments of [19], which are further strengthened by our analysis of section 7.5, the four-graviton amplitude is also finite in the presence of vector multiplets. The four-matter-photon amplitude would in principle be allowed to admit genuine two- and three-loop divergences should there be no full offshell formulation of the theory with sixteen supercharges realised linearly. It should be possible to check computationally whether or not these amplitudes diverge, which would in turn shed light on the validity of the conjecture regarding the existence of an off-shell formulation of the theory. In five dimensions there is a unique duality-invariant two-loop candidate, which can be written as a harmonic-superspacee integral over twelve of the sixteen fermionic coordinates. It can also be written as the full-superspace integral of the dilaton superfield. Assuming that there exists an off-shell D = 5 formulation of the theory with at least twelve supercharges realised linearly, one could already conclude that the finiteness of the four-graviton amplitude [21] implies the finiteness of all amplitudes at two loops. If the four-graviton amplitude were finite but if the four-matter-photon amplitude were to diverge in half-maximal N = 2 supergravity in five dimensions, one would then be able to infer that there is no off-shell formulation of the theory with more than eight supercharges realised linearly. What lesson do we learn from this analysis concerning N = 8 supergravity? First of all, note that the N = 4 theorys property that the duality-invariant (4, 1, 1) harmonicsuperspace integral can be rewritten as the sixteen-theta full-superspace integral of a function of the complex scalar is only possible because the latter is chiral. It therefore appears that the duality-invariant seven-loop candidate counterterm expressed as a (8, 1, 1) harmonic-superspacee integral [12] cannot be rewritten as a thirty-two-theta full-superspace integral in N = 8 supergravity. Although an off-shell formulation of the maximally supersymmetric theory with all thirty-two supercharges realised linearly would then permit one to conclude that the theory must be finite until eight loops, such a formalism is extremely unlikely to exist. We would like to thank Costas Bachas, Zvi Bern, Tristan Dennen, Emery Sokatchev, Piotr Tourkine and Pierre Vanhove for useful discussions. G.B. and K.S.S. would like to thank INFN Frascati, and K.S.S. would like to thank The Mitchell Institute, Texas A& M University, for hospitality during the course of the work. The work of G.B. was supported in part by the French ANR contract 05-BLAN-NT09-573739, the ERC Advanced Grant no. 226371 and the ITN programme PITN-GA-2009-237920. The work of K.S.S. was supported in part by the STFC under consolidated grant ST/J000353/1. Open Access. This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

This is a preview of a remote PDF:

G. Bossard, P. S. Howe, K. S. Stelle. Invariants and divergences in half-maximal supergravity theories, Journal of High Energy Physics, 2013, 117, DOI: 10.1007/JHEP07(2013)117