#### 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.
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