Minimal unitary representations from supersymmetry

We compute the supersymmetry constraints on the R^4 type corrections in maximal supergravity in dimension 8, 6, 4 and 3, and determine the tensorial differential equations satisfied by the function of the scalar fields multiplying the R^4 term in the corresponding invariants. The second order derivative of this function restricted to the Joseph ideal vanishes in dimension lower than six. These results are extended to the d^4 R^4 and the d^6 R^4 corrections, based on the harmonic superspace construction of these invariants in the linearised approximation. We discuss the solutions of these differential equations and analysis the consequences on the non-perturbative type II low energy string theory effective action.


Introduction
Type II string theory on R 1,9−d × T d is extremely constrained by supersymmetry and duality symmetries. The various formulations of the theory are conjectured to be related by U-duality, an arithmetic E d(d) (Z) subgroup of the split real form of the Lie group of type E d [1]. In particular, the exact low energy expansion of the effective action is expected to exhibit this symmetry [2,3,4]. However there is no non-perturbative formulation of superstring theory permitting to derive directly the low energy expansion of the amplitudes, and one must use perturbative string theory [5,6,7,8,9] and eleven-dimensional supergravity [3,10] altogether with U-duality to derive their non-perturbative completion. One can deduce the superstring effective action from the amplitude by inverse Legendre transform (up to field redefinition ambiguities), which can then be expressed in the low energy limit as the supergravity 1PI generating functional computed with the complete (appropriately renormalised) string theory Wilsonian effective action. The supersymmetric Wilsonian effective action admits the following expansion in the reduced Newton constant κ 2 in 10 − d dimensions where S (0) is the supergravity classical action, and S (n+3) [E (p,q) ] with 2p + 3q = n is a ∂ 2n R 4 type supersymmetric correction to the effective action depending on a function E (p,q) of the scalar fields parametrizing the symmetric space E d(d) /K d [11]; 1 although starting from n ≥ 5 one has independent corrections in ∂ 2n−2 R 5 and etcetera at higher orders [12]. It was shown in [13] that supersymmetry implies the function E (0,0) characterising S (3) [E (0,0) ] in type IIB supergravity in ten dimensions to be an eigenfunction of the Laplace operator with eigenvalue − 3 4 , consistently with the analysis carried out in [2]. As a consequence, supersymmetry and duality invariance entirely determine the function E (0,0) in ten dimensions. The constraints from supersymmetry have been computed for higher order invariants [14] and the same conclusion holds for the ∇ 4 R 4 type corrections [10]. The realisation of these functions as Eisenstein functions [2,4] has been generalised in lower dimensions [15], and to higher order ∇ 6 R 4 type corections [16], leading to more developments in lower dimensions [17,18].
In this paper we work out the constraints on the R 4 type corrections in lower dimensions. We carry out this program within the formalism of superforms in superspace developed in [19,20,21]. We concentrate in a first section on R 4 type invariants in N = 2 supergravity in eight dimensions. Computing the complete invariant is out of reach, and we concentrate on the components of the superform that carry the maximal R-symmetry weight representations, similarly as in [13,22]. We find in this way that the function of the scalar fields must satisfy to a tensorial second order differential equation consistent with the explicit Eisenstein function computed in [4].
We extend these results in dimension 6, 4 and 3 and exhibit that the function defining the R 4 type invariant satisfies to a unique tensorial second order differential equation associated to the minimal unitary representations of SO(5, 5), E 7 (7) and E 8 (8) , respectively. The function multiplying R 4 must satisfy that its second order derivative restricted to the Joseph ideal [23] vanishes J (D, D) E (0,0) = 0 . (1. 2) The relation between the minimal unitary representations and the R 4 type threshold function has been argued from several perspectives [24,25,26,27,28] and it is in particular conjectured that the function can be defined as the exceptional theta series associated to the minimal unitary representation of E d(d) [28]. Our results strongly support this conjecture by showing that supersymmetry indeed implies (1.2), which square integrable solutions define the minimal unitary representation of the corresponding exceptional group. Using the harmonic superspace construction of the higher order invariants in the linearised approximation [29,30,31,32], we extend these results to the ∇ 4 R 4 type invariants. In four dimensions we also determine the equation satisfied by the function defining the ∇ 6 R 4 type invariant, relying on properties derived in [33] to fix the free coefficients. We find that the threshold functions satisfy to higher order differential equations attached to certain nilpotent coadjoint orbits exhibiting their relation to next to minimal unitary representations as proposed in [28].
We study the corresponding differential equations in some detail in six and four dimensions, and find perfect agreement with the definition of the threshold functions as Eisenstein series [11,33,34,35]. We discuss in particular the two Eisenstein functions defining the ∇ 4 R 4 type correction in six dimensions [33], and exhibit that these two functions are associated to two independent invariants, and solve independent differential equations associated to the two next to minimal nilpotent orbits of D 5 (that both only include the closure of the minimal nilpotent orbit in their topological closure). Working out the general solutions to these differential equations, we extend the results of [34] on the structure of the Fourier modes of these functions.
Because the R 4 type corrections to the effective action are defined in the linearised approximation as superspace integrals over half of the Grassmann coordinates [30], the property that they only receive corrections from non-perturbative effects associated to 1/2 BPS instantons has been conjectured to be a consequence of supersymmetry [2]. The differential equation that we find to be a consequence of supersymmetry implies indeed strong restrictions on the possible perturbative corrections that the effective action can receive in string theory, and moreover implicates through the dependence on the scalar fields that the non-perturbative corrections associated to instantons must also be 1/2 BPS by supersymmetry. The generalisation of these results for ∇ 4 R 4 to only receive corrections from (at least) 1/4 BPS instantons go through as well, in agreement with the analysis carried out in [34], and the differential equation we propose for the ∇ 6 R 4 type invariant in four dimensions implies that it can only receive corrections from (at least) 1/8 BPS instantons, as expected from its harmonic superspace construction in the linearised approximation.
In this paper we distinguish the Wilsonian effective action that preserves local supersym-metry from the 1PI generating functional satisfying to the quadratic BRST master equation.
In particular we exhibit that the logarithmic contributions to the threshold functions responsible for the constant right-hand-side in the Poisson equation satisfied by these functions [36], do not appear in the Wilsonian effective action, but are consequences of duality anomalies. We discuss this property in particular in eight dimensions, where the R 4 threshold gets one contribution associated to the chiral 1-loop U (1) anomaly similarly as in four dimensions [37], whereas the second is associated to an incompatibility between supersymmetry and SL(3, R) duality invariance. We also exhibit that the ∇ 4 R 4 threshold function in six dimensions satisfies to a Poisson equation with a right-hand-side proportional to the R 4 threshold function, which is attributed to the duality transformation of the R 4 superform insertion (i.e. form factor) in the supergravity 1PI generating functional. The anomalies associated to the incompatibility between duality and supersymmetry Ward identities bypass the analysis carried out in [38] (although their possible existence was not overlooked), but they can only arise by construction when the threshold function is constrained to satisfy to the Laplace equation (i.e. with zero eigenvalue) from supersymmetry Ward identities. Therefore such anomaly can only arise when the supergravity amplitude exhibits a logarithm divergence [36], such that they do not affect the non-renormalisation theorems established in [39,40] regarding the absence of logarithm divergence in N = 8 supergravity before seven-loop order based on the absence of E 7(7) anomalies, consistently with the factorisation of eight additional external momenta in the explicit 4-loop four-graviton supergravity amplitude [41]. Our work does not give new insights on the ultra-violet behaviour of maximal supergravity amplitudes, but it does give predictions on the logarithmic divergences of supersymmetric densities form factors. The integrated invariants are observables of the theory, and therefore the zero momentum limit of the associated form factors are BRST invariant observables. Generalising the argument of [36] to these cases we find that the supersymmetric R 4 form factor should diverge at one loop in ∇ 4 R 4 in six dimensions, and similarly that the ∇ 4 R 4 form factor should diverge at one loop in ∇ 6 R 4 in four dimensions, whereas the R 4 form factor must be finite until 4-loop order by supersymmetry. The paper is organised in four sections devoted to the analysis of maximal supergravity in eight, six, four and three dimensions, respectively. It is in eight dimensions that we work out the supersymmetry constraints on the R 4 type invariants in most detail. For this purpose we start by deriving the superspace geometry, including cubic terms in the fermions that are relevant to our analysis. The latter can be found in Appendix C. From six dimensions and below, the algebraic constraints on the consistent second order differential equations on E d(d) /K d are so strong that it is enough to work out the supersymmetry constraints on the maximal Rsymmetry weight terms of order sixteen in the fermion fields to determine them. This is due to the property that (1.2) determines uniquely the eigenvalue of the Laplace operator.
More generally we find that the differential equation satisfied by the R 4 , ∇ 4 R 4 and ∇ 6 R 4 type invariants can be deduced from their harmonic superspace construction in the linearised approximation, up to a potential free parameter that is fixed for R 4 and ∇ 4 R 4 in dimension lower than six. The harmonic variables parametrize a homogeneous space where the U (1) factor determines the G-analytic superfield W as the component of the scalar field of highest U (1) weight. The harmonic superspace integrands are therefore in one to one correspondence with the symmetric order n monomials in the G-analytic superfield, that are associated to a set of irreducible representations R d,n,k of K d . The algebraic restriction on the symmetric monomials of the G-analytic superfield define a subspace of the vector space of monomials of a generic coset element. Assuming that the non-linear invariants are in one to one correspondence with the harmonic superspace integral invariants, the same restriction must hold on the jet space of n th order derivative acting on a generic function E defining these invariants, i.e. D n E (p,q) ∈ k R d,n,k .
This assumption is justified in four dimensions by the complete classification of SU (2, 2|8) chiral primary operators [31,42], which proves that all supersymmetry invariants are realised as harmonic superspace integrals. Although there is no theorem, is seems that all supersymmetry invariants can indeed be defined as harmonic superspace integrals in the linearised approximation in dimension lower than six. 2 This U (1) factor lies inside a GL(1, C) subgroup of the complexication of K d that determines a graded decomposition of the complex Lie algebra k d (C) as well as e d . The highest weight component of e d ⊖ k d (C) determines a nilpotent element, that characterises a unique nilpotent orbit of the real Lie group E d(d) according to the Kostant-Sekiguchi correspondence [43]. It follows that a nilpotent element Q satisfies to an algebraic constraint that is such that Q ⊗n ∈ k R d,n,k (C) .
(1. 4) We conclude that the same algebraic constraint satisfied by the nilpotent element Q is satisfied by the symmetrised product of derivatives acting on E (p,q) . For the R 4 type invariant, the relevant nilpotent orbit is always the minimal nilpotent orbit of E d(d) , and the quadratic algebraic constraint is the Joseph ideal [23]. In general the square integrable solutions to the corresponding differential equation define a quantisation of the corresponding nilpotent orbits. Because the nilpotent orbits are classified by the K d (C) weighted Dynkin diagram characterising the subgroup GL(1, C), it is straightforward to read of the nilpotent orbit associated to a given harmonic superspace in the classification [44]. For E 6(6) , E 7(7) , E 8(8) the 1/2 BPS and 1/4 BPS couplings correspond to the minimal and next to minimal nilpotent orbits, which K d weighted Dynkin diagram carry zeros on the maximal semi-simple H d subgroup Dynkin diagram and 1 on the other nodes. The 1/8 BPS couplings correspond to the nilpotent orbits which K d weighted Dynkin diagram carry zeros on the maximal semi-simple H d subgroup Dynkin diagram and 2 on the other nodes.

N = supergravity in eight dimensions
In this section we shall discuss the R 4 type invariants in N = 2 supergravity in eight dimensions, and prove that the R 4 threshold function must satisfy to differential equations consistent with the explicit SL(2, Z) × SL(3, Z) threshold computed in [4]. We will consider the problem in the superspace formulation of the theory, and we shall therefore compute the geometrical tensors of N = 2 supergravity in superspace in a first subsection. Our strategy is inspired from the idea proposed in [13] to concentrate on the fermion monomials of maximal weight, as was used in [22] in eight dimensions. However we will go beyond this results, and exhibit that the function satisfies to a stronger equation than the Laplace equation already exhibited in [4].

Supergravity in superspace
In order to determine supersymmetry invariants we shall use the superspace formalism. In this section we will derive the structure of the supergeometry in eight dimensions, following the same construction as in [45,46]. The R-symmetry group is U (2), and is represented such that the covariant derivatives D i α ,Dα i have respectively weights 1 and −1 with respect to the axial U (1), and the indices i correspond to the fundamental of SU (2), whereas α andα are respectively in the chiral and the anti-chiral Weyl representation of Spin(1, 7), which are complex conjugate. The complete set of fields is depicted in figure 1. The where T AB C is the torsion, and the Riemman curvature R ABC D t The consistency of the commutation relations implies the Bianchi identities where d ω is the covariant exterior derivative in superspace, with ω M B A itself valued in so(1, 7)⊕ u (2). The Bianchi identities read in components Figure 1: Structure of the supergravity supermultiplet in the linearised approximation. It includes a chiral superfield W and a tensor superfield L ijkl related through their second derivative. The symmetry with respect to the horizontal axe defines complex conjugation.
where denotes the sum over cyclic permutations of A, B, C. Moreover the internal connexion in u(2) is determined from the Maurer-Cartan superform of scalar superfields parametrizing the symmetric space SL(2, R)/SO(2) × SL(3, R)/SO(3), one complex superfield T and five real superfields φ µ . We represent SL(2, R) in terms of the SU (1, 1) matrices The Maurer-Cartan form defines the u(1) connexion and scalar momenta. Similarly one defines the SL(3, R) matrices with i = 1, 2 of the gauge group SU (2) and I = 1, 2, 3 of the rigid SL(3, R). We will not provide an explicit parametrization of this matrix in terms of the five scalars φ µ , because this will not be required in our analysis. One decomposes the Maurer-Cartan form as The momentum P and the su(2) connexion ω i j are defined in this way as where SU (2) indices are raised and lowered with the ε ij tensor. It follows from the Maurer-Cartan equations that and that the u(2) components of the Riemann tensor are determined as In components, these identities read To complete the definition of superspace, we enforce the existence of superform field strengths transforming in linear representations of SL(2, R) × SL(3, R). They are 6 1-form potentials A 1 I , A 2 I in the 2 ⊗ 3 that define the complex 2-forms F ij , 3 2-forms potentials B I in the 3 that define the three form field strengths H ij and one 3-form potential C that defines a complex 4-form G and its complex conjugate, transforming together in the 2 of SL(2, R) [47]. They satisfy to the Bianchi identities (2.13) Here we allow ourselves to fix the Chern-Simons couplings H ij ∧F ij and F k(i ∧F j) k , which determine the respective normalisation of the fields with non-canonical kinetic terms. One obtains in components where states for the sum over alternated permutations of all tangent indices ABC . . . , such that the result is a graded antisymmetric tensor.
The solution to these superspace identities determines the covariant superfields of the theory, which first components at θ = 0 (i.e. the pull back to the bosonic space embedded in superspace) correspond to the supercovariant fields of the theory in components. By construction, these fields satisfy to the equations of motion. In this paper we shall consider the classical superspace solution solving the classical (two derivatives action) equations of motion. Restricting ourselves to the classical superspace, one can use dimensional analysis to determine the various components of the superfields. Moreover, the dimension zero components must necessarily be invariant tensors. It follows for example that the only dimension zero components of the torsion are and its complex conjugate. One can use the same argument to restrict the decomposition of the superforms, such that no more than two of the tangent indices AB . . . can be fermionic. MoreoverF ij andḠ have an overall U (1) weight u = 2, whereas H ij is neutral. Using that the dimension zero component must be U (1) invariant, one gets the decompositions where we moreover used the property thatḠα ibcd = 0. This last condition is true because the only dimension 1/2 field of U (1) weight 1 is the fermion field with three symmetric SU (2) indices λ ijk α . In principle this property can be proved in general following [45,46], here we already assume the knowledge of the field content of N = 2 supergravity [47]. One computes that the dimension zero components of the form fields arē Indeed one straightforwardly checks that they are the only invariant tensors satisfying to the appropriate symmetry properties, and the specific coefficients are determined modulo an overall rescaling by the Bianchi identities (2.14), i.e. where the symbol jkl βγδ indicates the sum over cyclic permutations of the three pairs of indices. At dimension 1/2 one gets that there is no fermionic field of U (1) weight 5, such that T in (2.4) must be a chiral superfield, i.e.Dα i T = 0. Therefore, the scalar momenta decompose into with P m ijkl α and P ijkl αm having dimension 1/2 and U (1) weight 1, −1, whereasPα i has dimension 1/2 and U (1) weight 3. One computes that all components of U (1) weight 3 are determined in terms of one single fieldχ iα , as In the same way one use the Bianchi identities to show that all the dimension 1/2 component of U (1) weight 1 are determined in terms of a single field λ ijk α as The computation goes on then at dimension 1, with new independent fields associated to the scalar momenta P a , P ijkl a and the field strengthsF ij ab , H ij abc andḠ abcd − , although it turns out that the sefldual component of the 4-formḠ is determined in terms of the fermions as 3 This is consistent with the property that there is only one 3-form potential in eight-dimensions, and its complex selfdual and anti-sefldual components transform in the fundamental of SL(2, R). From dimension 1 and beyond the solution to the constraints is rather complicated, and we only display the dimension 1 and 3/2 components in Appendix B and C, respectively. Now we need to discuss the definition of supersymmetry invariants in superspace. In this section we will only consider the first corrections to the Wilsonian effective action, therefore it is enough to consider corrections to the action that are invariant with respect to supersymmetry subject to the classical equations of motion. In the superspace framework, such a correction 3 Note that in Minkowski signature γ abcd to the action is determined by a cohomology class in superspace, i.e. a d-closed superform in classical superspace, defined modulo the addition of a d-exact superform [20,21]. A superform decomposes in tangent frame as where each component will be referred to as L (m,n,p) , and for an order κ 2(ℓ−1) correction with u the U (1) weight. One understands that all bosonic indices are antisymmetrised whereas fermionic indices are symmetrised in pairs α k i k (respectivelyα k i k ). The condition dL = 0 ensures that the pull-back of this closed form to the bosonic subspace (2.29) is invariant with respect to supersymmetry, modulo a total derivative and the classical equations of motion [20,21]. In this form the components L (m,n,p) | θ=0 only depend on the supercovariant field strengths and their supercovariant derivatives. dL = 0 decomposes in tangent frame in dL (m,n,p) = T (2,0,0) (0,0,1) L (m-2,n,p+1) + T (2,0,0) (0,1,0) L (m-2,n+1,p) + T (1,1,0) (0,0,1) L (m-1,n-1,p+1) + D (1,0,0) + T (1,1,0) (0,1,0) + T (1,0,1) (0,0,1) L (m-1,n,p) + T (1,0,1) (0,1,0) L (m-1,n+1,p-1) + T (0,2,0) (0,0,1) L (m,n-2,p+1) + D (0,1,0) + T (0,2,0) (0,1,0) + T (0,1,1) (0,0,1) L (m,n-1,p) where we defined and together with their complex conjugate, and such that the indices of uppercase grades are understood to be contracted with indices of lowercase grades. Note that the components T a j β c , T aβ j c and T ab c vanish. In this paper we will only consider the component and its complex conjugate. We will indeed find out that these equations alone permit to determine the differential constraints on the function of the scalar fields characterising the d-closed superform.

The chiral R 4 type invariant
As explained in [30], one can define an invariant from an arbitrary holomorphic functions of the chiral superfield T ∼ W in the linearised approximation where t 8 is the standard tensor defined such that  [48]). Therefore one cannot directly rely on the chiral superspace integral to define the non-linear invariant, but one can still extract information from it as we are going to discuss. Supposing for simplicity that the invariant is SL(3, R) symmetric, such that it only depends on the scalar fields φ µ through the covariant derivative The covariance of the superspace constraints with respect to SL(2, R), implies that the derivatives of a function must necessarily be Kähler covariant derivatives (2.39) Expanding in the number of fields, one can consider the term in D mDn F(T , T ), as counting for −m − n fields, such that the linearised invariant corresponds to the 4-point approximation.
With this convention, one gets that the superform should take the form where I a 4n are SL(2)× SL(3) invariant monomials in the covariant superfields of U (1) weight 4n and dimension 8, and F a n (T,T ) are functions (or more precisely (0, n)-tensors on SU (1, 1)/U (1)) of the scalar T,T that multiply them in the invariant. The independent such monomials are labeled by the index a. In this section we shall consider the monomials of maximal U (1) weight in order to simplify the computation. To check the possible terms, it is convenient to consider the ratio of the U (1) weight by the dimension. The largest ratio is forχ i α , that has u = 3 and dimension 1/2, and therefore the maximal U (1) weight term is the uniqueχ 16 monomial as in (2.34). We define its normalisation such that The next field is the dimension 1 fieldP a that has u = 4, however note that a term of the formP a DF a n can always be eliminated by adding a trivial cocycle to the superform without modifying the invariant, and one can therefore disregard such terms. The next important fields are therefore the dimension 1 field strengthF ij ab ,Ḡ − abcd of U (1) weight 2 and the dimension 1/2 field λ ijk α of U (1) weight 1. There is a unique monomial inχ 15 and three inequivalent monomials inχ 14 , two isovectors in the irreducible SO(1, 7) representations 2 00 0 and 0 00 0 and one SU (2) singlet in the 0 01 0 . It is convenient to define their normalisation from the Grassmann derivative of (2.43) as a function of ordinary Grassmann variables (rather than fields) With these definitions, we write a general ansatz for the I a 44 , as (2.44) Note that we could also consider a term in (χ 13 ) ijk aα λ α ijkP a , but one can always remove such a term by adding to the superform L a d-exact form dΨ with Ψ (7,0,0) equal to while affecting only therms inŪ −20 . Therefore we will not consider such a term that would not lead to any constraints by construction, since G 10 (T,T ) is clearly arbitrary in Ψ (7,0,0) . One could also guess the appearance of a term inλ ijk χ 15 α k , but there is no SU (2) singlet such a monomial. Our ansatz for L (8,0,0) will therefore be L abcdef gh = ε abcdef gh Writing down (2.33), one sees however that the equation dL = 0 also includes mixing of L (8,0,0) with L (7,1,0) , L (7,0,1) , L (6,2,0) , L (6,1,1) , L (6,0,2) , so we must also consider an ansatz for these components. In the formalism in components (as opposed to superspace), this amounts to distinguish the terms that are written in terms of supercovariant field strengths, from the ones that carry naked gravitnino fields. Let us consider first L (7,1,0) , which is a spinor valued 7-form in the fundamental of SU (2) with U (1) weight u = 1. It can include two irreducible representations of Spin(1, 7), the 0 10 1 and the 1 00 0 . The maximal U (1) weight component one can get is u = 45, with the term χ 15 iα . We shall only check terms up to orderŪ −22 in dL = 0, and therefore this is the only term that will be relevant in our computation, so we consider the ansatz with again other functions F a n depending on T andT . L (7,0,1) has U (1) weight −1, and decomposes into the irreducible representations 1 10 0 and the 0 00 1 of Spin(1, 7), therefore it cannot include terms inχ 15 and the maximal U (1) weight terms one can have are inŪ −22χ14 λ and U −22χ13P . Moreover most of the latter can be reabsorbed in a trivial cocycle and lower U (1) weight terms such that one obtains the ansatz The same idea holds for L (6,2,0) , L (6,1,1) and L (6,0,2) of dimension 7, and of U (1) weight 2, 0 and −2, respectively. One checks that L (6,2,0) and L (6,1,1) carry at most terms inŪ −20 , whereas L (6,0,2) carries terms inŪ −22χ14 , i.e. The terms inŪ −24 in D (1,0,0) L (7,0,1) are computed using D a Ū −2n F a n =Ū −2(n+1) D F a n P a +Ū −2(n−1) 1 − TT 2 (DF a n ) P a , whereas D (0,0,1) L (8,0,0) does not depend onP a at this order, and we conclude that they must cancel by themselves. However they do not, and the functions F a 11 must be holomorphic forms for a = 6 , 7 , 8. Going further in the analysis one would in fact conclude that they vanish.
Therefore we can consider the equation D (0,0,1) L (8,0,0) = 0 at this order inŪ . The order U −26 term vanishes trivially whereas the orderŪ −24 terms give the equation Solving this equation requires to consider the explicit derivative of the fieldχ i α computed in Appendix B Using Fierz identities related to the uniqueness of (χ 15 ) iα and the property that the terms in (χ 16 )λ ijk α cancel by themselves because D (0,0,1) L (8,0,0) is in the fundamental of SU (2), one computes that (2.54) is satisfied if and only if Therefore in terms of a single function F 11 as Similarly, restricting ourselves to the terms of maximal U (1) weight, (dL) (8,1,0) = 0 simplifies to where we used moreover that the terms of orderŪ −22 of L (7,0,1) in (2.48) vanish. We start with the terms of orderŪ −24 that further reduce to (2.61) The covariant derivative Dχ is determined from (B.9) as so once again the terms in (χ 16 )λ ijk α cancel by themselves and we get the constraint Now we must consider the orderŪ −22 components of (2.60), however the computation involves many terms and we shall simplify the problem by neglecting all the terms that depend explicitly on λ ijk α andP a . This permits in particular to neglect terms of orderŪ −20 in L (7,1,0) that we have not computed. Using this simplification, one obtains To carry out this computation we need the covariant derivative D i α of bothḠ − abcd andF ij ab given in Appendix C in (C. 18) and (C.17), as well as the dimension 1 torsion T a j β γ k given in (B.11), for which we neglect all terms in λ ijk α andP a . Moreover, the equation can only be satisfied modulo the classical equations of motion, and we must distinguish in D aχ iα , its gamma trace that is equal to a polynomial in the other fields through the Dirac equation (C.15). We will write (D aχ iα ) ′ its component projected to the irreducible representation 1 10 0 of Spin(1, 7) (i.e. such that (γ a ) αβ (D aχ iβ ) ′ = 0). Combining all these terms one obtains finally We conclude therefore that the harmonic forms c 2 and c 5 vanish as expected, and the form F 11 satisfies to the differential equation It is rather clear that if we had computed the terms in λ ijk α one would have obtained similarly that c 3 = c 4 = 0, and we conclude therefore that It is important to note that this superform indeed reproduces the structure explained in the beginning of this section, i.e. each covariant combination of fields multiplyingŪ −2nDn F is approximated by the linearised invariant as The relation to the linearised invariant implies indeed that each covariant combination of fields multiplyingŪ −2nDn F must be of the formD n+4W n+4 W =0 such that (similarly as in [13]) Using the commutation relations between D andD, one computes in general that and therefore in particular that At each order inŪ −2nDn F(T,T ) one will get equations generalising the linearised equations of the form where the coefficient is determined to be the unique one consistent with (2.70), therefore we conclude that supersymmetry must imply eventually that the function F(T ) is anti-holomorphic.
There are two comments we would like to make on this computation, to be compared with the computations carried out in components in [13,22]. Here we implicitly used the Dirac equation satisfied byχ iα in several places, by removing the gamma trace appearing in D a χ iα when this term appeared explicitly, and when it appeared in the derivative of the field strengths F ij ab andḠ abcd . Indeed, in components one would consider instead the supersymmetry variation of their potentials. One concludes that considering ι * L as a correction to the effective action, the accordingly corrected covariant derivative D i α χ j β would be modified by terms of the form although we did not compute the coefficients explicitly. In components the correction to the Lagrange density takes the form whereF ij andḠ are supercovariant field strengths, that include respectively terms in −2ie a ∧ (ψ (i γ aχ j) ) and ie a ∧ e b ∧ e c ∧ (ψ i γ abcχ i ). There is therefore three different contributions to the term inŪ −22D11 F(T )(ψ ai γ a (χ 15 ) i ), and they must all be there with their respective coefficients.

The parity symmetric R 4 type invariant
In the linearised approximation, the scalar fields φ µ parametrizing SL(3, R)/SO(3) are conveniently represented by an isospin 2 field L ijkl , such that the covariant derivative and similarly for the complex conjugate. As explained in [30], one can define an invariant in the linearised approximation from an arbitrary holomorphic functions of the G-analytic superfield as the harmonic integral of (2.79) In this section we will repeat the computations of the last section to determine the dependence of this invariant in the scalar fields φ µ at the non-linear level. One can already infer from the linearised analysis that the function of φ µ must satisfy to the Laplace equation [49]. However, because the harmonic measure does not extend to the non-linear theory this construction had no reason to give the correct answer. To start with we need to discuss some properties of the differential operators on the symmetric space SL(3, R)/SO(3) that are perhaps less standard than for the special Kähler space SU (1, 1)/U (1).

Differential operators on SL(3, R)/SO(3)
The superfield momentum P ijkl defined in (2.8,2.9) determines the vielbein P µ ijkl on SL(3, R)/SO (3) in function of φ µ as Considering φ µ as coordinates rather than fields in this discussion, the Maurer-Cartan equation indeed gives the torsion free condition, and the definition of the constant Riemann tensor on SL(3, R)/SO(3) in tangent frame. One defines accordingly the metric and its inverse G µν such that the inverse vielbein read In these conventions one has where we use the symmetrised Kronecker symbol One defines the covariant derivative of a function and its subsequent covariant derivatives as and etcetera. For a generic symmetric tensor, the covariant derivative is defined accordingly as and one computes using (2.81) that In particular where the notation means that ijkl and tuvw are symmetrised in the first term of the righthand-side, and similarly in (2.88).
The covariant derivative D ijkl D pqrs E of a function E decomposes into irreducible representations of SU (2), as a singlet, an isospin 2 component and an isospin 4 component. We want to consider as a differential equation the property that the isospin 2 component is related to the first order derivative, i.e.
This equation can be rewritten for some function G to be determined. This equation implies that Because there is a unique scalar fourth order differential operator, one has the constraint for any function E, and one can therefore deduce from (2.91) that For s = 0 or 3 2 , one obtains immediately that the function E s satisfies to and in particular The reader might recognise at this point that this Poisson equation is satisfied by the Eisenstein series in the domain of absolute convergence of the series (i.e. for s > 3 2 ). One straightforwardly computes that the function (V ij I n I V ij J n J ) −s indeed satisfies to the quadratic equation (2.95) for any vector n I ∈ R 3 * , and one concludes that for s > 3 We are going to prove in this section that supersymmetry requires this equation to be satisfied for the function E multiplying the R 4 type term in the invariant for the value s = 3 2 , consistently with the string theory computation [4]. However, the series actually diverges for this value, and one must consider the regularised Eisenstein series [11] By continuity, and because the constant term drops out when acted on by the covariant derivative, one obtains that the regularised series satisfies to the inhomogeneous equation consistently with [11]. Note that the constant term is indeed consistent with (2.94), because for s = 3 2 the inhomogeneous term can in principe be any function satisfying to the Laplace equation ∆G 3 2 = 0. However the constraint from supersymmetry is by construction a homogeneous linear equation, and is in fact The inhomogeneous term in (2.100) is due to the logarithm log(V ij I n I V ij J n J ) that satisfies to and which appears explicitly in the expansion ofÊ [ 3 2 0] at large V ij I n I V ij J n J (for any chosen vector n I ),Ê We shall explain that this logarithm term is associated to an anomaly, and does not appear in the supersymmetric Wilsonian effective action. To prove that (2.101) is indeed required by supersymmetry, we shall consider the terms of maximal isospin. Because these terms will carry a large number of SU (2) indices, we will use the short-hand notation and repeated representations will be understood to correspond to contracted indices, as for example in D 12 [48] E (λ 8 ) [24] (λ 8 ) [24] Using the commutation relations (2.88), one computes that in general are respectively in the isospin 2(n − k) and 2n + 2 irreducible representations. Using this equation, one obtains that for a function E s satisfying to equation (2.95), one has moreover where D n+1 , D n and D n−1 are in the irreducible representations of maximal isospin 2n + 2, 2n and 2n − 2, respectively.

Constraining the superform
Similarly as for the chiral superform L[F] discussed in the last section, the linearised analysis suggests that the super-form L[E] admits the following expansion invariant isospin 2n tensors superforms, that coincide with the linearised invariant at 4 + n order in the fields Using this general structure, one is led to a general ansatz for L (8,0,0) ab λ 6ab [18]λ8 [24] + a 3 D 11 [44] E H [2] abc (λ 7 [21] γ abcλ7 [21] ) + a 2 D 11 [44] [44] E (λ 7 [21] γ abcλ7 [21] )(χ [1] γ abc χ [1] ) + b 6 D 11 [44] Eλ 6ab [18] (λ 9 [25] γ ab χ [1] ) + D 10 [40] E · · · (2.110) and similarly for L (7,1,0) bc γ abc αβλ 7β [21] + D 10 [40] E · · · (2.111) and its complex conjugate. Note that this ansatz is completely general provided one replaces each derivative term D n [4n] E by a generic isospin 2n tensor E a [4n] , and the computation we shall carry out does not require such an assumption. It particular, there is no candidate monomial in the fields of odd isospin at this order, and we did not avoid such terms in the ansatz. It will turn out to be enough to look at terms of isospin 24 in dL[E] = 0 to determine the properties of the function E, and because L (7,1,0) only contributes at this order through a space-time derivative, one can neglect the contribution from L (7,1,0) if one disregards terms including the momentum P ijkl a . At this order dL[E] = 0 simplifies drastically to (2.112) Moreover, the superform being real, these two equations are equivalent. Restricting ourselves to the components of D i α L (8,0,0) of isospin 24, the components of isospin 22 of L (8,0,0) only contribute through the derivative of their tensor E a [44] , and therefore only mix with the isospin component E [48] λ 8 [24]λ8 [24] through the covariant derivative acting on the fermions, but for the terms that are themselves in λ 8λ8 . It follows that most of these contributions simply constrain these tensors to satisfy to in agreement with the ansatz (2.110). Computing these terms one would determine the coefficients a k and b k for k ≥ 3 in (2.110), but one would not get any constraint on the function E. The only terms constraining the function itself are the ones in λ 9λ8 , and we will therefore focus on the restricted ansatz where we do not assume that the two other SO(3) tensors are also derivatives of the same function. At this point we need to precise the normalisation of the fermionic monomials (2.115) The first contribution comes from [24] (2.116) and using the property that the maximal isospin monomial in λ 9 is of isospin 25 2 , one gets that the isospin 24 contribution in D 13 [52] E cancels out such that where we neglect the terms of lower isospin. Using the covariant derivatives computed in Appendix B and concentrating on the terms in λ 9λ8 , one obtains after using Fierz identities These three combinations being linearly independent, one concludes that assuming that there is no inhomogeneous term satisfying to One can indeed convince oneself that there is no solution to this differential equation, which defines 4n + 1 independent first order equations for only 4n − 3 variables, i.e. 4 more equations at each order, equivalently as which only solution is a constant. Because there is no higher rank symmetric tensor, there is no solution for n > 1. The most important equation is the constraint It follows from the structure of the linearised invariants that the terms of lower isospin will be all related, such that they will satisfy to similar equations of the form such that one gets eventually Expanding dL[E] = 0 in the same way, one gets but because ∆E is necessarily a solution to the Laplace equation, i.e. ∆ 2 E = 0, the two terms must vanish independently. One deduces from the linearised analysis that L ijkl carries terms of the form and P ijkl ∧ L ijkl does not vanish, so we conclude that supersymmetry indeed requires and therefore (2.101) is satisfied. Using this constraint, the tensor superforms L [4n] satisfy to the differential equation (2.131) and the equation we have checked explicitly in this section is the λ 9λ8 component of Note moreover that this equation must satisfy the consistency condition One finds that the general solution to satisfying to (2.133) is determined up to an integration constant s, as One recognises that the coefficients are the same as in (2.107), and therefore they are the equations satisfied by a closed superform L[E s ] associated to a function E s satisfying to (2.95) in general. Equation (2.135) defines by construction a representation of sl 3 through the definition of the coset generators on the infinite sum ⊕ ∞ n=0 (4n + 1), which corresponds to the unitary representation of SL(3, R) on the set of square integrable functions satisfying to (2.95).

Anomalies
We have proved in this section that the function multiplying R 4 in the supersymmetry invariant is the sum of a harmonic function of the complex scalar T and a function of the SL(3, R)/SO(3) scalars solution to the quadratic equation (2.101). However, the string theory threshold function appearing in the four-graviton amplitude [4] does not solve these equations strictly, and solve inhomogeneous equations (2.100) [11]. The contributions responsible for these inhomogeneous terms come from the non-analytic component of the amplitudes, and are only captured by the supergravity 1-loop 1PI generating functional Γ 1-loop . Therefore these terms do not appear in the string theory Wilsonian effective action invariant with respect to local supersymmetry, but only in the 1PI effective action satisfying to the BRST master equation.
The discussion of the inhomogeneous term in the Laplace equation on SL(2, R)/SO (2) is very similar to the one of N = 4 supergravity in four dimensions [37]. The complex superform L[F(T )] discussed in section 2.2 admits by construction the R 4 type terms In this discussion it will be convenient to consider the upper complex half plan coordinate For the specific choice F(T ) = τ, the imaginary part of the superform (2.138) coincides with the dimensional reduction of the R 4 type invariant in eleven dimensions on T 3 , where the imaginary part of τ defines the T 3 volume modulus and its real part the pull-back of the 3-form potential on T 3 . This exhibits by consistency with gauge invariance in eleven dimensions that one must have Re where R ab is the Riemann tensor superform. One can prove this property directly in eight dimensions by studying the structure of the superform similarly as in [37] in N = 4 supergravity in four dimensions, although we will only report on this analysis in a forthcoming paper. It follows from [4] that the complete string theory Wilsonian action includes non-perturbative corrections in M-theory corresponding to Euclidean M2 branes wrapping T 3 such that the associated contribution to the Wilsonian effective action is (2.142) The logarithm of the Dedekind eta function admits the expansion in which the first term appears in the dimensional reduction of the eleven-dimensional R 4 type invariant on T 3 whereas the contributions in e 2πinτ are associated to M 2 branes wrapping altogether n times T 3 . This function is not whereb is an integer, and therefore the S (3) correction to the Wilsonian action is not duality invariant. However, the supergravity theory admits a U (1) anomaly in eight dimensions such that the supergravity 1-loop effective action is not SL(2, R) invariant, and neither does it preserve SL(2, Z). Using the family index theorem [50] for the chiral fields χ i α , λ ijk α , G + abcd and ρ ab i α , one computes the anomaly to the axial U (1) current conservation as in [51] Strictly speaking, the fermions contribute to the anomaly for the gauge axial U (1), but one can compensate for it [52] by introducing a correction to the effective action defined in term of the holomorphic function such that the supergravity 1-loop 1PI generating functional transforms with respect to SL(2, R) as It follows that the sum of the 1PI supergravity effective action and the string theory Wilsonian effective action Γ transforms with respect to SL(2, Z) as Therefore the complete effective action is indeed duality invariant in the eight-dimensional Minkowski background. It is a non-trivial consistency check that the same Pontryagin classes combination defining the U (1) anomaly (2.145) also supports the M5 brane gravitational anomaly [53], and it follows that on a general Riemmanian spin manifold whereÂ is the integral roof genus and σ is the signature. If one were to consider gravitational instanton corrections, SL(2, Z) invariance would require the effective action Γ to be invariant modulo 2π, and therefore the corresponding geometry to admit a signature multiple of four. This potential Z 4 obstruction is identical to the tadpole cancelation requirement studied on Calabi-Yau 4-folds in [54]. Note that the real part of the anomalous variation is the variation of a local functional because log Im The log of the dilaton is responsible for the inhomogeneous term in the Laplace equation Similarly, the regularised SL(3, R) Eisenstein functionÊ [ 3 2 0] includes a logarithm term (2.103) that cannot be part of the Wilsonian effective action by supersymmetry. To understand this, let us define the BRST-like nilpotent operator defining the sl 3 action where C J I is a constant anticommuting traceless matrix. The non-trivial consistent anomaly for the sl 3 Ward identities are in one to one correspondence with the su(2) anomalies in the bosonic theory [38]. Therefore there is no anomaly for the rigid SL(3, R) in the theory independently of supersymmetry. However, one must take care that a potential naively trivial anomaly can be removed by a local counter-term without violating supersymmetry Ward identities themselves. Consider for example the variation of the logarithm function By construction it satisfies to equation (2.101), and therefore one can define the supersymmetry invariant which satisfies by construction to the Wess-Zumino consistency condition However it cannot be eliminated by adding a supersymmetric counter-term because the logarithm function itself does not satisfy to (2.101). In this case one cannot compute the coefficient of the anomaly using the family index theorem because it is not related to a chirality anomaly, and one would need in fact to compute the soft limit of the 1-loop six point amplitude to compute the explicit coefficient. Nonetheless it is a consistent correction, and the string theory computation [4] indicates that it indeed appears. The appearance of these two anomalies is directly related to the appearance of a logarithm singularity in the four-point scattering amplitudes at 1-loop [55]. The relation between the logarithm of the dilaton and the logarithmic divergence is explained in string theory [36]. Rather naively, one can understand this property in field theory by noting that supersymmetry determines the power of the dilaton multiplying the R 4 type invariant counter-term in function of the dimension. Assuming the existence of some kind of supersymmetric regularisation valid at 1-loop order, one would naturally get an invariant counter-term in such that the finite term in ǫ would define the anomaly [37].

N = (2, 2) supergravity in six dimensions
In six dimensions, the Lorentz group is SU * (4) and the internal symmetry of maximal supergravity is Sp(2)× Sp (2). The scalar fields parametrize a symmetric space SO(5, 5)/(SO(5)× SO (5)) through SO(5, 5) matrices V ij I , Vî I satisfying to that are antisymmetric symplectic traceless in the pairs of Sp(2) indices ij andî, and I = 1, 10 is in the vector representation of SO(5, 5), such that η IJ is the SO(5, 5) metric and Ω ik Ω jl = δ j i is the Sp(2) symplectic matrix, and respectively is Ωî for the second Sp (2). Recall that the gamma matrices in five dimensions are such that both the conjugation charge matrix Ω ij and the gamma matrices are antisymmetric. They define the momenta and the sp(2) ⊕ sp(2) connexion through the coset decomposition of the Maurer-Cartan form (2), i.e. antisymmetric symplectic traceless in both pairs of indices, such that for any Sp(2) × Sp(2) tensor function of φ µ . The Dirac fermion fields are χ ik α andχ α ijk that are also symplectic traceless in the [ Here we write χ andχ for convenience, but recall that they are both symplectic Majorana-Weyl and not complex conjugate. The only non-vanishing dimension zero torsion components are where α = 1 to 4 is in the fundamental of SU * (4) and σ a αβ = 1 2 ε αβγδ σ a γδ . One computes that the non-zero dimension 1/2 components of the torsion are (3.6) We refer to [56,57] for the complete set of fields of the theory.

The R 4 type invariant
Let us recall in a first step the structure of the linearised R 4 type invariants. The relevant harmonic variables parametrize Sp(2)/U (2) with the split 4 ∼ = 2 (−1) ⊕ 2 (1) . We define u r i , u r i such that The linearised superfield L ijî satisfies to is then G-analytic, i.e. urîDî α W = 0 , u r iD αi W = 0 . (3.10) One can then define linearised invariants of the form is the 2n order monomial in the harmonic variables in the corresponding [0, n] representation of Sp(2), i.e.
and respectively is F However the corresponding measure does not exist at the non-linear level, and the G-analicity condition (3.10) admits obstructions, e.g. 14) The structure of the linearised invariant nonetheless suggests that the non-linear invariant admits an expansion in the derivatives of a It will be more convenient in the following to write the derivative D ijî in terms of vector indices of SO(5) × SO(5), i.e.
such that D ab Take care that we use the same letter a for the internal SO(5) vector representation, as for the Lorentz vector representation. There should be no confusion however, because we shall now on only use a as an SO(5) vector index. More explicitly, (3.18) read Altogether with the similar equation obtained using D αî L (6,0,0) instead, i.e.
The first equation implies that D 2 10 E = 1 10 1 10 ∆E in the vector representation, with the normalisation ∆ = 2D ab D ab . Using the spinor representation where A n are constants that remain to be determined and the first term is understood to be symmetric traceless in both sets of indices, i.e.
Using the Maurer-Cartan equation consistently with [33]. For completeness we give the equations satisfied by E in Sp(2) × Sp (2) representations but it will be more convenient in the following to write them as By construction (3.31) defines a representation of so (5,5), which corresponds to the unitary representation of SO(5, 5) on the set of square integrable functions satisfying to (3.35). This turns out to be the minimal unitary representation of SO(5, 5) as we are going to exhibit in the next section.

Minimal unitary representation
Let us solve these differential equations in the parabolic gauge associated to the decompactification limit. In this case one considers the decomposition The representative in the vector representation can be written (3.37) Here both a and I run from 1 to 5, and correspond respectively to SO(5) and SL(5) indices. We shall not consider a specific gauge for the SL(5)/SO (5) representative v a I . The associated momentum is (3.38) The metric on the symmetric space is where M IJ = v a I v a J and the coordinates on the symmetric space SL(5)/SO(5) are defined The corresponding differential operator is (3.41) The repeated action of the covariant derivative on a function, which we write formally as a square even if the left derivative includes a connexion component, reads (3.42) We shall also consider the derivative operator in the spinor representation. The coset representative is then (3.43) The associated momentum is (3.44) The derivative operator reads and acting twice on a function gives Here the notation we use is to exhibit that the corresponding Eisenstein function of SL(5) Assuming the function to be square integrable over SO(5, 5)/(SO(5) × SO(5)), and therefore to admit a convergent behaviour in the large radius limit e −2φ → ∞, the generic square integrable solution takes the form

Relation to BPS intantons
The differential equations (3.35) implies a non-renormalisation theorem such that the instantons that contribute to the R 4 type correction in the effective action are 1/2 BPS. To see this, let us consider a supergravity instanton determined by the scalar fields only. In this case we consider the Euclidean theory for which the SO(5, 5) symmetry requires to consider a non-compact complex real form of the divisor group, i.e. SO(5, 5)/SO(5, C). This real form is suggested in six Euclidean dimensions because there is no self-dual 3-form in Euclidean signature, and the five 3-form field strengths must decompose into complex selfdual and complex-antiselfdual in the complex five dimensional representation of SO(5, C) and is complex conjugate. In this case the instanton can decouple from gravity and the metric is chosen to be flat. The scalar fields then lie in a nilpotent subgroup, which is characterised by the number of preserved supersymmetries. For a 1/2 BPS solution, one splits Sp(4, C) into The fundamental representation in which lies the supersymmetry spinor parameters then decomposes as such that the grad 1/2 components carries the preserved half of supersymmetries. The coset component of SO(5, 5) decomposes accordingly such that The grad 2 component contains a single Lie algebra element that squares to zero in both the vector and the spinor representation. Defining the scalar fields with such a generator, the solution automatically preserves one half of supersymmetry because the Dirac spinors χ,χ do not carry a grad 5/2 component within this decomposition. The associated function is then simply a harmonic function on R 6 . More explicitly, the 1/2 BPS instanton with a charge q IJ satisfying to the condition ε IJKLP q IJ q KL = 0 defines a rank 2 antisymmetric tensor where the zero subscript indicates that this is the asymptotic value of the scalar at infinity. One can normalise it such that This tensor is a non-degenerate symmetric tensor J ij = 1 2 J ab γ ab ij in the spinor representation, that determines the preserved supersymmetry as the ones associated to spinor parameters satisfying to We consider the Euclidean Lagrangian density for which the scalars with negative kinetic terms have been dualised to 4-form potential B IJ , and that reduces to a sum of squares plus a total derivative as follows Cancelling the squares gives the equation One obtains the solution which action is determined by the total derivative term and gives It is therefore legitimate to believe that the next coupling in ∇ 4 R 4 will be a function satisfying to differential equations defining a quantisation of the algebraic equations associated to 1/4 BPS instantons. In so(5, 5), the next to minimal nilpotent orbit is not unique, and there are in fact three disconnected orbits connected to the minimal orbit associated to 1/2 BPS instantons. The two isomorphic smallest orbits are obtained by relaxing the nilpotency condition in the vector representation In this case however the instanton cannot be defined in the standard Euclidean formulation of the theory, and one must consider a real form of the divisor group that allows for an independent decomposition of the two factors. This is incompatible with the representation of the SO(5, 5) symmetry on the 3-form field strengths, and recovering the symmetry would require some analytic continuation of the Euclidean path integral in such a background. One can consider for example the coset SO(5, 5)/(SO(1, 4)×SO(4, 1)) such that only one Sp(1, 1) factor decomposes as In this case the instanton can be described within the scalar fields valued in the Riemannian symmetric space R * + × SO(4, 4)/(SO(4) × SO(4)) coupled to eight 4-forms in the 8 of SO(4, 4). The two orbits correspond to the choice of Sp(1, 1) factor. The coset component then decomposes as and a representative of the nilpotent orbit is a generic (time-like vector) element of the 5 ′ (2) component. 4 The associated solution preserves one half of the chiral (respectively antichiral) supercharges, depending on the choice of Sp(1, 1) factor. Note that in the decomposition of the vector representation, with a of SO(5) ′ andâ of SO(5), the charge satisfies moreover although Q 2 10 = 0. The third orbit is obtained by relaxing the nilpotency condition in the spinor representation In this case one can consider the standard formulation of the Euclidean theory with coset SO(5, 5)/SO(5, C) and the decomposition The fundamental representation in which lies the spinor then decomposes as such that the grad 1 component carries the preserved quarter of supersymmetries. The coset component of SO(5, 5) decomposes accordingly such that and a representative of the nilpotent orbit is a generic (time-like SO(1, 3) vector) element of (2 ⊗ 2) (2) R . The associated instanton preserves one quarter of supersymmetry (one quarter chiral and one quarter antichiral).

The ∇ 4 R 4 type invariants
We shall consider in a first place the linearised ∇ 4 R 4 invariants. There are three 1/4 BPS measures one can define in the linearised approximation [32], although none of them extend to the non-linear level as one straightforwardly checks using (3.6).
The function defining the closed superform (3.82) satisfies therefore to an equation compatible with the function defining the R 4 type invariant, consistently with the expected properties of the effective action in type II string theory [11,33]. Solving this spinor differential equation and its complex conjugate, where the upper complex half plan variable τ A 1 (q) parametrizes the v A 1 (q) component of v a I in the SL(2) subgroup of the stabiliser SL(2) × SL(3) ⋉ R 2×3 ⊂ SL (5) of q IJ . One computes moreover that with J ab defined as in (3.60), which plays the role of a complex structure such that for a holomorphic function and (3.90) is satisfied. The generic solution to these differential equations is therefore supported on a space of eight variables To prove that we note that their sum vanishes for n I m I = 0 whereas their difference is then obtained from the character generating e −6φ E [ 3 2 000] by an infinitesimal duality transformation of parameter q IJ = −q JI such that m I = q IJ n J , i.e.  appearing in the ∇ 4 R 4 coupling [11,33], but we should take care however that the Eisenstein function E 0 s00 0 diverges at s = 5 2 . Note that this function is generated by a specific character, and any covariant differential equation satisfied by the character is also satisfied by the Eisenstein function provided the series converges. Using this property one computes that it satisfies to One can use this property to constrains the Fourier modes of this function. Altogether with the constant terms computed in [11], we conclude that this Eisenstein function admits the expansion for some undetermined measure µ s (q). Using this expression, one recovers the singular limit (q) = n|q IJ n is the same measure as for s = 3 2 . Using the limit we compute that

The parity symmetric invariant
The third class of invariants can be obtained in the linearised approximation using harmonic variables parametrising Sp(2)/(U (1) × Sp (1)) with the split 4 ∼ = 1 (−2) ⊕ 2 (0) ⊕ 1 (2) [32]. We define accordingly u i ,ū i , u r i such that and respectively for the second Sp(2) factor. One can then define the G-analytic superfield Using this superfield one can define the class of invariants = E∇ 4 R 4 + · · · + D 16 [4,12], [4,12] E F 4 [4,0], [4,0]  The form of the linearised invariant therefore strongly suggests that the function E must satisfy to equation (3.111). In principle one could check this explicitly on the terms multiplying D 16 [4,12], [4,12] E, but this computation is rather involved and we shall not carry it out in this paper. Assuming (3.111), it remains to determine the eigenvalue of the Laplacian to fix the differential equation satisfied by the function. Note however that the 1/4 BPS condition discussed in the preceding section also requires Q 3 16 = 0, and considering the expansion (3.112) requires also that on a test function assumed to satisfied to these constraints, and the only solution is We checked this equation explicitly on a generating character of these Eisenstein functions.
Note that this series of functions do not satisfy to any quadratic differential equation in the 10 of SL (5)  according to [33]. However the M-theory limit also corresponds to the same decomposition of SO (5,5), with the opposite chirality, and the Eisenstein functions E [− 1 2 000] andÊ [0200] solving the same differential equation as E [00 1 2 0] do appear in this limit [33]. Let us now consider the Fourier modes. Note that the condition ε IJKLP q IJ q KL = 0 was coming from the quadratic equation in the spinor representation, and therefore does not hold in this case. It is therefore convenient to define the two functions The off-diagonal equation requires that the Fourier modes only depends on the SL(5)/SO(5) scalars through the central charge Z ab (q). Using these variables, one can rewrite the remaining differential equation as This provides three independent second order equations. One finds the solution Note that for a 1/2 BPS charge one has is the action associated to a 1/4 BPS instanton. Considering the central charge in the spinor representation 1 2 Z ab γ ab i k , the eigenvalues are and the BPS bound is defined by the largest. In fact they all define solutions to the equation (3.121), but only (3.122) is square integrable because the others exhibit exponential growth in the asymptotic. The generic square integrable solution of this form is therefore supported on a set of functions depending on ten variables 126) corresponding the other next to minimal unitary representation of SO (5,5).
We should also consider the sum over 1/2 BPS instantons. But because the solution is then singular by property of the function, one must rather consider the solution for a generic s. Because this class of Eisenstein function is associated to the decomposition of SO(5, 5) we use, the generating character of the function E 0 000 s restricted to the Cartan subgroup is simply e −10sφ . One straightforwardly checks that the latter function therefore satisfies to the equation of q IJ , which we shall write v A 2 (q) . For a 1/2 BPS charge q IJ one finds the solution Indeed In particular, we conclude that the 1/2 BPS instanton contributions to the ∇ 4 R 4 coupling in string theory combine into 1 2Ê It is rather striking that this combination ofÊ [1] andÊ [ 3 2 0] is precisely the one that defines the R 4 coupling in eight dimensions [11], for which the respective 1 ǫ poles cancel out.

The non-analytic terms
Similarly asÊ A ∇ 4 R 4 invariant does not have the right dimension to appear as a counterterm for logarithmic divergences in supergravity, and the non-analytic component of the effective action responsible for these corrections to the differential equations satisfied by the threshold functions must also include massive states contributions. From the supergravity perspective, this comes from the property that ∇ 4 R 4 has the correct dimension to be a counterterm for the 1-loop divergence of an R 4 invariant operator defined as an insertion. If we consider the low energy expansion of the effective action, the leading non-analytic components will match the supergravity effective action, but the next order correction will include the insertion of the exact R 4 string theory coupling. Schematically, the amplitude is determined by the supergravity path integral of the string theory Wilsonian effective S such that the corresponding Legendre transform decomposes as If one considers the perturbative string theory contribution as depicted in [33], one finds indeed a logarithm correction of the form 1 2Ê where the overall e −3φs corresponds to the Weyl rescaling to Einstein frame. According to the analysis displayed in [36], one understands that this logarithm of the dilaton comes from a logarithm of the Mandelstam variable s in the effective action. We see therefore that the tree-level and one-loop corrections to the R 4 coupling in string theory contribute respectively to a one-loop and a 2-loop correction to ln(s)s 2 R 4 in the effective action. In supergravity, this implies that the local operator L[E The consistency of this argument requires that the anomalous term in E . One also checks that the corresponding linearised invariants coincide because they have the same 1/2 BPS harmonic integral form (3.138)

N = 8 supergravity in four dimensions
We will now discuss the case of N = 8 supergravity in four dimensions [45,59]. The R-symmetry group is then SU (8) and the Lorentz group SL(2, C). In this section i = 1 to 8 is an SU (8) index. The same construction permits to determine the properties of the function defining the R 4 type invariant, and we will propose a conjecture for the equations satisfied by the functions defining the ∇ 4 R 4 and ∇ 6 R 4 type invariants.

The R 4 type invariant
One can define the linearised R 4 type invariants in the linearised approximation by using harmonic variables in SU (8)/S(U (4) × U (4)) as in [60]. One obtains that the scalar superfield is G-analytic with respect to (with r = 1 to 4 andr = 5 to 8) such that As in the preceding section, we will concentrate on the term with the maximal number of derivative carrying the highest weight SU (8) representation. Using representation theory and power counting, one obtains that the maximal weight term can only be the monomial in χ 8χ8 because one needs 48 open indices to get this representation. To show that this monomial exists and is unique, one can use the harmonic projection where the prime states that the [1, 0, 0, 5, 0, 1, 0] is not necessarily the same, because there exists two such combinations of χ 9 . Therefore one should also consider terms like could not mix with the terms we have been considering. Moreover the second can be eliminated by the addition of a total derivative, up to the addition of lower derivative terms in D 12 E. We conclude that there is nothing that can compensate for the first term in (4.8), and the function E must therefore satisfy to the equation (4.14) The derivative operator D 11 [0,0,0,11,0,0,0] includes all components (D ijkl ) 11 (without summation over the indices), and its kernel is the constant tensor. We conclude that the function E must satisfy to the quadratic equation one obtains the equality between the two quartic invariants Using this property one can conclude that E satisfies to The same argument as for SO (5,5) in the preceding section would permit to show that the only consistent solution satisfies to ∆E = −42 E, consistently with the analysis of [33]. Using the explicit form of the differential equation of the next section, one computes indeed that there is no non-trivial solution to (4.16) satisfying to the Laplace equation ∆E = 0. We conclude that E satisfies to 1 24

Minimal unitary representation
It is convenient to analyse Equation (4.20) considering an explicit coset representative in E 7(7) /SU c (8) in the parabolic gauge 0 000001 relevant to the decompactification limit. In this case we have where V ij I is a representative of E 6(6) /Sp c (4) in the fundamental representation, and t IJK is the invariant symmetric tensor of E 6(6) . The decomposition dV V −1 = P + B (4.22) in coset and subgroup components gives where all the antisymmetrisations are understood to be projected to the symplectic traceless component and we raise and lower Sp(4) indices as The metric on the coset space E 7(7) /SU c (8) is defined as and its inverse g −1 = 1 12 (4.28) Accordingly, we have on the symmetric space E 6(6) /Sp c (4). The reader should take care that we use the same notation for the differential operator D ijkl , that is associated to the 42 variables of E 6(6) /Sp c (4) in this subsection, whereas it was used for the 70 variables of E 7(7) /SU c (8) in the preceding one. The inverse vielbein on E 7(7) /SU c (8) are defined as We compute the different components of the differential equation D 2 E = − 9 2 1E to give The differential operator D clearly commutes with ∂ I , such that we can decompose the solution into Fourier modes e iq I a I . Let us consider in a first place the zero modes q I = 0. In this case equation (4.31) implies that By representation theory, the term in D ij kl in equation (4.34) cannot mix with the others, such that the function E ′ 5 (V ) must be a constant. One finds that the function e −12φ is indeed a solution to the complete differential equation D 2 E = − 9 2 1E. In order to define a solution, the other function E 5 (V ) must satisfy to the equation which is nothing but the supersymmetry constraint of the five dimensional R 4 threshold. Taking its trace, one obtain indeed the Poisson equation [33] ∆ E 6(6) Let us consider now the non-trivial Fourier modes. Equation (4.33) implies that which is the expected equation for a 1 2 -BPS scalar instanton. Equation (4.35) is very constraining, and implies that E q (φ, V ) only dependent on the E 6(6) /Sp c (4) coordinates through the invariant mass of the charge q I . So we define such that E q (φ, V ) = E q (φ, |Z(q)|). Because q I is a rank one vector [61], Equation (4.32) determines the dependence in |Z(q)| in terms of the one in φ, such that one obtains an ordinary differential equation. There are two solutions to this system (4.42) To check consistency, we use to compute that for a function E q (φ, |Z| 2 ) and The generic square integrable solution is therefore supported by a function of seventeen variables F (q), Spin(5,5)⋉R 16 (4.46) where the additional function E 5 [G] is a generic solution to (4.37) supported by a function G of eleven variables. The representation of E 7(7) on this space of functions is its minimal unitary representation.
We conclude that supersymmetry on its own already constrains the function E to have the expected structure for the string theory effective action, and using the explicit coefficients computes in [33] one gets the form of the Eisenstein series  µ(q) e −6φ |Z(q)| 3 1 + 2πe −2φ |Z(q)| e −2πe −2φ |Z(q)|+2πiq I a I .
The Fourier modes coincide with the analysis of [28,35].

∇ 4 R 4 and ∇ 6 R 4 type invariants
In the linearised approximation, the ∇ 4 R 4 type invariant can be obtained from a harmonic superspace integral based on SU (8)/S(U (2) × U (4) × U (2)) harmonic variables [60], and the G-analytic superfield with r = 3 to 6 of SU (4). W rs is therefore an SO(6) vector, one the most general integrand is a monomial in a symmetric traceless tensor of SO(6) suggesting that the non-linear invariant admits an expansion Consistently with this structure, the function E must satisfy to the constraints The two first define a condition on the differential operator to the third power in the fundamental of E 7(7) , whereas the last corresponds to a constraints on the differential operator to the third power in the adjoint representation. Indeed, the harmonic decomposition also defines the graded decomposition of e 7(7) associated to the next to minimal nilpotent orbit, for which the Lie algebra representative satisfies Q 3 56 = 0 and Q 3 133 = 0. It turns out that the eigenvalue of the Laplace operator is determined by these equations by consistency. Indeed, assuming that E satisfy to the equations  The first solution corresponds to the constraint satisfied by the R 4 threshold, and we conclude that the second solution is the relevant one for the ∇ 4 R 4 threshold, consistently with [33]. So ] to be closed. The ∇ 6 R 4 type invariant can be defined from a harmonic superspace integral based on SU (8)/S(U (1) × U (6) × U (1)) harmonic variables [60], and the G-analytic superfield with r = 2 to 7 of SU (6). In this case the measure extends to the complete theory [62]. The number of possible representations of SU (8) becomes rather large, but they are still self-adjoint by construction. It follows that the constraints still apply, although the second one is not satisfied. Using the closure diagram of E 7(7) [63], one finds that there is not a unique next to next to minimal nilpotent orbit. However the condition  The nilpotent orbit associated to the harmonic decomposition is in fact not the next one of dimension 64 that would also satisfy to Q 4 133 = 0, but the following one of dimension 66. Using harmonic superspace, one finds indeed a nonvanishing integral in the representation [2, 0, 0, 0, 0, 0, 2] by integrating the square of the quartic SU (6) invariant monomial in W rst with the appropriate function of the harmonic variables. Therefore the superform expansion must include terms as [2,0,0,0,0,0,2] E (0,1) L [2,0,0,0,0,0,2] + . . . (4.59) and the corresponding component of D 4 133 E (0,1) acting on the su(8) adjoint does not vanish. The determination of the eigenvalue of the Laplace operator does not follow straightforwardly from a group theory argument in that case, and one must moreover consider the corrections to the supersymmetry transformations at this order. Nonetheless, relying on the known Poisson equation satisfied by the function according to [33], we find that the function must moreover satisfy to (4.61) Let us now analyse these equations in the parabolic gauge as in the preceding section. We shall only analyse the solution for q I = 0, and for the homogenous equation in the fundamental representation. After some computations one obtains where we recall that D ijkl states for the covariant derivative on E 6(6) /Sp c (4) in these expressions. One finds indeed that the decompactification limit of the corresponding Eisenstein series [33] E 0 . (4.64) The latter equation must therefore define the differential equation satisfied by the function defining the ∇ 4 R 4 type invariant in five dimensions, and is indeed consistent with the associated Poisson equation [33]. For q I = 0, one computes straightforwardly that the equations D 3 56 E = −9D 56 E implies moreover t IJK q I q J q K = 0 . For E 0 5 2 00000 this is consistent with the property that the next to minimal unitary representation is defined on functions of 26 variables. Note that the sum of two vectors satisfying to t IJK q J q K = 0 necessarily satisfies (4.65), such that the complete function E (0,1) is supported on Fourier modes satisfying to this same constraint (4.65). The unitary representation on which E (0,1) is supported is however defined on functions of 33 variables, therefore the Fourier modes must depend on a non-trivial function of the scalar fields v B 4 (q) parametrizing the subgroup Spin(4, 5) ⊂ E 6(6) stabilizing q I [61]. Because 33 − 26 = 7 we expect the function E(v B 4 (q) ) to satisfy a differential equation restring effectively its dependence on seven variables. This suggests that the relevant function on SO(4, 5)/(SO(4) × SO (5)) should satisfy to the following differential equation associated to a coadjoint SO(4, 5) orbit of dimension 14, i.e.
We note moreover that the solution to (4.56) is also a solution to the homogeneous equation associated to (4.60), therefore the restriction of the Fourier mode function to the case in which the function on SO(4, 5) is a constant must also be solution. We conclude that the correct value of s must be s = 7 2 . This is precisely the value for which the Eisenstein series diverges in 5 N = 16 supergravity in three dimensions.
In three dimensions the only propagating degrees of freedom are the scalar fields parametrizing the symmetric space E 8(8) /Spin c (16) [64], such that the Maurer-Cartan form defines the scalar momentum P A in the Majorana-Weyl representation of the R-symmetry group Spin (16), whereas the fermion fields χ αȦ are defined in the opposite chirality Majorana-Weyl representation. Solving the superspace constraints [65] the momentum decomposes as The metric on the symmetric space is defined as  The superconformal symmetry OSp(16|4, R) of the linearised theory [67] suggests that all the supersymmetry invariants are defined by harmonic superspace integrals in the linearised approximation, such that the harmonic superspace integrals are indeed in bijective correspondence with the independent non-linear invariants. One confirms this property by looking at the monomial in the fermions of maximal weight. Using the harmonic decomposition, one gets directly that the 2 × 8 fermions χ r α to the sixteenth power carries a U (1) weight 48, just as does W 12 . Considering the action of the covariant derivative D i α , one cannot include one more χ r α , so the only non-trivial term appears to include instead a weight 1 fermion χ rst α . Projecting out the corresponding representations in D i α L[E] (3,0) using the harmonic variables, one gets where the two terms in the first line contribute to two independent terms in the second ∼ D (0) rstu (D (−4) ) 12 and ∼ D (−2) [rs D (−2) tu] (D (−4) ) 11 , such that they cannot compensate each other. To deduce the Spin(16) covariant expressions associated to these terms, we note that the rank p antisymmetric tensor representation of SO(16) admits as a highest weight component of weight p the rank p antisymmetric tensor in the anti-fundamental of SU (8). We conclude that χ stu(1) (χ 16 ) (48) is in the highest weight component of the There is no other contribution that could cancel this term, because the next terms of maximal weight are in the 0 000100 10 E and carry a maximal weight component in D 12 (−44) rstu E whereas and they cannot contribute to terms in D 13 (−48) rstu E. We conclude similarly as in the preceding sections that the function E must satisfy to the differential equation Using the property that the D n differential operator of maximal weight in the 0 000000 n has no kernel, one obtains that the function E must satisfy to the quadratic equation Using SO(16) Fierz identities (5.14) and the commutation relation (5.4), one computes that such that if E is canceled by the Laplacian, it must necessarily be a constant, and supersymmetry indeed implies ∆E = −120E , (5.18) consistently with [33].
Using these equations, one computes that the covariant derivative in the adjoint representation This equation defines a quantization of the algebraic equation for a Majorana-Weyl pure spinor of Spin * (16), which is a representative of the minimal nilpotent orbit of E 8(8) [68]. The square integrable solutions to the differential equation (5.13) with define the minimal unitary representation of E 8 (8) , and are supported on functions depending on 29 variables as explained in [25,26].

The ∇ 4 R 4 type invariant
The (∇ 2 P ) 4 type invariant can be defined in harmonic superspace [66] in the linearised approximation using harmonic variables parametrizing SO(16)/(SO(8) × U (4)) such that the Majorana-Weyl representations decomposes as  Assuming that all (∇ 2 P ) 4 type invariants are defined in this way, this exhibits that the function must have covariant derivatives restricted to these representations. This is the case if and only if he function E satisfies to the cubic equation (5.16). Moreover, acting with one more derivative on this equation one obtains using the Fierz rearrangements such that the function must then either satisfy to the quadratic equation (5.13) or to The two equations being incompatible, supersymmetry requires that the function defining the (∇ 2 P ) 4 type invariant satisfies to (5.27), consistently with [33]. Using the latter, (5.16) simplifies to which defines the quantisation of the algebraic equation Q 248 3 = 0 defining the next to minimal nilpotent orbit of E 8(8) [68]. We conclude that the square integrable solutions to (5.28) with appropriate boundary conditions define the next to minimal unitary representation of E 8 (8) associated to the next to minimal coadjoint orbit.

Acknowledgment
We thank Eric D'Hoker, Paul Howe, Ruben Minasian, Daniel Persson, Boris Pioline, K.S. Stelle and Pierre Vanhove for relevant discussions. This work was supported by the French ANR contract 05-BLAN-NT09-573739 and the ERC Advanced Grant no. 226371.

A Conventions in eight dimensions
The SU (2) invariant tensors ε ij and ε ij are defined respectively such that One raises and lower the SU (2) indices according to the rules The conventions for the SO(1, 7) invariant tensors are pletely antisymmetric tensor with the local metric are taken to be: and we define the antisymmetric Kronecker delta tensors We decompose the spinor representation into the Weyl representation of positive chirality with undotted indices and negative chirality with dotted indices, which are complex conjugate. We use the octonionic representation such that the charge conjugation matrix is the identity, and we have the following relations

B Dimension 1 solution to the superspace Bianchi identities
In this appendix we give the dimension 1 Bianchi identities of N = 2 supergravity in eight dimensions and solve them. The result are ordered in function of the U (1) weight.

B.1 Dimension 1 Bianchi identities
The components of d ω T = R of dimension 1 and U (1) weight 4, 3 and 0 are The Bianchi identity for the 2-form field strengthF decomposes in components of U (1) weight 4, 2 and 0 as

B.2 Dimension 1 solution
The only component of U (1) weight 4 is the covariant derivative of the fermion fieldχ From weight 2 and above, there are more components, and for convenience we will define the following basis of bilinear in the fermions in irreducible representations of SU (2) × Spin(1, 7) (λλ) ≡ λ ijk λ ijk , (B.10) The corresponding torsion component is  In our notations, the fieldF ab and H abc coincide with the corresponding (respectively (2, 0, 0) and (3, 0, 0)) components of their associated superforms, whereas the (4, 0, 0) component of the 4-form superform decomposes into a complex selfdual partḠ abcd and a complex antiselfdual part bilinear in the fermions, i.e.
We now consider the U (1) invariant components, with the following basis of bilinear in the fermions in irreducible representations of SU (2) × Spin(1, 7) The corresponding component of the torsion is

C Dimension 3/2 solution to the superspace Bianchi identities
In the core of the paper we use the dimension 1/2 covariant derivative of the dimension 1 fields and the equation of motion of the fermion fieldχ, which we derive from the dimension 3/2 Bianchi identities and the algebra of the covariant derivatives in this appendix. We do not derive the expression of the dimension 3/2 Riemann curvature that we do not need in this paper.

C.1 Dimension 3/2 Bianchi identities
The components of dimension 3/2 of d ωP = 0 of respective U (1) weight 5 and 3 are The Bianchi identity for the 2-form field strengthF gives at this dimension the following equations of U (1) weight 3 and 1 We will also make use of the following commutation relations between the covariant derivatives acting of the fermions, ordered with respect to their U (1) weight from 5 to 1

C.2 Dimension 3/2 solution
The number of linearly independent dimension 3/2 monomials in the fields is rather large, and we find it convenient to define the following basis in irreducible representations of SU (2), and filtrated with respect to Spin(1, 7) irreducible representations, such that the larger representations are not irreducible. It is indeed convenient to keep the gamma traces rather than to remove them systematically.  where we use again (B.10).
Within this basis, one computes the Dirac equation for the fermion fieldχ, solving the Bianchi identities displayed in section C.