The two D6R4 type invariants and their higher order generalisation

We show that there are two distinct classes of D6R4 type supersymmetry invariants in maximal supergravity. The second class includes a coupling in F2D4R4 that generalises to 1/8 BPS protected F2kD4R4 couplings. We work out the supersymmetry constraints on the corresponding threshold functions, and argue that the functions in the second class satisfy to homogeneous differential equations for arbitrary k>0, such that the corresponding exact threshold functions in type II string theory should be proportional to Eisenstein series, which we identify. This analysis explains in particular that the exact D6R4 threshold function is the sum of an Eisenstein function and a solution to an inhomogeneous Poisson equation in string theory.


Introduction
The determination of the exact string theory low energy effective action is a very difficult problem in general. In the case of type II string theory on R 1,10−d × T d−1 , the lowest order nonperturbative corrections could nonetheless have been computed [1,2,3]. Although there is no non-perturbative formulation of the theory, the constraints following from supersymmetry and U -duality have permitted to determine the non-perturbative low energy effective action from perturbative computations in string theory [4,5,6,7,8] and in eleven-dimensional supergravity [2,9,10,11,12]. The four-graviton amplitude allows in particular to determine the ∇ 2k R 4 type correction in the effective action, where the dots stand for other terms including the supersymmetric completion, (p, q) labels the different invariant combinations of derivatives compatible with supersymmetry according to the notations used in [13], and E (p,q) are automorphic functions of the scalar fields defined on E d(d) (Z)\E d(d) /K d . For (p, q) = (0, 0), (1, 0) and (0, 1), the complete effective action at this order is determined by these functions E (p,q) , which have been extensively studied [14,15,16,17,18,19,20,21,22,23]. E (0,0) is an Eisenstein series associated to the minimal unitary representation [15,17], E (1,0) is an (or a sum of two) Eisenstein series associated to the next to minimal unitary representation(s) [17], and both are therefore relatively well understood. They are nonetheless very complicated functions, and the explicit expansion of E (1,0) in Fourier modes is not yet determined [18,19,22]. E (0,1) is not even an Eisenstein series, and was shown in [10] to satisfy to an inhomogeneous Poisson equation in type IIB. A proposal for this function in eight dimensions [16], suggested a split of the function into the sum of an Eisenstein series and an inhomogeneous solution, which was subsequently generalised in seven and six dimensions [13,17], and recently clarified in [23].
In this paper we extend the analysis carried out in [20,21] to the study of E (0,1) . We show that this function indeed splits into the sum of two functions that are associated to two distinct supersymmetry invariants, and therefore satisfy to inequivalent tensorial differential equations. In particular, the second satisfies to a homogeneous equation, which is solved by the Eisenstein function appearing in [13,16,23]. One can distinguish the two functions by looking at specific higher point couplings that we identify. The new class of invariants generalises to an infinite class admitting a coupling in F 2k ∇ 4 R 4 , and we identify a unique Eisenstein function solving the corresponding tensorial differential equations in all dimensions greater than four. This function turns out to be compatible with perturbative string theory, and only admits three perturbative contributions in four dimensions, at 1-loop, (k + 2)-loop, and 2k-loop. However, the only amplitude that seems to unambiguously distinguish it from others is the (k + 2)-loop four-graviton amplitude in a non-trivial Ramond-Ramond background, which makes an explicit check extremely challenging.
We start with the analysis of the supersymmetry invariants in four dimensions. The two ∇ 6 R 4 type invariants in the linear approximation are associated to two distinct classes of chiral primary operators of SU (2, 2|8) discussed in [24]. We identify the corresponding representations of E 7(7) associated to nilpotent coadjoint orbits [25] that are summarised in figure 3. In the linearised approximation, the F 2 ∇ 4 R 4 type invariant does not carry a ∇ 6 R 4 coupling, but we explain that the structure of the linearised invariant allows for this mixing at the non-linear level, and that the latter must occur because the two classes of invariants merge in one single E 8(8) representation in three dimensions. We conclude that the exact threshold function in four whereÊ (8,1,1) is the solution to the inhomogeneous differential equation (2.143) that is consistent with perturbative string theory. The explicit relation between the tensorial differential equations and the associated nilpotent orbits permits us to determine the wavefront set of the associated functions, extending the results of [18,19] to the ∇ 6 R 4 threshold function. It appears, as can be seen in figure 3, that the two functions admit distinct wavefront sets. In particular we show that althoughÊ (8,1,1) is not an Eisenstein series, it admits the same wavefront set asÊ 0 00000 . We then consider the uplift of our results in higher dimensions, and exhibit that this general structure extends to all dimensions lower than eight, and is in perfect agreement with the exact threshold functions proposed in [13,16,23]. In each dimension, the supersymmetry invariants transform in irreducible representations of E d(d) , defined by the representation of E d(d) on the associated function on E d(d) /K d satisfying to the relevant differential equations implied by supersymmetry. The inequivalent invariants are summarised in figure 2. The tensorial differential equations satisfied by Eisenstein functions relevant to our analysis are reviewed in the appendices.

N = 8 supergravity in four dimensions
Maximal supergravity includes 70 scalar fields parametrising the symmetric space E 7(7) /SU c (8) [26], and can be defined in superspace by promoting these fields to superfields φ µ [27,28]. One defines the Maurer-Cartan form with P ijkl = 1 24 ε ijklpqrs P pqrs . (2. 2) The metric on E 7(7) /SU c (8) is defined as and the derivative in tangent frame is defined such that for any function The superfields satisfy to where χ αijk is the Dirac superfield in Weyl components, andχ ijk α its complex conjugate. The expansion of the scalar fields include the 28 Maxwell field strengths F αβij , the 8 Rarita-Schwinger field strengths ρ αβγi and the Weyl tensor C αβγδ , satisfying to N = 8 supergravity classical (two derivatives) field equations. The supervielbeins are the solutions to the Bianchi identities defined such that the Riemann tensor is valued in sl(2, C) ⊕ su(8) and the su(8) component is identified with the scalar field curvature [27,28], The covariant derivative on E 7(7) /SU c (8) in tangent frame satisfies to and the Laplace operator is In the linearised approximation, the scalar superfield W ijkl satisfies to the reality constraint (2.2) and to In this approximation the superfield W ijkl transforms in the minimal unitary representation of the superconformal group SU (2, 2|8) [29]. This property permits a complete classification of supersymmetry invariants in the linearised approximation in terms of irreducible representations of SU (2, 2|8) of Lorentz invariant top component [24,30]. In our analysis, we rely on the assumption of absence of supersymmetry anomaly, such that there is no algebraic obstruction to the extension of a linearised invariant to a full non-linear invariant. This implies a bijective correspondence between the set of linearised invariants and the non-linear invariants, such that one can deduce the explicit gradient expansion of the functions (or tensor functions) of the scalar fields on E 7(7) /SU c (8) that determine the invariants.

The standard ∇ 6 R 4 type invariant
One can define a ∇ 6 R 4 type invariant in harmonic superspace, using the harmonic variables , such that r = 2 to 7 of SU (6) [24,30,33]. In this case the harmonic superspace integral can be defined at the non-linear level [34], but we will only consider its linearised approximation. The superfield in the 20 of SU (6) satisfies to the G-analyticity constraints that decompose into the [0, 1, 0, 1, 0] and the singlet representation. One checks that these elements freely generate the general polynomials in W rst , such that the latter are labeled by five integers.
To integrate such a function in analytic superspace, one needs to consider these generating monomials with additional harmonic variables in order to compensate for the S(U (1) × U (6) × U (1)) representation, i.e.
One obtains in this way that the harmonic superspace integral of a general monomial of order n 1 + 2n 2 + 3n 3 + 4n 4 + 4n ′ 4 + 4 in the [n 2 , n 4 , n 1 + n 3 , n 4 , n 2 ] of SU (6) gives rise to a term where the function F (u) is the function of the harmonic variable defined as a product of the generating functions defined in (2.15,2.16). One needs at least one quartic singlet in the Ganalytic superfield to get a non-vanishing integral [24].
Referring to the one to one correspondence between linearised and non-linear invariants [24], one deduces that the non-linear invariant must admit the same gradient expansion, i.e.
This gradient expansion implies in particular that the third order derivative of E (8,1,1) in the [0, 2, 0, 0, 0, 0, 0] and its complex conjugate must vanish, i.e. (2.20) These equations imply all the higher order constraints on the function such that its gradient expansion is in agreement with (2.18). Defining the covariant derivative in tangent frame as a Lie algebra generator in the fundamental representation of E 7 (7) , this equation reads equivalently   [25], which defines the graded decomposition of SU (8) associated to the (8, 1, 1) harmonic superspace we consider in this section. The property that the linearised structure does not permit to determine the eigenvalue of the Laplace operator in this case, implies that the quantization of the associated nilpotent orbit is not unique, and depends on one free parameter. This property follows from the fact that a nilpotent element of this kind can be obtained as the appropriate limit of a semi-simple element satisfying to the characteristic equation Q 3 56 = 1 24 tr(Q 2 56 )Q 56 .
2.2 F 2 ∇ 4 R 4 type invariant and its relation to ∇ 6 R 4 Although the ∇ 6 R 4 type invariant provides the unique supesymmetric invariant preserving SU (8) one can write at this order, there is another class of invariants that can be defined form the chiral harmonic superspace defined in terms of the harmonic variables ur i , u r i parametrising SU (8)/S(U (2) × U (6)) [24,33], withr,ŝ equal to 1, 2 of SU (2), and r, s running from 3 to 8 of SU (6). One defines the superfield that satisfies to the G-analiticity constraint Similarly as in the preceding section, the most general function of W rs is freely generated by the three monomials One must supplement them with harmonic variables to preserve S(U (2) × U (6)) invariance, using One only gets a non-trivial integral if the cubic SU (6) singlet in W rs appears at least quadratically, which can be understood from the property that the associated chiral primary operator of SU (2, 2|8) is otherwise in a short representation [24]. Because the U (1) weight of the measure is compensated by a single factor of this cubic SU (6) singlet, it appears that there is no SU (8) invariant that exists in this class. For a general monomial, one gets an invariant of the form [0,n 2 +2n 3 +2,0,n 1 ,0,n 2 ,0] W n 1 +2n 2 +3n 3 +6 | [0,n 1 ,0,n 2 ,0] (2.28) [0,n 2 +2n 3 ,0,n 1 ,0,n 2 ,0]F 2 [0,2,0,0,0,0,0] ∇ 4 R 4 + · · · + W n 1 +2n 2 +n 3 −22 where the densities L [0,n 1 ,0,n 2 ,0](n 1 +2n 2 +3n 3 +3) (8,2,0) lin are of order n 1 + 2n 2 + 3n 3 + 6 in the fields and only depend on the scalar fields through their space-time derivative. The number n 1 +2n 2 +3n 3 +3 is the U (1) weight of the density. These densities determine by construction covariant superforms in the linearised approximation [35,36,37], such that d (0) L ij,kl (8,2,0) lin = 0 , d (0) L ij,kl,pqrs (8,2,0) lin + 3P pqrs ∧ L ij,kl We assume in this paper that the structure of superconformal multiplets implies the absence of supersymmetry anomaly, or equivalently that the fifth cohomology of d (0) is empty. Nevertheless, even if d (1) L ij,kl (8,2,0) lin only depends on the covariant superfields, nothing prevents its d (0) antecedent to depend explicitly on the scalar fields. This implies in this case that is the covariant correction to the superform, whereas M ij,kl,pqrs and M ij kl are superforms of order six in the fields in the [0, 2, 0, 1, 0, 0, 0] and the [0, 1, 0, 0, 0, 1, 0], respectively, that must satisfy to In order to have such corrections that could not be reabsorbed in a covariant correction as L ij,kl (8,2,0) (1) , one must have a corresponding short multiplet associated to a linearised invariant of the same dimension. The only candidate for a superform M ij,kl,pqrs is L ij,kl,pqrs (8,2,0) lin , but it is of order seven in the fields, and therefore M ij,kl,pqrs = 0 at this order. However, there is a candidate for M ij kl which is L ij (8,1,1) lin kl , the superform that appears in the ∇ 6 R 4 type invariant discussed in the last section. Following (2.17), we have and therefore is not the exterior derivative of a superform that does not depend on the naked scalar fields (uncovered by a space-time derivative). It follows that such a correction, if it appeared in (2.35), could not be reabsorbed in a redefinition of L ij,kl (8,2,0) (1) .
If this mixing between the (8, 2, 0) and the (8, 1, 1) superforms was not appearing at the non-linear level, then the action of the exterior derivative in the function of the scalar fields should not introduce lower derivative terms such that it should satisfy then to If the mixing did appear, then the unicity of the linearised invariants (2.37) would imply that the corresponding non-linear superform should be the same as in (2.18), such that once again the exterior derivative acting on D 3 [0,2,0,0,0,0,0] E (8,2,0) should not generate lower derivative terms and one would conclude again that (2.39) must be satisfied. Therefore this equation must be satisfied in either cases.
Using moreover the property that the gradient expansion of the linearised invariant is inconsistent with the presence of the third order derivative in the [1, 0, 0, 1, 0, 0, 1] of SU (8), one requires Using this equation one computes independently of (2.39) that This eigenvalue is such that the structure of the invariant is consistent with the mixing between the (8, 2, 0) and the (8, 1, 1) superforms. Only in this case can they reduce to the same invariant for a function E (8,2,2) satisfying to both (2.20) and (2.40), as for the ∇ 4 R 4 type invariant.
We are going to argue now that this chiral invariant must indeed include a ∇ 6 R 4 coupling, because the two classes of invariants reduce to one single class in three dimensions. But before to do this, let us mention that (2.40) can be rewritten as which defines a qantization of the algebraic eqation Q 3 133 = 0 associated to the complex nilpotent orbit of E 7 of Dynkin label 0 000002 with the real form defined with the SU (8) Dynkin label [0200000] [25], which defines the graded decomposition of SU (8) associated to the (8, 2, 0) harmonic superspace we consider in this section. In this case the choice of real form moreover implies that the complex charge in the 70 defining the nilpotent orbit through the Kostant-Sekiguchi correspondence satisfies to such that it admits a unique quantization, with the eigenvalue of the Laplace operator −60.
However, we will see in the following that the constraint (2.39) can be relaxed while keeping the property that the associated representation of E 7(7) is a highest weight representation.

Dimensional reduction to three dimensions
In three dimensions, the duality group is E 8 (8) , of maximal compact subgroup Spin(16)/Z 2 . We denote i, j the SO(16) vector indices and A, B the positive chirality Weyl spinor indices. The covariant derivative in tangent frame is a chiral Weyl spinor, i.e. in the 0 000000 1 representation. In the linearised approximation, the covariant fields all descend from the Weyl spinor scalar field, satisfying to [38] D Both four-dimensional (8, 1, 1) and (8, 2, 0) harmonic superspaces descend to the same (16,2) harmonic superspace in three dimensions, defined through the introduction of harmonic variables parametrising SO(16)/(U (2) × SO (12)) [39]. The Weyl spinor representation decomposes with respect to U (2) × Spin (12) as such that the grad 1 Weyl spinor W of Spin(12) satisfies to a G-analyticity constraint with respect to the positive grad covariant derivative in the 2 of U (2). The general polynomial in the Spin(12) Weyl spinor is parametrised by four integers, just as for the rank three antisymmetric tensor of SU (8) in section (2.1). 1 One computes in a similar way the general integral where ∇ 10 P 4 is a Spin(16) invariant quartic term in the scalar field momentum, that replaces the ∇ 6 R 4 type term that vanishes modulo the equations of motion in three dimensions. In three dimensions it is not established if there is a one to one correspondence between non-linear and linear invariants defined as harmonic superspace integrals. Nevertheless, the class of invariants we discuss descends from four dimensions, and we can therefore assume they admit the same structure, i.e.

(2.48)
This expansion implies that the fourth order derivative of the function E (16,2) restricted to the 0 100010 0 must vanish, i.e.
One can understand that the two kinds of 1/8 BPS invariants discussed in the preceding section dimensionally reduce to this single class. If one consider the decomposition of (2.46) with respect to U (6) ⊂ Spin (12), one obtains for one embedding such that the G-analytic superfield in the 32 + includes the four-dimensional (8, 1, 1) G-analytic scalar W rst as well as some components of the vector fields. A generic spinor of non-zero quartic invariant can be represented by W rst . For the other embedding U (6) ⊂ Spin (12), one gets such that the G-analytic superfield in the 32 + includes the four-dimensional (8, 2, 0) G-analytic scalar W rs as well as some components of the vector fields, and a Ehlers complex scalar parametrising the four-dimensional metric. The scalar field alone only parametrises a null spinor of Spin(12) of vanishing quartic invariant, and only together with the Ehlers scalar field it can provide a representative of a generic spinor. One could have naively concluded that the function E (8,2,0) should give rise to a function on E 8(8) /Spin c (16) satisfying moreover to excepted for the functions satisfying to the quadratic and cubic constraints that define the R 4 and ∇ 4 R 4 type invariants. We see therefore that this equation is incompatible with supersymmetry. It follows that both (8, 1, 1) and (8, 2, 0) type invariants dimensionally reduce to threedimensional invariants depending of functions on E 8(8) /Spin c (16) that belong to the same representation of E 8 (8) . Being in the same representation, they both carry a quartic component in the linearised approximation and they must both include a ∇ 6 R 4 type term in their uplift to four dimensions. This proves that the mixing between the two different linearised structures must occur such that the non-linearF 2 ∇ 4 R 4 type invariant cannot exist without including a ∇ 6 R 4 type term as well.
Before to end this section on the three-dimensional theory, let us discuss the modification of the supersymmetry constraint due to the completion of the R 4 type invariant at the next order. As it is argued in [10], the appearance of a R 4 correction with threshold function E (16,8) , will modify the Laplace equation with a non-zero right-hand-side, i.e. (16,8) . (2.56) Because the function E (16,8) satisfies to [20] DΓ ijkl D E (16,8) the second derivative of its square must necessarily vanish in the 0 100010 0 , and we get accordingly a modification of (2.49) to (16,8) . (2.58)

E 7(7) Eisenstein series
In this section we shall discuss some properties of Einstein series that solve the differential equations we have derived for the ∇ 6 R 4 type invariants.

Fundamental representation
As discussed in [15,21], one can define the Eisenstein series as a sum over the rank one integral charge vectors Γ in the 56 of E 7 (7) satisfying to the constraint that the quadratic tensor Γ ⊗ Γ vanishes in the adjoint representation. This formula is rather useful to identify the differential equations satisfied by the Eisenstein function, because one can simply consider the case of one charge Γ, with Z(Γ) ij = V ij I Γ I , such that the quadratic constraint becomes 60) and the differential operator acts on Z ij as an element of e 7(7) Using the definition |Z| 2 = Z ij Z ij , one computes that the function |Z| −2s satisfies to and to the Laplace equation For s = 2, 4, the function admits a generic gradient expansion in the irreducible representations [0, n 2 +2n 3 , 0, n 1 , 0, n 2 , 0] and their complex conjugate. To exhibit this property, it is convenient to consider a restricted set of indices as follows One computes moreover that for m ≤ n such that acting with a derivative operator in the conjugate representation [0, 0, 0, 0, 0, 2m, 0] does not produce an independent tensor. One has in particular for s an integer greater than 5 for integral s, and we would like to see that the function E 0 00000s also decomposes somehow into a "holomorphic" part F s and a "anti-holomorphic" partF s , satisfying respectively to and respectively for the complex conjugate. By consistency, this requires for instance that acting with further derivatives on this tensor does not permit to get back lower order tensors with n 3 < s − 4 in (2.65).
Through representation theory, one obtains that for some coefficients that are not specified. So the only way to reduce n 3 , is to increase n 2 by 1 unit. We will check this equation in the case n 1 = n 2 = 0. The restriction of the derivative D 3n |Z| −2s to the [0, 2n, 0, 0, 0, 0, 0] with two free indices reads and one computes that In particular we have that consistently with the assumption that no lower order tensor is produced out of the tensor function (2.69). We conclude therefore that the tensor is an E 7(7) (Z) modular form that is in some sense holomorphic, such that its gradient expansion is restricted to derivative of this tensor in the symmetric representations [0, n 2 + 2(n 3 + s − 4), 0, n 1 , 0, n 2 , 0], i.e. Using Langlands functional identity [17], one computes that the only integer values of s ≥ 5 for which the function diverges are 12 642 554 925ζ(5)ζ(9) 2 097 152π 9 ǫ +Ê 0 000009 (2.77) However, according to (2.62), the function E 0 00000s satisfies to and it follows that the tensor FE [0,2(s−4),0,0,0,0,0] (2.75) is finite for all s. However, we have argued in the preceding section that theF 2 ∇ 4 R 4 type invariant must include a ∇ 6 R 4 type term, which will be multiplied by the function itself. In this case the relevant Eisenstein function diverges, and one must regularise it such that the differential equation (2.42) will be modified to ∆Ê 0 000005 Such a correction is reminiscent of a 1-loop logarithm divergence of the ∇ 4 R 4 type invariant form factor.

Adjoint representation
One can also consider the Eisenstein series 2 as the sum over integral charges Q ∈ e 7(7) satisfying to the constraint Q 2 = 0, and such that the adjoint action of the coset representative V on Q decomposes into the anti-Hermitian traceless matrix Λ i j and the complex-selfdual antisymmetric tensor X ijkl satisfying to the constraints The action of the derivative on these tensors is defined as the e 7 (7) action which permits to derive that One gets therefore a solution to the equation showing that the function solves the cubic equation (2.40) for s = 5 2 . These functions satisfy the same equations as their analog Eisenstein functions defined in the fundamental representation, consistently with the property that 3 (2.88) One can also consider the restriction of the fourth order derivative to the [2, 0, 0, 0, 0, 0, 0, 2] to vanish, which defines a further restriction on solutions to (2.20). In this case one obtains  [17] with a factor of 2ζ(2s) in the definition of the Eisenstein function, one computes using Langlands formula The function is singular for various values of s, i.e. 9 2 , 11 2 , 6, 13 2 , 7, and 17 2 , and in particular for s = 6, which is the relevant value to solve equation (2.20) with (2.23). One should therefore consider the regularised series (2.92) However, we will see in the following that this function does not appear in string theory, similarly as the ∇ 6 R 4 threshold function is not described by an Eisenstein series in type IIB supergravity. Nonetheless, some components of this function should appear, as we will argue in the following.

F 2k ∇ 4 R 4 type invariants
The F 2 ∇ 4 R 4 type invariants we have discussed in section 2.2 have a natural generalisation to higher order invariants. Considering the same chiral harmonic superspace defined in terms of the harmonic variables ur i , u r i parametrising SU (8)/S(U (2) × U (6)) [24], one can define the G-analytic superfieldsF They do not permit to define directly chiral primary operators of SU (2, 2|8), becausē Chiral primary operators are annihilated by the special supersymmetry generators at the origin, One can enforce this property by defining a chiral primary as (2.96) for an arbitrary function F of the G-analytic superfield W rs . By construction such a chiral primary operator is never short, and defines a non-trivial integrand for the (8, 2, 0) measure. Because one can consider an arbitrary representative up to a total fermionic derivative, one can as well consider the first term in (2.96) as the integrand.
So similarly as in section 2.2, we get the general class of linearised invariants for an arbitrary positive integer k, We conclude that the corresponding supersymmetry invariants admit the same gradient expansion in for a function E (k) (8,2,0) satisfying to (2.40), and k ≥ 2. The coupling at the lowest number of points is then of the kind In principle one could expect to have a non-trivial mixing with another class of linearised invariant at the non-linear level, just as the one of theF 2 ∇ 4 R 4 type invariant with the ∇ 6 R 4 type invariant described in section 2.2. However, there is no higher order chiral primary operator that can define a non-trivial (8, 1, 1) harmonic superspace integral, and we did not find any linearised invariant with the right structure to define a possible cohomology class as does (2.38). Therefore we expect these invariants to have the same structure as the associated linearised invariants, i.e. to only contribute to (4 + 2k)-point amplitudes and higher. Independently of this assumption, the structure of these invariants requires that the action of the derivative D ijkl on D 3k E (k) (8,2,0) does not generate lower order derivatives of the function. This condition is precisely (2.74), and we conclude therefore that the eigenvalue of the Laplace operator is determined in the same way as such that the function satisfies to It is therefore tempting to conjecture that in the string theory effective action, and we will indeed show in section 2.7 that this function admits a consistent perturbative string theory limit. Moreover, we will see in section 3 that it also admits an appropriate decompactification limit.

Wavefront set and Poisson equation source term
We have seen that there are two classes of ∇ 6 R 4 type invariants in four dimensions, that preserve tree-level supersymmetry modulo the classical field equations. However, considering that the effective action already includes an R 4 type correction, we must take into account the action of the accordingly modified supersymmetry transformation on the R 4 type invariant itself. This is a very difficult task to carry out in practice, but one can nonetheless show general properties on these corrections. We recall that the R 4 type invariant admits the following gradient expansion in derivatives of the function E (8,4,4) L (8,4,4) [E (8,4,4) ] = , with E (8,4,4) satisfying to (2.85). The first order modification of the supersymmetry transformations will therefore necessarily admit the same gradient expansion in the function E (8,4,4) , such that to twelve, and the differential equations (2.20,2.23) must be modified in the presence of the R 4 type correction. Following the analysis carried out in [10,17], we conclude that (2.106) As explained in [20], this requires then to modify (2.20) to (8,4,4) , (8,4,4) . one shows that We therefore conclude that no higher derivative correction in (E (8,4,4) ) 2 can consistently modify (2.107) without contradicting (2.106). These properties of the source term in the Poisson equation (2.106) can also be understood through the structure of the Fourier modes of these functions. In the decompactification limit, the Fourier modes of a function are the coefficients, functions on E 6(6) /Sp c (4), of e 2πi(q,a) , with the axion field a in the 27 of the E 6(6) subgroup. We have shown in [20] that (2.85) implies then that the associated momenta are rank one vectors, i.e. using the cubic Jordan norm 3 det (q) = tr q(q × q), consistently with the properties of the R 4 threshold function. If we consider the square of E (8,4,4) , it admits by construction Fourier modes of momenta q 1 + q 2 where q 1 and q 2 satisfy to (2.110), such that det (q 1 + q 2 ) = det (q 1 ) + tr q 1 (q 2 × q 2 ) + tr q 2 (q 1 × q 1 ) + det (q 2 ) = 0 , (2.111) As one can see in [20], equation (2.106) implies that the Fourier modes of the function E (8,1,1) must indeed carry momenta satisfying to the rank 2 constraint det (q) = 0, whereas the Fourier modes of the function E (8,2,0) are generic by construction in the parabolic decomposition. The nilpotent orbit associated to E (8,2,0) is indeed defined from the graded decomposition e 7(7) ∼ = 27 (−2) ⊕ gl 1 ⊕ e 6(6) (0) ⊕ 27 (2) , such that a representative of the nilpotent orbit is a generic element of the grad two component in the 27. Considering instead the string theory limit, the non-abelian Fourier modes are defined over a Heisenberg algebra with 32 momenta in the positive chirality Weyl spinor representation of Spin(6, 6) associated to Ramond-Ramond D-brane charge Q, and an additional momentum associated to the Neveu-Schwarz 5-brane charge N 5 . The nilpotent orbit associated to E (8,1,1) is defined from the associated graded decomposition A representative of the nilpotent orbit is defined as a generic Weyl spinor in the grad 2 component [40], Q ∈ Spin(6, 6)/SU (2, 4) , or Q ∈ Spin(6, 6)/SL (6) ,  A generic rank 1 charge vector can always be rotated to the grad 3 component. Considering the sum of two rank one charges, respectively in the grad -3 and the grad 3 components, one obtains a generic rank 4 spinor of stabilizer SL(6) ⊂ Spin(6, 6). All the rank four charges defined as the sum of two rank 1 charges with a non-trivial symplectic product can be written in this form. Therefore the right-hand-side in (2.107) indeed sources generic Fourier modes of E (8,1,1) . More precisely, all the Fourier modes with a negative quartic invariant I 4 (Q) ≤ 0 (belonging to the second orbit in (2.114)) are sourced by the function E 2 (8,1,1) , whereas the Fourier modes with a strictly positive quartic invariant I 4 (Q) (belonging to the first orbit in (2.114)) satisfy to a homogeneous equation.
On the contrary, a representative of the nilpotent orbit associated to E (8,2,0) satisfies that its third power in the adjoint representation vanishes, which according to (2.113) implies that [40] ( The relation with the Fourier modes is not completely straightforward in the presence of a non-trivial NS5-brane charge, because in that case the nilpotent subgroup is a non-abelian Heisemberg group, such that the corresponding Killing vector where C αβ is the antisymmetric charge conjugation matrix of Spin(6, 6). For a Fourier mode of vanishing NS5-brane charge, k 5 E Q,0 = 0, and one can define the spinor charge Q such that κ α E Q,0 = iQ α E Q,0 , and Q must satisfy to the same algebraic equations as the representatives of the nilpotent orbits associated to the differential equations. For a non-zero NS5-brane charge the relevant equations are more complicated, but still involve the Killing vector κ α to the third order in the same combination.
Let us now consider the M-theory limit, for which one considers the decomposition In this case the nilpotent subgroup also generate a non-abelian Heisenberg type algebra For a Fourier mode of vanishing M5-brane charge, k m E q,0 = 0, and one can define the M2-brane charge κ mnp E q,0 = iq mnp E q,0 . For a non-zero M5-brane charge k m E q,p = ip m E q,p the relevant equations are more complicated, but still involve the Killing vector in a way similar as does the corresponding nilpotent orbit characteristic equation involves the algebraic charges. For a 1/2 BPS charge satisfying to the quadratic constraint, one obtains [41] ε mnqrstu q pqr q stu = 0 , q mnp p p = 0 , (2.123) giving 17 = 13 + 4 linearly independent solutions, with typical representative 1 6 q mnp dy m ∧ dy n ∧ dy p = q 1 dy 1 ∧ dy 2 ∧ dy 3 . (2.124) The cubic constraint in the adjoint representation implies ε nrstuvw q rst q uv[p q qm]w = 0 , ε mnqrstu q pqr q stu p p = 0 , (2.125) that gives 27 = 21 + 6 linearly independent solutions, with typical representative 1 6 q mnp dy m ∧ dy n ∧ dy p = dy 1 ∧ q 1 dy 2 ∧ dy 3 + q 2 dy 4 ∧ dy 5 + q 3 dy 6 ∧ dy 7 . (2.126) The cubic constraint in the fundamental implies instead ε mnrstuv q mnr q st(p q q)uv = 0 , (2.127) that gives 33 = 26 + 7 linearly independent solutions, such that the SL(7) M2-brane charge orbits are either with typical representative 1 6 q mnp dy m ∧dy n ∧dy p = q 1 dy 1 ∧dy 2 ∧dy 3 +q 2 dy 1 ∧dy 4 ∧dy 5 +q 3 dy 2 ∧dy 6 ∧dy 4 +q 4 dy 3 ∧dy 5 ∧dy 6 , (2.129) and such that the orbit (2.128) is determined by the sign of the eigenvalue of this rank one symmetric tensor, I 4 = q 1 q 2 q 3 q 4 . The generic sum of two rank one charges takes the form 1 6 q mnp dy m ∧ dy n ∧ dy p = q 1 dy 1 ∧ dy 2 ∧ dy 3 + q 2 dy 4 ∧ dy 5 ∧ dy 6 , (2.131) and is a generic solution to (2.127) associated to the second orbit (i.e. I 4 < 0), and violates equation (2.126). Therefore we confirm that a quadratic source in E (8,4,4) is in contradiction with the cubic equation satisfied by E (8,2,0) , whereas it is consistent with the one satisfied by E (8,1,1) .
Let us now argue that all the invariants of the infinite series ofF 2k ∇ 4 R 4 do not get modified at the same order by lower order modifications to the supersymmetry transformations. By power counting, the next order correction to the R 4 type invariant and aF 2k ∇ 4 R 4 type invariant can in principle contribute to a right-hand-side for the classical supersymmetry variation of ā F 2k+6 ∇ 4 R 4 type invariant. So in principle one could expect that the function E (k) (8,2,0) satisfies to a Poisson equation of the kind However, the solutions to the differential equation (2.101) admit restricted Fourier modes in the string theory limit, satisfying to (2.117) for a vanishing NS5-brane charge. As we have already explained, the product of two functions including non-perturbative corrections admits generic Fourier modes in the string theory limit, because the sum of two pure spinors can be a generic spinor. We see therefore that a source term modifying (2.101) would necessarily involve the third order differential operator such as to source these Fourier modes. Such a modification would destroy completely the structure of the equations, which would reduce then to some kind of Poisson equation.

String theory perturbation theory
In order to deduce constraints on the contributions that can possibly appear in perturbative string theory, it is important to solve the differential equations satisfied by the threshold functions in the parabolic gauge with manifest T-duality symmetry (2.113). In this section we will solve these equations on an ansatz function depending only on the string theory dilaton e 2φ and the scalar fields parametrising SO(6, 6)/(SO(6) × SO (6)). We have not computed explicitly the decomposition of the differential equations, but using the manifest covariance, and the known solutions for the R 4 and the ∇ 4 R 4 threshold functions [17], we can determine unambiguously all the unknown coefficients. We define the covariant derivative D ab on SO(6, 6)/(SO(6) × SO (6)) in tangent frame, such that a = 1 to 6 of one SO(6) andb = 1 to 6 of the other. It is convenient to define the covariant derivative as an SU (4) × SU (4) tensor  (6)) as gives the components in the 32 of Spin(6, 6) The ∇ 4 R 4 threshold function solves (2.134) for s = 3 2 and (2.137) for s = 4. One reads directly from these equations, that a solution of type e aφ E D 6 on R * + × SO(6, 6)/(SO(6) × SO (6)) must be such that E D 6 satisfies to the quadratic equations in all fundamental representations (i.e. the vector and Weyl spinor of positive and negative chirality), unless a = −6 or a = −8. The only other solutions are therefore such that E D 6 is either a constant, or solves (B.24). One finds the unique solution e −10φ . For the values a = −8, the function E D 6 satisfies to a quadratic equation in the spinor representation, and solve (B.30) for s = 4 (or 1 which is equivalent). For a = −6, E D 6 satisfies to a quadratic equation in the vector representation, and cubic equations in the two spinor representations, and must therefore satisfy to (B.25). We find therefore that supersymmetry and T-duality alone already determine the ∇ 4 R 4 type corrections in perturbative string theory, up to three free coefficients, that are given in [ It is quite remarkable that the only solutions we get all correspond to a strictly positive number of loops in perturbative string theory. After implementing the Weyl rescaling, one obtains indeed that c (k) ℓ is a coefficient for a ℓ-loop correction in string theory for theF 2k ∇ 4 R 4 threshold function. For k = 1 and k = 2, equation (2.137) is exact for the Wilsonian effective action (not taking into account linear corrections associated to logarithms in the complete effective action). U -duality therefore implies that E (8,2,0) must be an Eisenstein function as in (2.102). Assuming that our argumentation in the preceding section is correct, and that equation (2.137) is satisfied for all k, we arrive at the conjecture that theF 2k ∇ 4 R 4 threshold function is defined by the Eisenstein series E 0 000004+k for all k. It is rather remarkable that this coupling would only get three corrections in perturbation theory, at 1-loop, k + 2-loop and 2k-loop.
This Eisenstein function diverges precisely for k = 1, corresponding to theF 2 ∇ 4 R 4 threshold related to the ∇ 6 R 4 threshold function by supersymmetry. One must therefore consider the regularised Eisenstein series Because it is a three-loop contribution, it cannot come from the completion of the R 4 type invariant and it must appear as a solution to equation (2.134) for s = 6. Considering the general SO(6, 6, Z) invariant solution of (2.134), one finds indeed (2.141) where the coefficients c ℓ are constants that would correspond to ℓ-loop contributions for the ∇ 6 R 4 threshold function in string theory. Note that the first two terms do not make sense in perturbative string theory. The corresponding Eisenstein function E 0 s00000 includes generically all these terms, and therefore cannot define the string theory threshold function, consistently with the property that (2.134) is corrected by a source term (2.107). However, the three-loop contribution is not affected by the source term, and one can take seriously the last contribution, which is precisely the one required to restore O(6, 6, Z) invariance for s = 6. This is indeed confirmed by the expression obtained in [23] for the ∇ 6 R 4 threshold function, and using these results we conclude therefore that the exact threshold function for the ∇ 6 R 4 coupling is defined as where the function E (8,1,1) solve the differential equation Here the anomalous right-hand-side is determined such as to coincide with the one obtained in [23] for the complete function E (0,1) . These coefficients can also be directly computed from the properties of the Eisenstein functions and the structure of the differential equations [42].

Supergravity in higher dimensions
In this section we will consider the extension of the results of the preceding section in five, six, seven and eight dimensions. We will see that the two ∇ 6 R 4 type invariants both lift to higher dimensions, even if they cannot be defined as harmonic superspace integrals in the linearised approximation in general.

N = 4 supergravity in five dimensions
Let us recall in a first place some properties of maximal supergravity in five dimensions. The scalar fields parametrise the symmetric space E 6(6) /Sp(4) c . We use i, j = 1, ..., 8 as indices in the fundamental representation of Sp(4), and Ω ij defines the symplectic form with the normalisation Ω ik Ω jk = δ i j . The covariant derivative in tangent frame D ijkl is a symplectic traceless rank four antisymmetric tensor in the representation [0, 0, 0, 1] of Sp(4).
The consistency of this ansatz requires that the function E (4,1) must be an eigenfunction of the Laplace operator, and that its third order derivative restricted to the [0, 2, 0, 0] is proportional to its second derivative in the same representation. This linearised analysis is consistent with the one of the (8, 1, 1) type invariant in four dimensions, and we are going to see that the relevant equation is However, the (8, 2, 0) type invariants cannot be defined in the linearised approximation through a harmonic superspace integral, and we shall instead consider the uplift of the general invariant to five dimensions.

Decompactification limit from four to five dimensions
We are therefore going to solve the differential equations (2.40) and (2.143) for a function depending only of the Levi subgroup R + * × E 6(6) of the parabolic subgroup associated to the decompactification limit, such that e 7(7) ∼ = 27 (−2) ⊕ gl 1 ⊕ e 6(6) (0) ⊕ 27 (2) , For this purpose we use the same conventions as in [20], such that the coset representative in where V ij I is the coset representative in E 6(6) with the Sp(4) pair ij being antisymmetric symplectic traceless and the index I in the fundamental of E 6(6) . t IJK is the E 6(6) invariant symmetric tensor normalised as in [20]. We have already computed the decomposition of the cubic equation (2.21) in [20], which is (8,4,4) (3.12) (8,4,4) , (3.13) and indices are raised and lowered with the symplectic matrix Ω ij . Because of the Weyl rescaling required to stay in Einstein frame, the relevant radius power in the decompactification limit for a ∇ 2n R 4 threshold function is such that E E 7 = e −(6+2n)φ E E 6 , and because we are interested in the constraint on the ∇ 6 R 4 threshold function, we use the ansatz (8,1) , E (8,4,4) = e −6φ E (8,4) , (3.14) where E (8,1) and E (8,4) are functions on E 6(6) /Sp(4) c . Using this ansatz, one derives and These equations are satisfied by the 1/8 BPS threshold functions in the Wilsonian effective action, but the U-duality invariant function appearing in the 1PI effective action satisfied to anomalous equations with additional terms linear in E (8,4) in the right-hand-side [23].
We shall now consider the uplift of the F 2k ∇ 4 R 4 type invariants, but for this purpose it will be more convenient to consider directly the decompactification limit of the Eisenstein function E 0 00000k+4 . We shall only consider the term with the correct power of the compactification radius r to lift to a diffeomorphism invariant in five dimensions, as computed in appendix B.1, (3.17) We conclude in this way that the threshold function E (k) 1 8 defining the F 2k ∇ 4 R 4 type invariants satisfies to It follows from representation theory that such equations are indeed implied by (2.136), and this explicit example permits to determine them uniquely.
We therefore obtain that the threshold function is the regularised Eisenstein serieŝ such that the exact ∇ 6 R 4 threshold function E (0,1) is (3.20) The series E 0 0000s admits a pole at s = 9 2 proportional to the series E 0 3 2 0000 defining the R 4 threshold, exhibiting that the R 4 invariant form factor diverges at two loop into the ∇ 6 R 4 form factor associated to the same function. This is in agreement with [23], where the explicit coefficient is computed.
From this structure one deduces that supersymmetry requires the function E (4,1,0) to be an eigenfunction of the Laplace operator, and to satisfy equations of the form for some integer k. We will see in the next section that the standard ∇ 6 R 4 type invariant threshold function indeed satisfies to these equations for k = 2.

Decompactification limit from five to six dimensions
We are now going to solve the differential equations (3.16) and (3.18) for a function depending only of the Levi subgroup R + * × SO(5, 5) of the parabolic subgroup associated to the decompactification limit, such that The covariant derivative on E 6(6) /Sp c (4) acting on such a function takes the block diagonal form To check the differential equations (3.16) and (3.18) we need to compute the block diagonal decomposition of the higher order differential operators. In order to do this computation we consider a general ansatz and determine all the free coefficients by consistency with the various differential equations displayed in appendix A. We obtain in this way and where ∆ ≡ Tr D 2 10 . In order to determine the constraints on the threshold function is six dimensions, we consider an ansatz with the appropriate power of the radius modulus e −3φ such as to compensate for the Weyl rescaling to Einstein frame, i.e. which is indeed consistent with (3.14). Working out spinor and the vector equations, using the Poisson equation (3.36), one obtains similarly The only Eisenstein function that solves this homogenous equation (for E (4,2,2) = 0) is the Eisenstein seriesÊ 0 0 7 2 0 0 , but its expansion in the string theory limit is inconsistent with perturbation theory. As is computed in appendix B.3, one has moreover which defines an integrability condition for the function to decompose into the sum of two functions satisfying to (3.30) and its conjugate obtained by exchange of the two Sp(2) for k = 2.
Let us now consider the differential equation (3.18) for the function E defining the F 2k ∇ 4 R 4 threshold function in five dimensions. Diffeomorphism invariance in six dimensions requires an ansatz of the form E (k) 39) and using this ansatz, one obtains from the singlet component of (3.18) the Laplace equation where we removed the D 5 label for simplicity. The spinor and the vector equations give then where we used that the even and odd powers of D 10 lie in different irreducible representations of Sp(2) × Sp (2), and must therefore vanish separately. The unique Eisenstein function satisfying to this equation is E We conclude therefore that the exact ∇ 6 R 4 function is defined as whereÊ (4,1,0) satisfies to (3.37), and to an anomalous Poisson equation with an additional constant source term. This function is consistent with [23], where the second Eisenstein function appears with this normalisation, and the 2-loop five dimensional threshold function must indeed solve (3.37), because the equation is parity invariant with respect to O(5, 5).

N = 2 supergravity in seven dimensions
None of the ∇ 6 R 4 type invariants can be defined in the linearised approximation as harmonic superspace integrals in seven dimensions. We will therefore consider the uplift of the fourdimensional invariants in the decompactification limit. In seven dimensions the scalar fields parametrize the symmetric space SL(5)/SO (5), and the covariant derivative D ij transforms as a symmetric traceless tensor of SO(5), with i, j = 1, . . . , 5 of SO (5). We consider therefore the parabolic subgroup of E 7(7) of semi-simple Levi subgroup SL(5) × SL(3) associated to the decomposition (5) . (3.45) We will use the same conventions as in [21], where the decompactification limit of equations (2.40) and (2.143) is already discussed in details. We consider the ansatz Using this equation in the (3 ⊗ 5) (1) component of (2.143), one obtains Using moreover these equations to simplify the 10 (3) components, one obtains The solution to the homogenous equation (with E (4,2) = 0) can be written as the Eisenstein function E [3,0,0, 5 2 ] , using the formulae of Appendix A.5. We consider now the F 2k ∇ 4 R 4 threshold function, with the ansatz Appropriate linear combinations of the grad 6 and 4 components of (2.40) in (3.45) give the two equations For k ≥ 1 we conclude that the function must satisfy to the quadratic equation where the Eisenstein series is normalised with an extra 2ζ(2s) factor with respect to the Langlands normalisation. We conclude therefore that the exact ∇ 6 R 4 threshold function is

N = 2 supergravity in eight dimensions
We shall now consider the oxidation of the seven-dimensional ∇ 6 R 4 and F 2n ∇ 4 R 4 type invariants to eight dimensions. Because there is a 1-loop divergence in eight dimensions, the exact R 4 threshold function differs from the Wilsonian effective action function. In the dimensional reduction, the divergence appears to be absorbed into the infinite sum of Kaluza-Klein states over the circle such that the function is finite in seven dimensions, but involves a logarithm of the radius modulus in the decompactification limit [32]. In order to consider the non-analytic terms in eight dimensions, we will take these logarithms into account in the decompactification limit. We shall use the same conventions as in [21], i.e. the complex scalar field τ parametrises the coset representative v α j ∈ SL(2)/SO(2), with α, β = 1, 2 of SO(2) and i, j = 1, 2 of SL(2), whereas the five real scalar fields t parametrise the coset representative V a J ∈ SL(3)/SO (3), with a, b = 1, 2, 3 of SO(3) and I, J = 1, 2, 3 of SL (3). The corresponding covariant derivative in tangent frame are then traceless symmetric tensors D αβ and D ab , respectively. In the decompactification limit, one writes the SL(5)/SO(5) coset element in the parabolic gauge associated to the graded decomposition In this way one computes that the covariant derivative over SL(5)/SO (5) in tangent frame acts on a function of φ, τ, t as [21] (3.58) One computes that the higher order derivative operators obtained as products of D 5 in the 5 of SL(5) decompose similarly as and We can now solve equation (3.48) in the docompactification limit, with E 1 8 (φ, τ, t) and [13] E (4,2) = e −6φ 2Ê [1] where φ 0 is a constant that depends on the renormalisation scheme. Using the property that one shows that the general solution to (3.48), as a function of φ, τ and t takes the general form where F [4] (τ ) and F [4,−2] (t) are solutions to ∆F [4] (τ ) = 12F [4] (τ ) − (Ê [1] (3.64) Here the notation is used to emphasize that these solutions are defined modulo the homogeneous solutions E [4] (τ ) and E [4,−2] (t), respectively, as one can see using the formulae of Appendix A.6. Note nonetheless that these homogeneous solutions are inconsistent with the string theory perturbation expansion, and the exact threshold function is uniquely determined by these equations and consistency with string theory [16]. The structure of the threshold function exhibits that there is a 1-loop divergence of the R 4 type invariant form factor proportional to the ∇ 6 R 4 type invariant. This implies the presence of an addition renormalisation scheme ambiguity in the definition of the analytic part of the effective action. It appears that the renormalisation scheme used in [13,16], cannot be obtained by simply neglecting the terms in φ − φ 0 , but one finds nonetheless that the threshold function only differs by terms proportional to the linear and the quadratic term in φ − φ 0 , i.e.
Let us now consider the oxidation of the F 2k ∇ 4 R 4 type invariants, i.e. solve the differential equation (3.52) for a function of the form e −(10+2k)φ E (k) 1 8 (τ, t), as required by diffeomorphism invariance in eight dimensions. One obtains straightforwardly Using the results of Appendix A.6 one obtain that the solution can be written in terms of Eisenstein functions as consistently with the decompactification limit of the SL(5)/SO(5) Eisenstein function [21], The sum of the two functions reproduces correctly the threshold function obtained in [13,16].
In this appendix we collect the differential equations satisfied by the Eisenstein functions that are relevant in the analysis of BPS threshold functions in string theory, and their related coadjoint nilpotent orbits. We write them in terms of the covariant derivative in tangent frame valued in the Lie algebra in some particular representations, which are specified by their dimension.
A.1 E 8 (8) The Eisenstein function in the adjoint representation is associated to the nilpotent orbit of Dynkin label 0 0000002 , with D 8 Dynkin label 0 · 0000020 , and satisfies in general to the differential equation For the following three special values of s, the function is associated to lower dimensional nilpotent orbits, and satisfies moreover to A.2 E 7 (7) The Eisenstein function in the adjoint representation is associated to the nilpotent orbit of Dynkin label 0 200000 , with A 7 Dynkin label [2000002], and satisfies in general to the differential equation For the following three special values of s, the function is associated to lower dimensional nilpotent orbits, and satisfies moreover to The Eisenstein function in the fundamental representation is associated to the nilpotent orbit of Dynkin label 0 000002 , with A 7 Dynkin label [0200000] and its conjugate, and satisfies in general to the differential equation . (A.5) The function moreover satisfies to highest weight representations constraints for integral s, and is associated to lower dimensional nilpotent orbits for s = 2 and 4. The relation of these Eisenstein functions with nilpotent orbits can be summarised in the following truncated closure diagram [25] • The Eisenstein function in the adjoint representation is associated to the nilpotent orbit of Dynkin label 2 00000 , with C 4 Dynkin label [2, 0, 0, 0] [43], and satisfies in general to the differential equations The function is associated to the next nilpotent orbit in the closure diagram for s = 5 2 , to the next to minimal nilpotent orbit for s = 3 2 and to the minimal nilpotent orbit for s = 1. However, there is a 1-parameter family of Eisenstein functions associated to the next to minimal nilpotent orbit. It is the Eisenstein function in the fundamental representation, that satisfies to and its third order derivative restricted to the [2, 0, 0, 1] of Sp(4) vanishes. It is functionally related to the Eisenstein function in the anti-fundamental representation at 6 − s, and reduces to the unique Eisenstein function associated to the minimal nilpotent orbit at s = 3 2 .
A.4 SO(5, 5) The Eisenstein function in the adjoint representation is associated to the nilpotent orbit of Dynkin label 0 020 0 , with C 2 × C 2 Dynkin label [2, 0] × [0, 0] and [0, 0] × [2, 0], and satisfies in general to the differential equations The function is associated to lower dimensional nilpotent orbits for s = 3 2 , 1, 1 2 . The Eisenstein function in the Weyl spinor representation is associated to the largest next to minimal nilpotent orbit, and satisfies in general to the differential equations It is functionally related to the Eisenstein function in the conjugate representation at 4− s. The Eisenstein function in the vector representation is associated to the smallest next to minimal nilpotent orbit, and satisfies in general to the differential equations and the former satisfies to the differential equations The antisymmetric tensor Eisenstein function is associated to the next to minimal nilpotent orbit, and satisfies to the differential equation whereas the vector Eisenstein function is associated to the minimal nilpotent orbit, and satisfies to both B Some additional computations on Eisenstein series B.1 E 6(6) Eisenstein series in the fundamental representation In the decompactification limit, the series definition (2.59) of the Eisenstein series E 0 00000s as a sum over the rank 1 charges in the 56 or E 7 (7) , decomposes into the four components (3.10) p 0 , p I , q I , q 0 of grad −3, −1, 1, 3 respectively, with the rank one constraint The E 7(7) invariant norm then reads where e −2φ is the radius moduli, whereas |Z(q)| 2 now represents the E 6(6) invariant norm. At large e −2φ we will only consider the sum over the maximal weight charges q 0 , q I The other terms are more complicate to obtain explicitly, but they follow the same pattern such that the perturbative terms reduce to sum over the charges of grad −1 and −3 after Poisson resumation. The complete perturbative expansion in e −2φ is then determined by compatibility with the Langlands functional identity to be is defined in terms of the central charge of q I Z ij (q) = V ij I q I , |Z(q)| 2 = Z ij (q)Z ij (q) , (B.5) and the quadratic constraint t IJK q J q K = 0 is equivalent to The covariant derivative in tangent frame acts on the central charge as

B.2 SO(6, 6) Eisenstein series
We define first the series associated to anti-chiral spinors. The associated 'central charge' is Z iî and its complex conjugate Z iî , where i = 1, 4 of one SU (4) factor andî = 1, 4 of the other. The rank one constraint on the spinor is the pure spinor constraint, that reads One computes then that the covariant derivative over SO(6, 6)/(SO(6) × SO (6)) acts on the central charge as One computes then that a homogeneous function of Z a Z a satisfies to The second equation implies that this function always satisfies to a quadratic equation in the two spinor representations, whereas it only satisfies to a quadratic constraint in the vector representation for the critical value s = 2.
As in the preceding section, one can define the series

B.3 SO(n, n) Eisenstein series in the adjoint
For SO(n, n) the adjoint representation decomposes with respect to SO(n) × SO(n) with a running from 1 to n of the first SO(n) andâ running from 1 to n of the second. We decompose therefore the adjoint into the coset component X ab and the two antisymmetric tensors Λ ab and Λâb. The minimal representation is such that a charge Q ∈ so(n, n) is nilpotent in all three fundamental representations, which reads explicitly Λ a c Λ bc = X aĉ X bĉ , such that the representation s = 1 2 is special, and then reduce to a vector representation Eisenstein function. Using representation theory, one straightforwardly check that there is no possible rank 3 antisymmetric tensor that one can write, such that Similarly as for the E 7(7) Eisenstein series in the fundamental representation, we expect this property to generalise to s = n 2 + k for any integer k, such that D 2+2k E α 2 , n 2 +k restricted to the symmetric rank 2 + 2k representation of SO(n) vanishes. For k = 1 on computes indeed that D dâ X (ab X b|b X cĉ X d)ĉ |X| −2s−4 − 4 n + 4 δ (ab X cb X d)b |X| −2s−2 + 2 (n + 2)(n + 4) δ (ab δ cd) |X| −2s = n(n + 2 − 2s) 2(n + 4) X (ab X b|b X c)ĉ Λâĉ|X| −2s−4 − 3 n + 2 δ (ab X c)b Λâb|X| −2s−2 . (B.47)