ε∇4R4 type invariants and their gradient expansion

We analyse the constraints from supersymmetry on ∇4R4 type corrections to the effective action in N=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N}=2 $$\end{document} supergravity in eight dimensions. We prove that there are two classes of invariants that descend respectively from type IIA and type IIB supergravity. We determine the first class as d-closed superforms in superspace in eight dimensions, whereas we obtain the second class by dimensional reduction down to four dimensions, in which there is a single class of invariants transforming in the next to minimal unitary representation of E7(7).


Introduction
The low energy effective action of Type II string theory on R 1,9−d × T d is extremely constrained by supersymmetry and U -duality [1][2][3]. Although there is no non-perturbative formulation of the theory, duality invariance permits 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]. At low orders in the derivative expansion, the effective action is completely determined by the four-graviton amplitude, and one can in principle reconstruct the effective action at these orders from the functions E (p,q) of the moduli parametrizing the symmetric space E d(d) /K d that define the amplitude [11], The functions E (0,0) , E (1,0) and E (0,1) are strongly constrained by supersymmetry, and are in particular eigenfunctions of the Laplace operator on the scalar manifold [9,[12][13][14].
We have shown in [26] that these functions moreover satisfy to tensorial differential equations that determine their egenvalues for all Casimir operators. The function E (0,0) satisfies for example that its second-order derivative vanishes when restricted to the Joseph ideal [27], constraining it to lie in the minimal unitary representation of E d(d) , in accordance with [19][20][21][22]. We have shown that E (1,0) satisfies to an equivalent equation associated to the JHEP03(2015)089 The lines refer to their connection by dimensional reduction. The • refers to parity symmetric invariants that can be defined in harmonic superspace in the linearised approximation, while • indicates that they are complex chiral invariants in the linearised approximation. • refers to invariants that cannot be written in harmonic superspace in the linearised approximation.
This paper extends the analysis of the ∇ 4 R 4 type invariant at the non-linear level in eight dimensions. To carry out this program, we concentrate on terms of maximal Rsymmetry weight, similarly as in [12,14,26]. We find in this way that the function of the scalar fields must satisfy to a tensorial second-order differential equation consistent with one of the explicit Eisenstein functions conjectured in [17] to define the non-perturbative threshold function E (1,0) . The second function does not depend on the type IIB torus complex structure, and is not constrained by this analysis that only considers Kähler derivatives of the function. However, we prove that the two sets of differential equations satisfied by the two functions defining E (1,0) , are in the same E 7 (7) representation in four dimensions. We show moreover that they are the unique differential equations satisfying to this criterium. We conclude therefore that there is two classes of ∇ 4 R 4 invariants in eight dimensions, consistently with the two functions appearing in the string theory effective action. Combining these results with the ones obtains in [26], we conclude that there is a unique ∇ 4 R 4 invariant in dimension five and lower, that splits into two different invariants in dimension 6, 7 and 8. They descend respectively from type IIA and type IIB 2-loop corrections to the supergravity effective action.
We provide an overview of the results in the first section, that combines results already obtained in [26], and new ones that are derived in this paper. It exhibits the structure of JHEP03(2015)089 the R 4 and ∇ 4 R 4 type invariants as gradient expansions in the covariant derivative of a defining function E of the scalar fields parametrising E d(d) /K d . In section 3 we discuss in details the structure of the ∇ 4 R 4 type invariant in eight dimensions that lifts to type IIA in ten dimensions, in the formalism of superforms in superspace [32][33][34]. Because the associated function depends on both the complex scalar parametrising SL(2)/SO (2) and the scalar fields parametrising SL(3)/SO(3), one must consider the gradient expansion in terms of both the Kähler derivative and the isospin 2 tangent derivatives on SL(3)/SO (3). This permits to distinguish terms of maximal U(1) weight and isospin, that are uniquely determined as monomials of order twenty-four in the fermion fields.
In order to show the existence and the uniqueness of the other class of ∇ 4 R 4 type invariants in eight dimensions, we use the uniqueness of the ∇ 4 R 4 type invariant in four dimensions, due to the bijective correspondence between supersymmetry invariants and superconformal primaries of Lorentz invariant top component in four dimensions [30,35]. Any supersymmetry invariant that can be defined in eight dimensions, clearly descends to four dimensions by dimensional reduction on T 4 . Starting from the type IIA invariant we study in section 3, one can consider the corresponding four-dimensional invariant, and the differential equations satisfied by the associated function on E 7(7) /SU c (8). Any other solution to these differential equations is also supersymmetric in four dimensions, and for a function defined on R * + × SL(2)/SO(2) × SL(3)/SO(3) with the appropriate power of the Kaluza-Klein dilaton, it must lift to an invariant in eight dimensions. The invariance of the supersymmetry invariant with respect to the nilpotent subgroup of E 7 (7) defining the shift of the axions, indeed implies that the dependence in the gauge fields and the axions is defined in such a way as to ensure gauge invariance and diffeormorphism invariance in eight dimensions.
We show that this line of arguments is indeed valid in section 4, although the proof is not formulated in this order. We rather start by solving the relevant differential equations derived in [26] in four dimensions on a function of the seven-dimensional scalar fields. This way we exhibit the existence of two classes of ∇ 4 R 4 type invariants in seven dimensions, which are then shown to lift to corresponding invariants in eight dimensions. We also discuss the properties of the solutions with respect to E d(d) (Z) invariance, and we prove that the functions conjectured to define the type II exact low energy effective action components in R 4 and ∇ 4 R 4 [15,23] are indeed solutions to the equations derived in [26].

Overview of the results in various dimensions
In this section we review the E d(d) multiplets of supersymmetric corrections to the supergravity effective action in various dimensions. We will concentrate ourselves on R 4 and ∇ 4 R 4 type invariants in maximal supergravity (see figure 1). Such corrections are invariant modulo the classical field equations, and are determined by closed superforms within the superform formalism of Bates [32][33][34]. A closed superform depends in general on the scalar fields parametrising E d(d) /K d through a function E and its covariant derivative in JHEP03(2015)089 tangent frame, and takes the form where R refers to irreducible representations of K d such that the superforms LR are E d(d) invariant and transform with respect to K d in the conjugate irreducible representation R. For BPS protected invariants such as the ones of type R 4 and ∇ 4 R 4 , the appearing irreducible representations R are generally determined from the linearised analysis, and the function E satisfies to the constraints that its derivative D n R ′ E in irreducible representations that do not appear in the gradient expansion either vanish or are related to lower order derivatives of the function in the same representation.
All along the paper we use the convention that the function E w for a weight w of a * (e d(d) ) is the Eisenstein function on E d(d) (Z)\E d(d) /K d associated to this weight [15,23], whereas a function E w refers to any function on E d(d) /K d solving the same differential equations as E w . Supersymmetry is preserved for any such a solution E w with the appropriate weight w, and requiring moreover E d(d) (Z) invariance only then distinguishes the Eisenstein function E w .

N = 2 supergravity in eight dimensions
In eight dimensions, maximal supergravity admits for duality group SL(2) × SL (3), and the scalar fields parametrise the symmetric space SL(2)/SO(2) × SL(3)/SO (3). The Kähler derivatives on SL(2)/SO (2) are denoted with D andD, while the SU(2) isospin 2 tangent derivatives on SL(3)/SO(3) are defined as D ijkl , with i, j, k, l running from 1 to 2 of SU (2). The theory includes two 1/2 BPS R 4 type invariants and two 1/4 BPS ∇ 4 R 4 type invariants, which are supersymmetric up to the classical equations of motion. These invariants decompose in a gradient expansion of a given function E of the scalar fields as follows where the L (4k)[4n] are SL(2)×SL(3) invariant 8-superforms in the isospin 2n representation of SU(2) with U(1) weight 4k. The indices of the function E (n,p,q) refers to the harmonic superspace construction of the associated invariant in the linearised approximation, whereas the notation E ′

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in agreement with [3]. The two classes of invariants coincide in trivial topology when the function is a constant, and define the 1-loop counter-term for the supergravity logarithm divergence [36]. The invariant associated to E (2,2,0) is chiral and complex, and its associated complex conjugate associated to the function E (2,0,2) satisfies to the complex conjugate constraints. The functions E (2,1,0) and E ′ 1 4 defining the ∇ 4 R 4 type invariants are discussed in this paper, and are proved to satisfy to consistently with [11,17]. These equations are indeed satisfied by the Eisenstein functionŝ ,0] , (2.8) which determine the exact R 4 and ∇ 4 R 4 thresholds in type II string theory [3,17], up two inhomogeneous terms associated to the chiral anomaly and the SL(3) anomaly produced by the 1-loop divergence [26,37]. Here the hat overÊ (2,2,0) andÊ (2,0,2) indicates that their sum satisfies to the inhomogeneous equation with a constant right-hand-side [11], and similarly forÊ (2,1,1) .

N = 2 supergravity in seven dimensions
In seven dimensions maximal supergravity has for duality group SL(5), with maximal compact subgroup SO (5). We label the vector indices a, b, c of SO(5) and the covariant derivative D ab is symmetric traceless, i.e. transforms in the [0, 2] of Sp (2). The R 4 and ∇ 4 R 4 type invariants have the following gradient expansion in the function E of the scalar fields does not admit a Lorentz invariant harmonic superspace integral form in the linearised approximation. The function E (4,2) defining the R 4 type invariant satisfies to

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consistently with [3]. It is important to remark that the two possible functions multiplying R 4 in eight dimensions E (2,2,0) and E (2,1,1) are related by SL (5) in seven dimensions. The functions appearing in the ∇ 4 R 4 type invariants satisfy instead consistently with [11]. The two invariants coincide for a constant function, and define the counter-term for the 2-loop logarithm divergence in supergravity [38]. The SL(5, Z) invariant Eisenstein functions 14) which are conjecture to define the exact low energy effective action in string theory [3,11], indeed solve these differential equations, up to an inohomogenous right-hand-side for the ∇ 4 R 4 type invariants that comes from the anomaly associated to the 2-loop divergence.
Once again the hat on the functions refers to these anomalous corrections.

N = 4 supergravity in five dimensions
In five-dimensional supergravity the duality group is E 6(6) , with maximal compact subgroup Sp(4)/Z 2 . The covariant derivative in tangent frame is a symplectic traceless rank four antisymmetric tensor of Sp(4), i.e. in the [0, 0, 0, 1] irreducible representation. The 1/2 and 1/4 BPS invariants admit the following gradient expansion in the function of the scalar fields The functions E (8,2n) satisfy to the tensorial equations where δ kl ij is the projector to the antisymmetric symplectic traceless irreducible representation of Sp(4). 1 E 6(6) (Z) invariant solutions to these differential equations are defined by the Eisenstein series of the type that are conjectured to define the non-perturbative low energy effective action in type II string theory [15,23].
We denote i, j . . . the indices in the fundamental of SU (8), and the covariant derivative in

N = 16 supergravity in three dimensions
In three dimensions, the duality group is The support of the E 8(8) (Z) invariant Eisenstein functions conjectured to define the low energy effective action in type II string theory [23], on BPS instantons in the decompactification limit [24], indicates that they must indeed satisfy to the differential equations (2.32), (2.33) such that

The ∇ 4 R 4 invariant in eight dimensions
In this paper we investigate the second order corrections of type S (5) ∼ E∇ 4 R 4 + . . . , which can appear in N = 2 supergravity in eight dimensions We denote i, j the SU(2) indices, a, b the vector SO(1, 7) indices, and α, β andα,β the Weyl spinor indices of positive and negative chirality, respectively. The field content of the theory in the linearised approximation is summarised in figure 2, [29]. We will perform this analysis within the superform formalism defined in [33,34]. In this context, a supersymmetry invariant modulo the classical equations of motion is defined as the integral as well as The explicit action of the covariant derivative and the torsion components have been computed up to mass dimension 3/2 in [26]. The complete set of equations (3.4) fixes uniquely the components L (8−m−n,m,n) up to d−exact terms. But it is enough to enforce some of them to determine the differential equations satisfied by the function of the scalar fields, as was shown for the R 4 type invariant in [26]. Here we will extend these results for one class of ∇ 4 R 4 type invariants.

Invariant in the linearised approximation
In the linearised approximation, the scalar superfields are defined as the chiral superfield W of U(1) weight −4 and the isospin 2 real superfield L ijkl that satisfies to the constraint [29] from which it follows that D 1 α (L 1111 ) 2+nW 2+m = 0 .

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One can therefore define a supersymmetry invariant in the linearised approximation, as where the coefficients are not specified, and one understands that the terms in W m−k (L 1111 ) n−l always vanish for k > m or l > n. However, this construction cannot be extended to the non-linear level because of the torsion terms that prevent the derivatives D 1 α to define vector fields closing among themselves in harmonic superspace. The analysis of these linearised invariants is nonetheless very useful to understand the structure of the corresponding invariant in the full non-linear theory. Considering a linearised invariant defined for an arbitrary analytic function F of L 1111 (which we write L for simplicity) andW , we havē are densities of order 4 + p + q in the fields, that do not depend on the naked scalar fields uncovered by a space-time derivative, as for example According to this structure, we expect the non-linear invariant to decompose in the same way in components of U(1) weight multiple of 4 and even isospin, such that , for p prositive. These superforms must satisfy to covariant differential equations in superspace in order for the complete superform L to be d-closed. BecauseŪ −2p F p[4q] are tensor functions of the scalar fields, the only covariant quantities that can enter these equations are the scalar field momenta superforms P,P and P ijkl . If we assume that there is a unique superform L (4p)[4q] for given p and q, as suggested by the linearised analysis, the most general linear equation consistent with U(2) representation theory is determined up to a rescaling of these superforms as = a p,q P [2] ij ∧ L (4p)[4q−2]ij + b p,q P ijkl ∧ L (4p)[4q]ijkl + c p,q P ∧ L (4p+4) [4q] (3.14)

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for some coefficients a p,q , b p,q , c p,q . In this notation [4q] refers to 4q symmetrised SU (2) indices that are not written explicitly, and identically for a partition [2][4q − 2], etc. . . The closure of the covariant derivative implies moreover the integrability condition [26] This equation admits for general solution for some integration constants s and s ′ . It is natural to define a normalisation of the superform such that the complex conjugate forms do appear with the same coefficient, such as to make manifest the reality condition on the superform. Therefore the definition (3.16) holds for strictly positive p only, whereas we will have for p = 0 and the complex conjugate condition for strictly negative −p, i.e.
The range of p can only be bounded if there is a minimal p solution to such that the exterior differential of the superform set to zero indeed vanishes. This is clearly possible if and only if s is an integer such that p min = 1 − s or s. For simplicity we will assume that s is indeed a strictly positive integer (we will eventually prove that s = 2). Because it is possible in principle to have L (4p)[4q] = 0 for all p < s, but this is generally not the case, and we will see that for the ∇ 4 R 4 , the gradient expansion rather stops at p = 1 − s.

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Using the explicit exterior derivative (3.16), one finds that the closed superform is necessarily defined in terms of a unique function such that (3.21) and the function must moreover satisfy to In the linearised approximation, (3.16) reduces to which is automatically satisfied using the definition (3.11) and In the next section we will consider the full non-linear superform, concentrating attention on the terms of maximal weight with respect to U(1) × SU(2). This will permit to determine the value of the integration constants s and s ′ . Considering the possible terms allowed by representation theory, one obtains that the components of maximal weight are uniquely fixed up to an overall coefficient as [24] (λ 8 ) [24] , [3] α , L (56) [24] (8,0,0) ∝ (χ 16 )(λ 8 ) [24] , (3.25) where there is always a unique way to define a Lorentz invariant such that the contraction of the indices should not be ambiguous. All these terms already appear in the linearised invariants as depicted in (3.9), suggesting that they multiply the corresponding derivative of the functionD 6+k D 14−k [56−4k] E s,s ′ for k = 0 to 8, as anticipated in (3.21), see figure 3. However, in eight dimensions it is not true that all linearised invariants can be written as harmonic superspace integrals, and it is not clear if all linearised invariants do extend to full non-linear invariants. Therefore one cannot rely blindly on the linearised analysis, and we will not assume the closed superform defining the invariant to admit the gradient expansion (3.21) in the following section. Our computation will retrospectively confirm that the structure of the invariant is indeed the one suggested by the linearised analysis, and we will be able to conclude that the invariant admits indeed the gradient expansion (3.21) for s = 1 and s ′ = − 1 2 .

Constraints on highest R-symmetry weight terms
We will consider a completely general ansatz for the components of the closed superform 2q] are tensor functions of the scalar fields of U(1) weight −4p and isospin q, whereas a labels the possible SL (2) (8−m−n,m,n) have U(1) weight m − n + 4p and isospin j such that q − m+n 2 ≤ j ≤ q + m+n 2 , depending of the specific tensor structure for the symmetrised pairs of fermionic indices. Note that we do not assume q to only take even values, as suggested from the linearised analysis in the preceding section, although we will eventually conclude that it must indeed be even.
We will concentrate on the maximal mass dimension components of the d-closure equations (3.4), i.e.

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In order to simplify further these equations we will moreover restrict ourselves to the analysis of the terms of highest U(1) weight and carrying the maximal amount of symmetrised SU(2) indices, which correspond to the terms with maximal value of p and q in (3.26).
Let us consider first the components of L (8,0,0) , that are by construction Lorentz scalars of mass dimension 12. Each I is therefore a Lorentz scalar of mass dimension 12, U(1) weight 4p and isospin q. The terms of maximal weight depends only on the fermions fields, because they have the lowest mass dimension while carrying the largest weight representation. However, Fermi statistics requires to limit the number of them to maximise the weight. For example, there are only eight different λ 111 α , so a term in (λ 9 ) α will necessarily includes at least one λ 112 α , such that the maximal SU(2) representation one obtains for an octic term is (λ 8 ) [24] is of isospin 12, while for nine fermions one only gets (λ 9 ) [25] α of isospin 25 2 . A term with ten fermions (λ 10 ) [26] ab has therefore the same mass dimension and U(2) representation as a term in (λ 8 ) [24]F [2] ab . The same argument applies to the sixteen fermion fieldsχ iα . The terms of maximal weight involving scalar momenta can always be eliminated in favour of lower weight terms through the addition of a d-exact term, and will therefore be disregarded in our analysis.
The maximal weight terms are therefore the terms of order 24 in the fermions depicted in (3.25). We shall here concentrate on the two monomials I 1 (24)[56] = (χ 8 ) [8] (λ 8 ) [24] (λ 8 ) [24] , The next-to-maximal contribution with a lower isospin could have been I a (24)[54] , however the only possible terms must also be of order 24 in the fermions and one checks that there is no Lorentz scalar in this representation. Indeed, lowering the isospin of one of the octic monomial (χ 8 ) [8] , (λ 8 ) [24] or (λ 8 ) [24] requires to consider only seven among eight of the Spin(1, 7) indices to be antisymmetrised, such that they cannot be scalars. The same reasoning applies to the terms of order nine and seven in (χ 9 ) [7]α and (λ 7 ) [21] α , respectively, such that there is no candidate components I

28[52]
(8,0,0) either. It is also clear that one cannot reduce the U(1) weight by 2 only, since the difference of the U(1) weights of the fermion fields of identical chirality is zero modulo four.
We must also consider the other components of the superform, corresponding to the terms involving naked gravino fields in the formalism in components. The superform component L (7,1,0) is in the Spin(1, 7) representation tensor product of the 7-form times the positive chirality spinor representation, i.e.
We will not need to specify any of these terms in our analysis. The L (6,1,1) component is of mass dimension 11 and U(1) weight 0. Chirality implies that the highest weight terms one can build in the relevant representations of the Lorentz group are inχ 5 λ 9λ8 orFχ 4 λ 8λ8 , as for example γ (λ 9 ) [25] δ (λ 8 ) [24] , (3.36)

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and other terms of the same weight, such that the ansatz is of the form .
Let us now describe the action of the fermionic covariant derivatives on a general tensor functionŪ −2p F a 4p [2q] . Since the tensor transforms covariantly with respect to U(2), one obtains where the field T is the unit disk coordinate on SL(2)/SO (2), and U is the U(1) weight −2 variable satisfying to The momentum components were derived in [26] to be where we denote with (i 1 . . . i 2n ) the symmetrisation of 2n indices, while the numbers into brackets sum up to the total number of symmetrised indices i 1 , i 2q that are not written JHEP03(2015)089 explicitly. One understands that the uncontracted indices of the terms in DF 4p[2q] are symmetrised first, and all the indices i 1 , i 2q that are not written explicitly are symmetrised afterward.
We are now ready to solve equation (3.27) in terms of our ansatz (3.31), (3.33), (3.35), i.e.Dα We shall only consider the mixings between the terms involving tensor functions of U(1) weight lower or equal to −24 and of isospin 28. As a consequence of (3.35), there is no mixing contribution coming from T (1,0,1) (0,0,1) L (7,0,1) at this weight, and these terms can be disregarded. However, there are contributions from D (1,0,0) L (7,0,1) , because the application of the derivative to the tensor functions can increase the weight. Those mixings are nonetheless either proportional toP or to P ijkl , and we can neglect them as long as one does consider terms involving explicitly the scalar momenta. Disregarding these terms will allow us to simplify drastically the computation in the following. Because the maximal weight terms in the ansatz (3.33) are associated to tensor functions of U(1) weight −20 and isospin 28, the terms in T (1,0,1) (0,1,0) L (7,1,0) do not contribute either in the computation. Because the isospin 28 terms in L (6,1,1) and L (6,0,2) are all associated to tensor function of U(1) weight greater than −20, we can also disregard the terms in T (2,0,0) (0,1,0) L (6,1,1) and We get therefore that equation (3.44) simplifies drastically tō when restricted to the terms involving tensors functions of isospin 28 and of U(1) weight less or equal to −24.
In order to solve (3.45), it will be convenient to define an explicit basis of fermion fields monomials as follows

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Let us first consider the action of the fermionic derivativeDα i on the tensor function [8] (λ 8 ) [24] (λ 8 ) [24] (3.47) where the dots state for lower isospin terms in (λ 9 ) [23] and (λ 9 ) [21] that we neglect at this order. The first term can only be canceled by the one coming from the application ofDα i onŪ −14 F 2 28[52] , leading tō α (λ 8 ) [24] (λ 8 ) [24] + . . . (3.48) where the dots state for lower isospin terms that we neglect at this order. We conclude that the two tensor functions must be related through Note that in principle the two tensor functions F 24[52] could differ by an inhomogeneous term such thatDD [4] c 24[52] = 0. However one argues that the equation admits no solution, and such inhomogeneous term can only be a holomorphic tensor on the symmetric space SL(2)/SO(2), i.e. c 2 (T,T ) = (1 − TT ) −12c 2 (T ). Considering other constraints from supersymmetry one would get to the conclusion that such inhomogeneous terms must vanish because the supersymmetry constraint is linear in the tensor functions. For simplicity we shall assume from the beginning that all such terms vanish.

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where D (i [2] j F 1 24[55])j is of isospin 58, and therefore cannot be canceled by any other term since there is no components with a tensor functionŪ −12 F a 24[54] as we discussed above. Therefore we conclude that The first term vanishes because the commutator of two covariant derivative involves the contraction of three of their respective indices, such that and therefore Now it remains to cancel the second term in (3.52), for which we will need to consider the action of the covariant derivative on next to maximal weight terms (3.30), i.e.

20[56]
F [2] ab (χ 6 ) ab [6] (λ 8 ) [24] (λ 8 ) [24] +Dα i Ū −10 F 4 20[56] (χ 7 ) [7] α (λ 8 ) [24] (λ 9 )α [25] +Dα i Ū −10 F 5 20[56] (χ 6 ) ab [6] (λ 10 ) [26] ab (λ 8 ) [24] + · · · = 0 (3.56) where we have already computed the two first terms to simplify to the second term in (3.52). The corresponding tensor function has U(1) weight 28 and isospin 56. Therefore it also gets contributions from the action of the covariant derivative on tensor functions of U(1) weight 28 and isospin 52. However, there is a large number of terms like that, and analysing them all would be rather cumbersome. In order to bypass this difficulty, we remark that their contributions only arise as an isospin 56 tensor function times a combination of the fields of isospin 55, whereas the term we want to cancel in (3.52) includes a combination of the fields of isospin 57. Therefore we will be able to neglect the contribution from the isospin 52 terms in L (8,0,0) . In the same way, the action of the covariant derivative in the order 24 term in the fermions of maximal isospin decomposes into a term of isospin 57 and a term of isospin 55 that we will neglect, i.e.
After investigation, it turns out that the only terms that can contribute to cancel the terms of isospin 57/2 inχ 8 λ 9λ8 in (3.69) are the ones coming from the action of the covariant derivative on the fermions of the maximal weight term itself. Using the action of the covariant derivative on the fermion λ andλ (3.58), as well as This equation is one of the main results of this section, that will allow us to determine the differential equation satisfied by the function that defines the invariant. To summarise the results obtained so far, the expression of L (8,0,0) subject to these constraints takes the following form

The gradient expansion of the invariant
The structure of the maximal weight terms of L (8,0,0) derived in the preceding section 3.73, together with the constraint (3.65) reproduces precisely the structure of the invariants defined in the linearised approximation, such that we conclude that we can indeed trust the gradient expansion (3.21). Extending the computation of the last section indeed necessarily implies that the tensor function F 20[52] is itself determined as the covariant derivative of a lower weight tensor functions, according to the constraints implied by supersymmetry in the linearised approximation (3.11). We conclude therefore that there is a function E (2,1,0) of the complex scalar field T and the five scalars t µ parametrising SL(3)/SO (3), such that Note that the general solution to this equation can be obtained from an anti-holomorphic function F[τ ] and its complex conjugate as where τ is the upper complex half plan coordinate τ = i 1−T 1+T . One computes that which implies that the terms in D n E (2,1,0) only depend on the holomorphic function of τ for n ≥ 2, whereas the terms inD n E (2,1,0) only depend on the anti-holomorphic function F[τ ].
to prove that As explained in [26], this equation moreover implies that E (2,1,0) is an eigen function of the Laplace operator which is precisely equation (3.22) for s ′ = − 1 2 . The closed-superform defining the invariant, admits therefore the gradient expansion for an arbitrary solution to (3.93) and (3.99). Of course one has the complex conjugate invariant, defined such that , (3.101) and the associated function multiplying ∇ 4 R 4 is E (2,1,0) + E (2,0,1) , which is defined to be a real function of τ andτ . This is consistent with the appearance of the threshold function in the low energy effective action of type II string theory compactified on T 2 [11,17]. The Eisenstein function E [s,0] satisfies in general to the differential equation [26] D ij pq D klpq E s = − 4s − 3 12

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such that E [− 1 2 ,0] is indeed a solution to (3.99), whereas E [2] solves (3.90). Using the explicit expansion of the Eisenstein series E [2] , (3.104) one finds indeed that E [2] = E [2] + E [2] for the complex function However this complex function is not modular invariant, and in order for the supersymmetry invariant to preserve SL(2, Z), it is necessary that such that the whole invariant only depends on the gradient expansion of the modular invariant function E [2] . This reality condition is indeed compatible with the linearised analysis, because there is only one linearised invariant for each values of p and q, and (3.107) must therefore be satisfied in the linearised approximation. We know indirectly that this reality condition must be satisfied at the non-linear level, because the term in 2ζ(4)τ 2 2 lifts to type IIA supergravity in ten dimensions [11], where it is known to appear in the 2-loop string theory effective action [8], which is by construction invariant with respect to the B field gauge transformations.

Decompactification limit in lower dimensions
We have derived in the last section the structure of the chiral ∇ 4 R 4 type invariant in eight dimensions, however the same analysis does not apply directly to the second real ∇ 4 R 4 type invariant (2.10). To understand the two invariants, we are going to analysis the corresponding invariant obtained by dimensional reduction in four dimensions. We will see that these two invariants are related through the action of E 7(7) in four dimensions. Solving the differential equation satisfied by the function E (4,2,2) defining the ∇ 4 R 4 type invariant (2.28) in four dimensions in the decompactification limit, we will indeed obtain that it lifts to the two independent invariants (2.10) in eight dimensions.
We must warn the reader that considering explicit decompositions of E 7(7) and SL(5) forced us to use the same indices for various representations. Each subsection in this section uses a different definition of the indices that is recalled in the beginning.

R 4 and ∇ 4 R 4 type invariants in four dimensions
In this subsection we shall review the results displayed in section 2.5, which were originally derived in [26]. In N = 8 supergravity the scalar fields parametrise the symmetric space JHEP03(2015)089 where SU c (8) is the quotient of SU(8) by the Z 2 kernel of the antisymmetric rank two tensor representation, and the covariant derivative D ijkl on E 7(7) /SU c (8) in tangent frame are in the rank four antisymmetric complex selfdual representation of SU (8), with i, j, k, l running from 1 to 8 are in the fundamental representation of SU (8).
In four dimensions there is a bijective correspondence between the supersymmetry invariants and the linearised invariants defined as superspace integrals in harmonic superspace, due to the enhanced superconformal symmetry SU(2, 2|8) of the theory in the linearised approximation [30,35].

The R 4 type invariant
One defines R 4 type invariants in the linearised approximation using harmonic variables u r i and ur i parametrising SU(8)/S(U(4) × U(4)), where r runs from 1 to 4, andr from 5 to 8. One defines the G-analytic superfield [39] such that one can define the supersymmetric Lagrangians  Using the bijective correspondence one concludes that the R 4 type invariant is unique in four dimensions, and admits the following gradient expansion in a function E (8,4,4) L[E (8,4,4) ] =
The ∇ 4 R 4 type invariant One defines ∇ 4 R 4 type invariants in the linearised approximation using harmonic variable parametrising SU(8)/S(U(2) × U(4) × U(2)). We define the G-analytic superfield [39] W rs = u 1 i u 2 j u r k u s l W ijkl , (4.14) where r, s are now SU(4) indices running from 1 to 4 and W rs is in the [0, 1, 0] representation. Since SU(4) ≃ SO(6), W rs is a vector of SO (6), and the general monomials in W rs are the symmetric traceless monomials times an arbitrary power of the scalar product of W rs with itself. The general invariant Lagrangian is defined as the harmonic superspace integral over 24 Grassmann variables of such monomials as Using the bijective correspondence, one concludes that the non-linear invariant admits the following gradient expansion in the function E (8,2,2) L[E (8,2,2) ] = One computes similarly that (4.18) implies Using the property that the function satisfies to all (4.18), (4.19), (4.20), one gets the following integrability condition in the [0, 1, 0, 0, 0, 1, 0], where we only used (4.19) in the last step. Because the function E (8,2,2) does not satisfy to the quadratic constraint (4.7), we conclude that it must satisfy instead ∆E (8,2,2) = −60E (8,2,2) . These constraints can be rewritten in terms of the e 7(7) valued differential operator D 56 and D 133 in the fundamental and the adjoint representations, respectively, as [26]  One can define solutions to these differential equations in terms of Eisenstein series defined as constrained Epstein series in the fundamental representation [15]. Let us consider a rank one charge vector Γ in the 56 of E 7 (7) such that the second derivative of the quartic invariant restricted to the adjoint representation vanishes. Acting with the scalar field one obtains that the central charges Z(Γ) ij = V ij I Γ I satisfy to

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The action of the covariant derivative on the central charges gives One computes then that with |Z| 2 = Z ij Z ij . Using moreover the intermediate step one computes that One gets therefore a solution to the second order equation (4.11) associated to the R 4 type invariant for s = 2. One computes then that and therefore the third equation (4.18) is automatically satisfied by |Z| −2s . One computes moreover (4.32) One concludes therefore that the function |Z| −2s solves to the cubic equation (4.8) for s = 4. In general one has moreover ∆|Z| −2s = 3s(s − 9)|Z| −2s . (4.33) One formally obtains E 7(7) (Z) invariant functions by considering the sum over all integral charges satisfying to the rank one constraint However this series does not converge for s ≤ 9, which includes the cases of interest. Using the theorem of [40], the rank 1 integral charge vectors Γ are in the E 7(7) (Z) orbit of an integer element of grad 3 in the parabolic decomposition of e 7(7) e 7(7) ∼ = 27 (−2) ⊕ gl 1 ⊕ e 6(6) (0) ⊕ 27 (2) 56 i.e. that {Γ ∈ Z 56 | I ′′ 4 (Γ)| 133 = 0} ∼ = Z * × E 7(7) (Z)/ E 6(6) (Z) ⋉ Z 27 . (4.36) Using the property that the rank 1 charge vector (with unit grater common divider of all components) defines a character of E 7(7) whose restriction to the Cartan subgroup is the exponential of the generator such that these functions indeed satisfy to the differential equation associated to the R 4 and ∇ 4 R 4 type invariants consistently with the conjecture that they define the exact low energy effective action in type II string theory [15,23].

Decompactification limit to seven dimensions
Any supersymmetry invariant in seven dimensions, dimensionally reduces to a well defined supersymmetry invariant in four dimensions. It follows that the structure of the invariants in seven dimensions must be compatible with the differential equations we have derived in four dimensions. In this section we will solve these differential equations in the parabolic gauge associated to the dimensional reduction from seven to four dimensions, to exhibit the differential equations satisfied by the seven-dimensional scalar fields. But before to do this, let us review shortly some properties of the theory in seven dimensions.

Maximal supergravity in seven dimensions
In seven dimensions the scalar fields parametrise the symmetric space SL(5)/SO(5), and the double cover Sp(2) of SO(5) is the R-symmetry group. The SL(5) representative V is defined such that it transforms with respect to rigid SL(5) on the right and local Sp(2) on the left where i, j, · · · = 1, . . . 4 are the indices in the fundamental representation of Sp (2). The theory is defined in the linearised approximation in terms of the real scalar superfield L ij,kl L * ij,kl = Ω ip Ω jq Ω km Ω ln L pq,mn , The R 4 type invariant can be defined in the linearised approximation in harmonic superspace, using harmonic variables u r i , u ri parametrising Sp(2)/U(2), with r = 1, 2 of U(2) [31], such that the superfield

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One ∇ 4 R 4 type invariant can be defined in the linearised approximation in harmonic superspace, using harmonic variables u 1 i , u r i , u 4 i parametrising Sp(2)/(U(1)×Sp(1)), with r = 1, 2 of Sp(1) [31], such that the superfield satisfies to the G−analyticity constraint One can write generic invariants [24,16] , We define the nilpotent component of the E 7(7) coset representative in the fundamental representation JHEP03 (2015)089 and its semi-simple component 59) such that the coset representative is (4.60) We use the notation that the SL(5) indices K, L and the SL(3) index C are contracted on the right-hand side through the left action of E 7(7) , whereas I, J and A are contracted on the left-hand-side through the right action of E 7 (7) . The same convention is used for the SO(3) indices a, b contracted on the left and c, d on the right, and for the SO (5) indices i, j contracted on the left and k, l on the right, through the respective right and left actions of SU (8). We apologise to the reader for using now on i, j as vector indices of SO(5), whereas we were using them as Sp (2)  and the metric on E 7(7) /SU c (8) reads where the matrices M IK = V i I V iK and µ AB = υ a A υ aB are symmetric by construction, and ∇A, ∇B, ∇C are defined in (4.56). The derivatives dual to these differentials satisfying to We are interested in finding functions of the scalar fields defining invariants in four dimensions that lift to seven dimensions. Therefore we will consider functions that depend only on the seven-dimensional scalar fields t parametrising SL(5)/SO(5) and the Kaluza-Klein dilaton φ that must appear at a specific power determined from for the invariant to be diffeomorphism invariant in seven dimensions. With this restricted ansatz for the function, the differential operator D 56 = E µ (∂ µ − B µ ) is block diagonal in the decomposition (4.57), i.e.
Now we want to compute the action of the second order derivative D 2 on a function of φ, t defined on GL(5)/SO (5). Note that the spin-connexion decomposes into where the matricesM are also constant tensors. Defining D (0) µ , the covariant derivative with respect to the grad zero so(3)⊕so(5) spin-connexion, one obtains therefore that (4.67) simplifies to

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On a function of φ, t on GL(5)/SO(5), one computes in this way that D 2 56 reduces to We can use this expression to solve the differential equation (4.13) for a function E (8,4,4) = e aφ E(t) on GL(5)/SO (5). These equations give (4.73) Using the commutation relation one computes that Substituting (4.73) in (4.75) and taking the trace over i and k one obtains and therefore a = −18 only is possible, as required for a R 4 type invariant (4.65). Therefore we obtain This function is one example of the generic class of functions for which the second order derivative restricted to the [2,0] vanishes, and we write them where the notation refers to the property that the Eisenstein series E [s,0,0,0] satisfies to these equations whenever it converges. The result is consistent with the conjectured exact low energy effective action in type II string theory. We just note here that the general solution depending on R * + ×SL(5)/SO(5)× SL(3)/SO (3) is such that one should have the expansion of the Eisenstein series at large volume modulus V (T 3 ) = e −6φ , We will now analysis the differential equations (4.25) relevant for the ∇ 4 R 4 type invariant. For this, we need in particular to compute the third order differential operator D 3 56 (4.82) One computes that on a function of φ and t on GL(5)/SO (5), it reduces to where the dots stand for the conjugate representations that are identical to the ones written explicitly up the sign. Using the same ansatz E (8,2,2) = e aφ E(t), one obtains combining these equations that

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which implies that the tensor structure of the first term must necessarily reduce. Considering the general solution of such a system depending on four variables associated to a maximal abelian subgroup of SL(5), we find that there is a two-parameter family of equations with this structure, such that for s = 5 2 − s ′ . Again the notation we use refers to the property that the corresponding Eisenstein series E [s,s ′ ,0,0] and E [0,0,s,0] satisfy to the same equations when they converge. This way we find only three independent solutions, i.e. We already see that the two first solutions correspond to the seven-dimensional ∇ 4 R 4 type invariant, whereas the second would correspond to the ∇ 6 R 4 invariant. The first equation in (4.25) is indeed also satisfied for the ∇ 6 R 4 type invariant that descends from ten dimensions, and the type IIB 3-loop invariant in ten dimensions indeed defines a function solving (4.85) for s = 4, s ′ = − 1 2 . Now we want to check the second equation in (4.25). However the computation of the commutator terms of the M matrices (4.82) becomes rather tedious in the adjoint representation, and we will only fix the coefficients on the general covariant ansatz by demanding that the knows solutions indeed satisfy the equations, i.e.

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It appears that the system of equations for the coefficients is over-constrained, and it is a non-trivial check that one can indeed find a solution. The positive grad component (4.55) of the differential operator D 133 restricted to a function on GL(5)/SO (5) is and we obtain after calibrating the coefficients such that (4.88) are all satisfied that The dots stand for the zero and negative grad components. In the same way, one finds that the component in the adjoint of SL(5) of D 3 133 − λD 133 admits the two components

threshold function in seven dimensions
Consistently with the analysis in [26], we find that there are only two classes of ∇ 4 R 4 type invariants in seven dimensions. The first class is associated to the linearised invariants discussed above, with the gradient expansion (4.54), with the function E (4,1) = E [0,0, 5 2 ,0] , that admits the correct gradient expansion as a consequence of (4.86). This equation indeed implies that the order n derivative is only non-vanishing in the representations [4p, 2q] for 2p + q ≤ n, and is related to lower order derivatives when 2p + q < n. The second solution E [ 5 2 ,0,0,0] was shown in [26] to correspond to a chiral invariant in six dimensions, which explains that the corresponding seven-dimensional invariant cannot be defined as a harmonic superspace integral in the linearised approximation.
The solution to (4.80) can be defined in terms of a vector Z i = V i I n I such that such that the relevant function to define the string theory Wilsonian action in (2.2) is As explained in [26], this additional function defines a consistent anomaly for the continuous SL(5) Ward identity, because the sl 5 variation of this function solves (4.80) for s = 5 2 by construction, whereas the function itself does not. We therefore conclude that this contribution comes from the 2-loop supergravity amplitude [38]. Similarly, the solution to (4.86) can be defined in terms of an antisymmetric tensor Z ij = V -1  (Z ij (n)Z ij (n)) −s , (4.102) diverges at s = 5 2 , and one must consider the regularised series [11] E [0,0, In the same way, the additional function defines a consistent anomaly for the continuous SL(5) Ward identity. The specific m IJ , m I that define the logarithm function of the scalar appearing in the 2-loop supergravity amplitude must depend of the specific parametrisation of the symmetric space SL(5)/SO (5), and this ambiguity amounts to a choice of renomalisation scheme.

Decompactification limit to eight dimensions
To make link with the analysis of section 3, we will now solve equations (4.80) and (4.86) in the parabolic gauge associated to the large compactification radius limit, with the graded decomposition sl 5 ∼ = (2 ⊗ 3) (−5) ⊕ gl 1 ⊕ sl 2 ⊕ sl 3 (0) ⊕ (2 ⊗ 3) (5) . (4.107) We consider therefore the SL(5) representative V in this gauge such that with the indices α, β running from 1 to 2 of the local SO(2), i, j from 1 to 2 of the rigid SL(2), a, b from 1 to 3 of the local SO(3) and I, J from 1 to 3 of the rigid SL(3). We decompose the Maurer-Cartan form into symmetric and antisymmetric components dVV −1 = P + B (4.109) These results are in agreement with the constant term formula for the corresponding Eisenstein series [11], and one computes using Poisson summation formula Note that the dependence in the specific integral vector p i = rp ′i does not depend on the scalar fields, and defining p ′i , the solution to (4.121) such that p ′1 and p ′2 are relative primes, one has which is also an exact solution to (4.116). For s = 5 2 , we get the function associated to the invariant that cannot be written as a harmonic superspace integral in the linearised approximation E = e −10φ E 5 2 (t) + E 1 (τ ) . (4.129) The regularised Eisenstein function decomposes as [11] E [ 5 2 ,0,0,0] = e −10φ E [1,0] (t) + 4π 3Ê [1] (τ ) + 8π 2 φ  In the same way we get that the two sides of the first equation must vanish separately, such that either the function does not depend on the complex scalar field τ parametrising SL(2)/SO(2) and E = e −12sφ E(t) or E = e 4(2s−5)φ E(t) , or E = e −(2s+5)φ E(τ, t) and ∆ SL(2) E(τ, t) = (2s − 1)(2s − 3) 4 E(τ, t) .

Decompactification limit to ten dimensions
The decompactification limit to type IIB supergravity in seven dimensions can be obtained in the same way as in (4.108) for the inverse matrix However and there is no solution that lifts to ten dimensions such that no E [0,0, 5 2 ,0] ∇ 4 R 4 type invariant in seven dimensions does lift to type IIB supergravity.
To understand the decompactification limit to IIA supergravity, it is more convenient to take an explicit basis for the diagonal elements of the matrix V ∈ SL(5), i.e. with arbitrary coefficients, which shows that the eight-dimensional threshold E 3 2 (t)R 4 includes both type IIA and IIB tree-level R 4 thresholds, and the 1-loop type IIB R 4 threshold, whereas E 1 (τ )R 4 includes the type IIA R 4 threshold that lifts to eleven dimensions. Similarly, the only solutions that lift to ten dimensions are with arbitrary coefficients, which shows that the eight-dimensional threshold E 5 2 (t)∇ 4 R 4 includes both type IIA and IIB tree-level ∇ 4 R 4 thresholds, and the 2-loop type IIB ∇ 4 R 4 threshold, whereas E 2 (τ )E − 1 2 (t)∇ 4 R 4 includes the type IIA ∇ 4 R 4 threshold. In type IIB, supersymmetry implies a second order Poisson equation on SL(2)/SO(2), such that the two invariants must be in the same SL(2) representation, whereas in type IIA supergravity there is only one scalar, and they are independent.