Saturating unitarity bounds at U-duality symmetric points

It has recently been shown that the leading Wilson coefficient in type II string theory can take (almost) all values allowed by unitarity, crossing symmetry and maximal supersymmetry in D = 10 and D = 9 dimensions. This suggests that string theory might define the unique consistent quantum theory of gravity with maximal supersymmetry. We study the minima of the leading Wilson coefficient in D = 6, 7 and 8 dimensions and find the global minimum at the point in moduli space with maximal symmetry. The minimum value turns out to always be negative for D ≤ 7.


Introduction
Superstring theory on asymptotically flat spacetimes defines a large number of scattering theories including gravitons. In the low energy limit, the scattering amplitudes behave as quantum field theory amplitudes. In spacetime dimensions larger than four, the corresponding S-matrices are well defined and must therefore satisfy the usual quantum field theory conditions of analyticity, crossing symmetry and unitarity. A string theorist may wish that all consistent quantum gravity S-matrices could be obtained in superstring theory. Assuming this is the case, one can in principle derive constraints on consistent effective field theories that would not be visible in perturbative quantum field theory. This is often used in the swampland conjectures, see [1] for a review. A more humble conjecture, and maybe more realistic, is that all consistent quantum gravity theories with extended supersymmetry could be formulated as superstring theories. In this paper, we wish to study this question for maximally supersymmetric theories.
Type II superstring theory on the Cartesian product of D-dimensional Minkowski spacetime and a compact torus admits the maximal number of supersymmetries. The low energy effective theory is then maximal supergravity in D spacetime dimensions and the scattering amplitudes of massless states can in principle be described in supergravity using the Wilsonian effective action obtained by integrating out massive string states. In practice one first computes the perturbative string amplitudes and compare them with the supergravity amplitudes with operator insertions to deduce the Wilsonian effective action [2][3][4]. The effective action is highly constrained from supersymmetry and U-duality [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]. The leading Wilson coefficient is completely determined by supersymmetry, U-duality and anomaly cancelations, provided one requires consistency in the decompactification limits.
It is commonly believed that type II superstring theory on a torus is the unique consistent quantum theory with maximal supersymmetry. One may therefore expect that the superstring S-matrix of massless states in a maximally supersymmetric vacuum covers all possible S-matrices satisfying analyticity, crossing symmetry, unitarity and maximal supersymmetry, as well as all the required anomaly cancelations. These consistency conditions can be analysed within the S-matrix bootstrap initiated in [23][24][25]. A lower bound on the leading Wilson coefficient was computed in [26,27] in maximal supergravity in D = 9, 10, 11 dimensions, using the constraints from the two-to-two S-matrix. The unitarity bound was found to be close below the minimal value these Wilson coefficients can take in string theory and eleven-dimensional supergravity. The unitary bounds derived in [26,27] are not sharp since they neglect non-elastic contributions to the optical theorem. Only integrability in two dimensions provides non-trivial examples of purely elastic S-matrices [28]. So one may conclude from this result that string theory does indeed seem to saturate the sharp unitarity bound.
The case of D = 11 is particular because there is no moduli and the leading Wilson coefficient is a fixed number in M-theory. It is determined by the cancelation of the M5-brane anomaly [29,30]. One may argue that this anomaly inflow argument is independent of string theory [31], so that there should not be any consistent theory in eleven dimensions with a different value of the leading Wilson coefficient. Supersymmetry also fixes the next-to-leading Wilson coefficient and the first Wilson coefficient to be determined by unitarity and crossing symmetry multiplies In D ≤ 10 the leading Wilson coefficient is a function of the moduli and can take arbitrary large values, such that all the values consistent with the S-matrix unitarity bound seem to be covered by the string theory amplitude in D = 9 and D = 10 dimensions.
The analysis of [27] can in principle be generalised to all spacetime dimensions 5 ≤ D ≤ 11. However, one must modify the amplitude ansatz in dimension D ≤ 8 to include the contribution from the supergravity one-loop amplitude in the low energy limit. In D = 5 one would furthermore need to include the contribution from the two-loop amplitude. Although it is technically challenging to include the one-loop correction in the S-matrix bootstrap method, it is a priori doable, see for example [32].
On the string theory side one needs to find the minimum value of the leading Wilson coefficient. It is a maximal parabolic Eisenstein series of the U-duality group in D dimensions [6,11,17,33,34]. In particular for the type IIB superstring amplitude in ten dimensions it is a real analytic Eisenstein series 2ζ(3)E 3/2 (S) on the upper complex half-plane [6]. The minimum is known to be at the Z 3 -symmetric point S = 1+i √ 3 2 [35]. In dimension D ≤ 8 the leading Wilson coefficient is again a specific Eisenstein series of higher rank groups associated to their minimal automorphic representation [16,17]. The SL(3) Epstein series relevant for the leading Wilson coefficient in D = 8 dimensions has been studied numerically in [36]. The global minimum was found to be at the point in moduli space defined by the unimodular symmetric matrix proportional to the Gram matrix of the lattice A 3 . Finding minima of Eisenstein series is generally an open problem, and is the subject of this paper.
We provide strong evidence that the global minimum of the SL(N ) Epstein series Ep N s (H) is obtained at the unimodular symmetric matrix H = H dlp proportional to the Gram matrix of the densest lattice sphere packing in N dimensions for all s > N 4 . It is proved for asymptotically large s in [37]. We prove that the densest lattice sphere packing Gram matrix H dlp is a local minimum of the Epstein series for all s and N ≤ 8. We identify candidates for local minima as symmetric points and we checked that the lowest minimum is at H dlp numerically. For N = 5 we study the leading Wilson coefficient in D = 7 dimensions on several surfaces containing H dlp in moduli space and find each time that it is a global minimum on these surfaces.
We generalise this analysis to the Spin(5, 5) Eisenstein series appearing as the leading Wilson coefficient in D = 6 dimensions and find strong evidence that the global minimum is at the W (D 5 ) × W (D 5 ) symmetric point. This point is the analogue of the point of enhanced Spin (10) symmetry in perturbative heterotic string theory on T 5 .
One striking feature is that the leading Wilson coefficient can always be negative in dimension D < 8. This is not in contradiction with unitarity because the leading Wilson coefficient is subleading with respect to the supergravity one-loop correction for D ≤ 8. One therefore expects the Wilson coefficient to possibly be negative, and comparable to the one-loop correction at Planck scale [38].
The Eisenstein series appearing in the Wilson coefficients are absolutely convergent when they are dominant compare to the loop corrections, and defined by analytic continuation when they are subleading [39,40]. It follows that the Wilson coefficient are necessary positive when they are dominant, and can always be possibly negative when they are subleading, consistently with unitarity. We will show indeed that the global minimum of an Eisenstein series is always negative in the critical strip, where it cannot be defined as an absolutely convergent sum.
The paper is organised as follows. In the second section we give some notations and summarise our results. We define a fundamental domain for the various moduli spaces of interest in Section 3. In Section 4 we give the main results leading to the conjectured minimum of Epstein series at symmetric points and in particular prove they are local minima. We discuss the specific case of dimension 8 in Section 5, where the splitting of the string amplitude into analytic and non-analytic pieces is ambiguous due to a logarithmic divergence. In Section 6 we expose numerical checks of our conjecture.

Notations and summary of the results
The four-graviton superstring amplitude on R 1,9−d × T d factorises in the form where A(s, t, u, φ) is invariant under permutations of the three Mandelstam variables and is a function on the moduli space K(E d+1 )\E d+1(d+1) /E d+1 (Z) [41]. We define the Planck length in D = 10 − d spacetime dimensions as and the D-dimensional effective string coupling g D = e ϕ √ υ d in terms of the ten-dimensional dilaton ϕ and the volume In the low energy limit, one can write so that the leading Wilson coefficient ℓ 6 P E (0,0) (φ) is between the one-loop and the two-loop supergravity corrections for D = 6, 7, 8. It is equal to the maximal parabolic Eisenstein series where Λ d+1 is the fundamental weight associated to the electric charges representation in D dimensions and ξ(s) = π −s/2 Γ(s/2)ζ(s) is the completed zeta function. In D ≥ 7 it can be written in terms of SL(N ) Epstein series defined by analytic continuation of the sum 1 With this definition one has in seven dimensions [11]. We conjecture that the global minimum is at the densest sphere lattice packing point D 5 Ep In this paper we use Ep N s to distinguish the Epstein series normalisation used in the original papers [6,11,33] from the Langlands Eisenstein series normalisation E SL(N ) sΛ 1 more commonly used since [17,34].
In eight dimensions one must take into account that the one-loop supergravity amplitude is logarithmically divergent, and so are the Epstein series appearing in E (0,0) (φ). One must therefore introduce an appropriate renormalisation where the renormalised Epstein series Ep and Ep 2 1 are defined as in [34]. The renormalisation scale µ cancels in the complete amplitude, and the specific finite number is determined by our definition of the 1-loop box integral, as we explain in Section 5. Choosing µ = 1/ℓ P , we find the minimum value In eight dimensions it also makes sense to include the next-to-leading Wilson coefficient

Fundamental domain of K\G/G(Z)
In order to determine the minium of an automorphic function on K\G/G(Z), it is useful to find an appropriate fundamental domain F for the action of the arithmetic subgroup G(Z) on the symmetric space M = K\G. Here we assume that G is a simple group of real split form different from E 8 , F 4 or G 2 , and G(Z) is its Chevalley subgroup associated to the weight lattice. This includes in particular the locally symmetric spaces relevant for the type II string theory effective action in dimensions greater than four.
Let us first recall the definition of a fundamental domain. One defines a free regular set F ⊂ M as an open set in M such that any point in M can be mapped under the action of and for any element γ ∈ G(Z) acting non-trivially on M one has It is required that ∂F = F ∖F is of measure zero in M. We then call F a fundamental domain of G(Z) in M. Equivalently one can define F to satisfy There is generally no canonical fundamental domain. A first fundamental domain was introduced by Minkowski for G(Z) = GL(N, Z) acting on SO(N )\SL(N ) [42]. Grenier then defined a different fundamental domain for GL(N, Z) that is easier to generalise to arbitrary simple groups [43]. In this section we shall first review Grenier's construction in the case of SL(N, Z). Then we will show that there is a natural generalisation of Grenier's fundamental domain for all simple groups of split real form but E 8 , F 4 or G 2 . We will finally discuss SO(N, N, Z) in more detail as an example.

Fundamental domain of SL(N, Z)
Following Grenier, we parametrise the symmetric space M = SO(N )\SL(N ) in the Iwasawa decomposition with N − 1 variables The group representative in SL(N ) can then be written as the upper triangular matrix For short we do not distinguish the group representative from the corresponding point in M.
To any element V ∈ M we can associate a positive definite symmetric bilinear form over Z N The Grenier fundamental domain is defined by induction by writing first V in the maximal where x ⊺ = (x 12 , . . . , x 1N ) and V 1 ∈ SL(N −1), and then recursively for each V i ∈ SL(N −i). To avoid taking care of the factors of y, Grenier introduces the non-unimodular symmetric matrix is a symmetric bilinear form over Z N −1 , so its argument n is implicitly (n 2 , . . . , n N ). One defines recursively the bilinear forms and by convention Y 0 [n] = Y [n]. The condition for V to be in the Grenier fundamental domain F of M is defined recursively by the two conditions: 1. Y i [n] ≥ 1, for all non-vanishing n ∈ Z N −i of greatest common divisor gcd(n) = 1.
Grenier proved that one can always restrict the first condition to a finite set of vectors such that this gives a finite number of inequalities. This follows from the property that Y i [n] is positive definite and there is a finite number of lattice points in the ball Y i [n] ≤ 1.

Fundamental domain of SL(2, Z)
For the case N = 2 the Grenier fundamental domain agrees with the standard SL(2, Z) fundamental domain. The only vector for which the first condition is non-trivial is n = (0, 1) ⊺ and together with the second condition one gets

Generalisation to G(Z)
It appears that Grenier's construction of a fundamental domain is based on a sequence of maximal abelian parabolic subgroups. In this section we therefore consider G to be the split real form of a simple group of rank r admitting an abelian parabolic subgroup, which excludes E 8 , F 4 and G 2 . We shall discuss the explicit example of SO(N, N ) in the next section. One can probably generalise this construction to Heisenberg parabolic subgroups to encompass all exceptional groups, but this requires further analysis and will not be relevant for us. In what follows the group G is defined from its fundamental representations and G(Z) ⊂ G is the Chevalley subgroup associated to the weight lattice.
Let us first describe an appropriate coordinate system on M = K\G. Let P 1 be a maximal abelian parabolic subgroup where G 1 is itself the split real form of a semi-simple group and U 1 is an abelian unipotent subgroup. One calls respectively GL(1) × G 1 the Levi subgroup and U 1 the unipotent radical of the parabolic subgroup P 1 . In the last section, for G = SL(N ), we had G 1 = SL(N − 1) and U 1 = R N −1 the additive group for example. We can then apply the same decomposition to G 1 and all G i semi-simple subgroups successively with U i an abelian unipotent group for all i = 1 to r. By construction P r = GL(1) ⋉ U r and the Borel subgroup can be defined as 13) and the Iwasawa decomposition of G is compatible with this succession of abelian parabolic subgroups 14) The Iwasawa decompositions provides coordinates on M, with y i ∈ R + for i = 1 to r, parametrising each GL(1) factors of the Cartan torus and a vector x i ∈ u i the Lie algebra of U i for i = 1 to r parametrising each U i unipotent group. By construction the Borel subgroup B i ⊂ G i can be decomposed accordingly into By definition each maximal parabolic subgroup P i ⊂ G i−1 is determined by a fundamental weight Λ (i) of the subgroup G i−1 . The associated set of roots of G is called an abelian enumeration in [22]. We write R (i) = R(Λ (i) , R) the associated highest weight representation, which highest weight vector e (i) ∈ R (i) admits as stabiliser the subgroup (3.16) For example in the case of SL(N ), the maximal parabolic subgroup (GL(1)×SL(N −1))⋉R N −1 is associated to the fundamental weight Λ 1 of SL(N ) and the corresponding highest weight representation is the fundamental representation R N . The orbit of the highest weight vector is then dense in the module By convention we will call R (i) the fundamental representation. In general the orbit of the highest weight vector is not necessarily dense in the module R (i) , but is defined instead as the set of non-zero elements v (i) satisfying the quadratic constraint [44] v where κ αβ denotes the Killing-Cartan form of g i−1 and T α its representation matrices in R (i) . One can then write (3.16) as where some further positivity condition L(v (i) ) > 0 may be necessary for specific groups G i−1 and representations R (i) . For example for SO(N, N ) one can take the fundamental representation as the vector representation, and v For the connected component SO 0 (1, 1) one would moreover demand that the non-zero lightcone coordinate is positive.
in the representation R (i+1) . We can then define the symmetric bilinear form The matrix V i can be written as where ρ Λ (i+1) is the representation homomorphism and h (i+1) the Cartan generator of G i associated to the weight Λ (i+1) with the normalisation 2 Note that the coordinates y i defined above are not the same as the coordinates introduced in (3.4). 3 In the following we shall use that a point in M i can be parametrised equivalently by all {y j , x j } for j ≥ i + 1 or by the symmetric matrix H i . One can also equivalently use the coordinates {y i+1 , x i+1 } and the symmetric matrix H i+1 , etc...
The idea behind Grenier's construction of a fundamental domain is that one can construct recursively the fundamental domains F i for M i under the action of G i (Z) from i = r − 1 to 0. At each step, one can use the bilinear form (3.20) to define a set of inequalities that determines F i .
Before generalising Grenier's construction, let us give some definitions. We define for 1 ≤ i ≤ r − 1 (3.24) Note that with this definition G i (Z) may not be a subgroup of G i but includes the entire discrete Levi subgroup. For SL(N, Z) we would then define G i (Z) = GL(N − i, Z) for example.
The construction of the fundamental domain is defined by induction. In order to show that F i is a fundamental domain of M i under the action of G i (Z) we have to show that for any element H i ∈ M i there exists an element γ ∈ G i (Z) such that γ ⊺ H i γ ∈ F i and conversely that for any non-trivial element γ ∈ G i (Z) and any element H i ∈ F i , either γ ⊺ H i γ / ∈ F i or both H i and γ ⊺ H i γ are on the boundary of F i .
We first consider the fundamental domain in M i for the action of the parabolic subgroup P i+1 (Z). Let p be an element of P i+1 (Z) = G i+1 (Z) ⋉ U i+1 (Z), we can decompose it as The adjoint Levi subgroup G i+1 (Z)/Z(G i+1 (Z)), with Z(G i+1 (Z)) the centre of G i+1 (Z), acts freely on a dense open set in M i+1 , and determines the fundamental domain To find a fundamental domain of P i+1 (Z) in M i , it remains to act with Z(G i+1 (Z)) ⋉ U i+1 (Z) on u i+1 . By construction U i+1 (Z) acts by translation and Z(G i+1 (Z)) acts either trivially or by multiplying x (i+1) ∈ u i+1 by −1. One gets therefore where d i+1 is the dimension of U i+1 and µ = 2 if Z(G i+1 (Z)) acts as Z 2 on u i+1 and µ = 1 if it acts trivially.
To define the fundamental domain of G i (Z), we introduce the Chevalley lattice R (i+1) (Z) ⊂ R (i+1) , which is preserved by the action of G i (Z).
The equivalent of (3.19) for the discrete subgroup G i (Z) gives where n × n is defined as in (3.18) and an additional positivity condition L(n) may be required when the only elements in G i (Z) that change the sign of n are trivial in G i (Z)/G i+1 (Z). One defines the positive set S i > ⊂ M i as Where H i [n] is the positive definite bilinear form (3.20). This domain is non-empty. For fixed (H i+1 , x i+1 ) in F P i , one finds that H i ∈ S i > for y i+1 > L i+1 (H i+1 , x i+1 ) sufficiently large. We will show that a fundamental domain F i = M i /G i (Z) can be defined as the intersection By construction, any element γ ∈ G i (Z) can be decomposed as with p ∈ P i+1 (Z), l ∈ G i+1 (Z) and exp(b) ∈ U i+1 (Z). For each point in G i (Z)/P i+1 (Z) one can find a representative h ∈ G i (Z) that preserves F P i . Using the isomorphism (3.29), one has therefore for each n ∈ S i a unique h(n) ∈ G i (Z) such that We have already checked that for any point H i ∈ M i , there is an element p ∈ P i+1 (Z) such that p ⊺ H i p ∈ F P i . For any such H ′ i ∈ F P i there is a smallest norm element n 0 ∈ S i such that It may not be unique, but because R (i) (Z) is discrete in R (i) , there is a finite number of discrete points in any ball and therefore a finite number of n 0 satisfying (3.34). Choosing h(n 0 ) one obtains 1 Now we need to prove that for any point in the interior of F i and any non- In the appropriate coordinate system this reads for y(α i+1 ) the associated Cartan torus coordinate in the Levi subgroup G i+1 . In particular one can always choose the succession of parabolic subgroups (3.14) such that y(α i+1 ) = y i+2 . It follows by induction that there exists l i+1 > 0 independent of (H i+1 , x i+1 ) such that any point One can therefore restrict the conditions defining S i > to the vectors n ∈ S i in the ball H i [n] ≤ 1 l i+1 2 and there is only a finite number of those.
As a consequence S i > is defined by a finite intersection of closed sets and its interiorS i > is defined by the finite intersection of open sets It follows directly that for any point H i ∈F i , the action of a non-trivial γ = h(n)p ∈ To see this, note that any non-trivial p ∈ P i+1 (Z) moves for this specific n, which shows that In the second case it means that there is a finite set of vectors n ∈ S i not equal to e (i+1) such that H i [n] = 1 , and the corresponding h(n) map ∂S i > ∩ F P i to itself. This concludes the proof that F i is a fundamental domain of M i /G i (Z), and by induction that F is a fundamental domain of M/G(Z).

Fundamental domain of SO(N, N, Z)
In this section we shall illustrate how this construction applies to the symmetric space M = SO(N ) × SO(N ) \SO(N, N ) and the arithmetic subgroup SO(N, N, Z) preserving the even self-dual lattice of split signature II N,N . In string theory, the group of T-duality on the torus We will use the convention that the split signature metric is and we parametrise points in M by a symmetric matrix H in the vector representation of SO(N, N ), which satisfies For short we will use the symbol H for the point in moduli space and for the symmetric matrix in SO(N, N ) that represents it. In this case it is convenient to choose the succession of parabolic subgroups (3.14) such that We decompose accordingly H = H 0 into each H i for 1 ≤ i ≤ N as For the quotient by O(N, N, Z), one derives that . The sets S i in (3.29) can be defined as 3. For any m, n ∈ Z, q ∈ II N −i,N −i such that gcd(m, q, n) = 1 and (q, q) + 2mn = 0 we have: The first two conditions ensure that For the fundamental domain M/SO 0 (N, N, Z) one must further take into account that SO 0 (1, 1) is trivial and (x N −1 ) 1 ∈ − 1 2 , 1 2 instead of 0, 1 2 and the last condition y N > 1 is not imposed.

Inductive proof
For simplicity we consider the fundamental domain M/O(N, N, Z) in this section. Let us first show that for any one obtains that such transformation gives This only fixes l up to sign, and there is therefore enough symmetry together with the shift in b to fix for any m, n ∈ Z, q ∈ II N −1−i,N −1−i with gcd(m, n, q) = 1 and 2mn + (q, q) = 0. These components are determined by the action on a vector Q = (1, 0, 0) up to right multiplication by an arbitrary element in P i+1 (Z). Using the result above, one can always determine h such that moves an element H i ∈F i outside of the fundamental domain. By induction we assume that γ = ph = lbh and any non- Choosing y i+1 sufficiently large compared to all eigenvalues of H i+1 ,, it is clear that The open setF i is therefore defined by the strict inequality and it follows that any non-trivial h moves H i ∈F i outside the fundamental domain. Because the action is continuous it follows that any non-trivial element γ ∈ O(N −i, N −i, Z) acts on a point in the boundary of the fundamental domain F i to give another point in the boundary. It follows by induction if H i+1 is in the boundary of F i+1 , and it is rather obvious if one of the (x i+1 ) k = ± 1 2 . If there exists a non-trivial vector Q ̸ = (±1, 0, 0) such that the inequality (3.48) is saturated, the corresponding element h also preserves the boundary.
This establishes the definition of the fundamental domain described in this section for O(N, N, Z). To prove the result for SO 0 (N, N, Z) only require to study the case of SO(2, 2), which we describe now.

The example of SO(2, 2)
In this case we write explicitly the bilinear form as The induction starts with the condition that q 2 1 + y 4 2 q 2 ≥ 1 for any (q 1 , q 2 ) in the orbit of (1, 0). For O(1, 1, Z) one gets the four vectors (q 1 , q 2 ) = (±1, ±1) and so one obtains the condition y 2 ≥ 1. For SO(1, 1, Z) or SO 0 (1, 1, Z) one does not get q 2 = ±1 and there is no further condition on y 2 > 0.
At the next step the action of the unipotent subgroup allows to fix both x i ∈ [− 1 2 , 1 2 ]. For O(1, 1, Z) and SO(1, 1, Z) one can use the element −1 to constrain x 1 ≥ 0. For SO 0 (1, 1, Z) the trivial group, we do not get this further restriction.
Using now the third condition, one obtains for m = n = 0 the two conditions and all the other conditions one obtains are consequences of these two. We conclude that a fundamental domain of O(2, 2, Z) is defined by the conditions and a fundamental domain of SO 0 (2, 2, Z) by This is consistent with the isomorphism 60) and T = x 1 + iy 1 y 2 and U = x 2 + i y 1 y 2 in the standard fundamental domain of SL(2, Z). The further condition x 1 ≥ 0 appears for the fundamental domain of SO(2, 2, Z) which includes the further generator SO(2, 2, Z) = S(GL(2, Z) × Z 2 GL(2, Z)) that changes the signs of x 1 and x 2 . The additional condition y 1 ≥ 1 appears for the fundamental domain of O(2, 2, Z) which further includes the generator that exchanges T and U .

Minima at symmetric points
The main purpose of this paper is to find the global minimum of Eisenstein series. The SL(2) real analytic Eisenstein series 2ζ(2s)E s is known to have a global minimum at the symmetric point [35,45] for any value of the parameter s > 0, with the appropriate regularisation at s = 1. There is no general result for the global minimum of the Epstein series Ep N s (H) for N ≥ 4 and generic s > 0. The sum (2.6) is absolutely convergent for s > N 2 , but admits an analytic continuation to a meromorphic function of s ∈ C with a single pole at s = N 2 [46]. One can define the regularised value of the Epstein function at the simple pole by minimal subtraction. For s → ∞, the global minimum is the solution to the densest lattice sphere packing in N dimensions [37]. One can understand this result intuitively using an expansion of Ep and so H[n H min ] maximal also gives the densest sphere packing in the lattice. For 2 ≤ N ≤ 8, the densest lattice sphere packings are known to be the following rank N root lattices [47] such that that the minimum of the Epstein series at large s is at However, we are interested in small values of s for the string theory couplings, in particular s ≤ N 2 . It was observed in [36] that the densest lattice sphere packing bilinear form H dlp cannot be the global minimum of the Epstein series for all s > 0 when the lattice and its dual do not define the same point in SO(N )\SL(N )/SL(N, Z), i.e. when H −1 dlp ̸ ≈ H dlp modulo the action of SL(N, Z). This is a direct consequence of the functional relation In particular the Epstein series is the same for the two dual points at One may therefore argue at most that the densest lattice packing H dlp is the global minimum for s ≥ N 4 , which implies by the functional relation above that its dual lattice H −1 dlp is the global minimum for 0 < s < N 4 . The case of N = 3 was analysed numerically in [36]. It was shown that the densest lattice sphere packing point H A 3 is a global minimum for s > 3 4 , and in particular for the regularised series Ep (4.6) They also demonstrate that the minimum of Ep N s must be strictly negative in the critical strip 0 < s < N 2 . However, it was proved in [48] that Ep > 0 if the minimal length vector n H min of the associated quadratic form H has length bounded from above as This excludes the possibility that the minimum be at H g for an ADE lattice for N > 24 and suggests instead that the densest lattice sphere packing will be the minimum [36]. The densest lattice sphere packing has been proved to be the global minimum for N = 8 and 24 and all values of s > N 2 [49].
It would be very difficult to carry out a complete numerical analysis of the Epstein series for N ≥ 4. Based on the results above we shall therefore concentrate on symmetric points in moduli space for which the bilinear form H is invariant under a finite subgroup of SL(N, Z). In this section we define these symmetric points and determine a criterion for them to be local minima of Eisenstein series.
We then generalise the same analysis to SO(N, N, Z) and prove that symmetric points are local minima of Eisenstein series provided they are negative in the critical strip.

Taylor expansion at symmetric points
The hyperplanes defining the boundary of the fundamental domain are mapped to themselves under elements of G(Z), and their intersections are invariant under non-trivial finite subgroups of G(Z). The maximal intersections define isolated points V 0 ∈ M that are invariant under maximal finite subgroups When the G V 0 (Z)-fixed points in M are isolated, we call them symmetric points. All the symmetric points are maximal intersections but some maximal intersections may not be symmetric points.
An automorphic form Φ on M in a representation ρ of K transforms as Here ρ generalises the weight for SL (2). By definition, the automorphic form is invariant under the linear action of ρ Φ (k γ ) at V = V 0 for any γ ∈ G V 0 (Z) and k γ satisfying (4.9). Indeed, one checks that We can use this property to constrain the covariant derivatives of an automorphic function at a symmetric point. We define the covariant derivative from the left-action of p ∈ p = g ⊖ k as where a labels the components of p in p. Note that the differential operators D a can equivalently be defined as the covariant derivative in tangent frame on the symmetric space The Taylor expansion of an automorphic function is therefore highly constrained by the symmetry group G V 0 (Z). At order n in the Taylor expansion, one can classify the order n polynomials in p that are invariant under all such elements k γ = V 0 γV −1 0 ∈ K. If V 0 is a symmetric point as we define above, there is no invariant linear polynomial and V 0 is an extremum of any automorphic function.
For any symmetric point V 0 we must check if the extremum is a minimum. In practice we shall find that the symmetric points admit a small number of G V 0 (Z)-invariant quadratic polynomials, and that it is sufficient to evaluate the automorphic function on a small dimension hypersurface to determine if it is a minimum.
If there is a single invariant quadratic polynomial, by construction it must be proportional to the quadratic Casimir such that where κ ab is the restriction to p of the Killing-Cartan form on g. The Hessian of the function at V 0 is then completely determined, and the symmetric point is a local minimum provided ∆f (V 0 ) > 0. In particular the function must be negative at a local minimum if the eigenvalue of the Laplace operator is negative. We will be interested in maximal parabolic Eisenstein series E G sΛ i associated to a fundamental weight Λ i , which satisfy the Laplace equation , the Eisenstein series is integrable and therefore such that E G sΛ i (V) must be negative at its global minimum. We find therefore that when the moduli space K\G admits a symmetric point V 0 with a unique invariant quadratic polynomial in p, the symmetric point V 0 is a local minimum of any absolutely convergent Eisenstein series. Moreover, V 0 is a local minimum of any integrable automorphic function that is negative at V 0 .
When there are two G V 0 (Z)-invariant quadratic polynomials we need to compute the two independent eigenvalues of the Hessian matrix at V 0 . For this purpose we shall define a twodimensional subspace of SO(N )\SL(N ) that includes the relevant symmetric points. There also exist symmetric points with more invariant polynomials, but we find that they never correspond to global mimima and shall disregard them.
We will now discuss the cases of SL(N ) and SO(N, N ).

SL(N ) symmetric points
The The reducible maximal subgroups stabilise the reducible bilinear forms A 1 +A 2 , A 1 +A 3 , A 1 +A * 3 . One easily checks that reducible bilinear forms that are extrema are always saddle points. To prove this, one considers coordinates associated to the maximal parabolic subgroup with n ∈ Z k , m ∈ Z N −k and X ∈ R k×(N −k) . The Fourier expansion of the Epstein series in the maximal parabolic P k can be obtained by Poisson summation and reads The second derivative with respect to X at X = 0 is negative definite as the absolutely convergent sum of positive terms. Therefore any extremum at X = 0 is a saddle point. We will therefore only analyse symmetric points for which the bilinear form H is irreducible.
It will be convenient to introduce the following parametrisation of V ∈ SO(N )\SL(N ) in the maximal parabolic subgroup P N −1 where V 1 ∈ SO(N − 1)\SL(N − 1), x ∈ R N −1 , and y ∈ R + has been rescaled. When V 1 is at In this paper we define the A N point in the fundamental domain as the symmetric matrix or equivalently In this way the symmetry under the subgroup S N ⊂ S N +1 is manifest. One gets the A N , D N , E N representatives by fixing V 1 at the A N −1 lattice point, and all x i components equal such that One moreover checks that the Gram matrix C A +2 5 and C A +3 5 of A +2 5 and A +3 5 are given by 4 This ansatz for y > 0 and a single x ∈ [0, 1 2 ] therefore describes many symmetric points. In particular it covers all the symmetric points (up to inversion) for SL(3, Z) and SL(5, Z), the groups relevant in string theory. The dual lattice points can be studied in the same way with the inverse matrix or using the functional relation (4.4). This representative of the A N symmetric point is in the Grenier domain, but not the others, see Appendix A.
The densest sphere packing criterion can be checked using the density (4.2) so the densest sphere lattice packing is indeed the lattice D 5 , and A +2 5 is denser than A 5 . The dual lattices are less dense, with On the surface where H(y, x, C A N −1 ) is parametrised by y > 0 and − 1 2 ≤ x ≤ 1 2 as in (4.28), one writes the explicit formula for the SL(N ) Epstein series as   We will now study the different symmetric points.

The A N symmetric point
The automorphism group of the A N lattice is Z 2 ⋉ S N +1 , but only the alternating group Alt N +1 embeds inside P SL(N, Z) for N even and S N +1 ⊂ P SL(N, Z) for N odd. The Eisenstein series are nonetheless invariant under the action of P GL(N, Z), so their symmetry group is always S N +1 . The automorphism group Z 2 ⋉ S N +1 is realised on the lattice vectors such that Z 2 acts as 43) and S N +1 acts as the permutation of the N elements n i and a fictitious N +1th element − N j=1 n j , i.e. σ ∈ S N acts as σ : n i → n σ(i) , It is useful to split this group into S N and where Z N +1 is the cyclic group generated by the transformation To determine the S N +1 -invariant quadratic polynomials, it is convenient to definẽ for which the traceless condition reads For any invariant polynomial f V 0 (p), one defines the polynomial for any γ ∈ G A N (Z) representing S N +1 . The symmetric matrixp simply transforms under S N +1 as a traceless bilinear form in the n i 's, i.e. under σ ∈ S N as and under σ f as σ f :p 11 →p N N , (4.53) The vector n transforms in the standard representation of S N +1 associated to the partition (N, 1). The symmetric tensor product decomposes into irreducible representations as such thatp splits into the irreducible components q ∈ (N, 1) andp ⊥ ∈ (N − 1, 2). The vector q in the standard representation is defined as and transforms by permutation under S N and as and transforms under S N +1 asp in (4.53). It follows that there are two independent invariant quadratic polynomials inp, one quadratic in q and one quadratic inp ⊥ , such that the Killing-Cartan form splits into For completeness, we give the explicit proof that there are only two invariant quadratic polynomials in Appendix B.
To study the Taylor expansion of Epstein series at the symmetric point A N , we use the coset representative (4.28) and its vielbeins (4.59) At the symmetric point we take V 1 constant and such that N 1 N −1 V ⊺ 1 V 1 = C A N −1 and we define the pull-back momentum such that P 1 * = 0. One computes then , where x * i = 1 N . This gives the standard representation components of P * and one checks that the (N − 1, 2) components P ⊥ * only depend on dx We define the pull-back of an automorphic function to the two-dimensional subspace parametrised by y and x. The Taylor expansion of the pull-back function must be consistent with the S N +1 symmetry, and therefore for two coefficients a + and a − determined by the second derivatives of the automorphic function at the symmetric point. In particular Hence the condition for the symmetric point A N to be a local minimum of the automorphic function f is that a + > 0 and a − > 0, which is equivalent to the condition that It is therefore sufficient to study the pull-back function f * (y, x) to determine if the symmetric point A N is a local minimum of the automorphic function f (y, We have carried out this computation numerically for the Epstein series and found that the A N symmetric point is always a local minimum for s large enough, but the eigenvalue a − becomes negative for small s for N ≥ 4, giving a saddle unstable along the corresponding It is always a local minimum for N = 3, and is in fact the global minimum for s > 3 4 [36]. We find that it is a local minimum for s > 1 for SL(4), and for s ≳ 3.16603 for SL (5). For SL (6) and SL(7) one finds similarly that it is not a local minimum for s < s A N with s A N slightly above the critical value N 2 . Most importantly for us, it is not a local minimum of the Epstein function Ep that defines the leading Wilson coefficient in type II string theory on T 3 .

The D N symmetric point
The D N symmetric point is realised with the ansatz (4.34) as the bilinear form It is easy to show that it is related to the identity matrix by a GL(N, Q) transformation that determines for N integers m i such that i m i = 0 mod 2. One then gets that preserve the condition that i m i = 0 mod 2. For N = 4, one has moreover the triality automorphisms. This determines the action of the automorphism group on the momentum p ∈ p for generic N as For N = 4 the triality symmetry implies that there is a unique invariant quadratic polynomial, while for generic N one straightforwardly computes that with and a + and a − the two eigenvalues of the Hessian of f at the symmetric point. One finds by numerical evaluation that the D N symmetric point is indeed a local minimum of the Epstein series Ep N s for all s and N ≤ 7. We do not expect this to be true for arbitrary large N , but this is not relevant for the study of the type II string theory effective action. We find in particular that among all symmetric points, the D N symmetric point gives the minimum value of the Epstein series Ep N s for N = 4, 5 and s ≥ N 4 . 6 We did not find any other local minimum for N 4 < s < N 2 and N = 4, 5 except for the dual point D * N by numerical evaluation, leading to the conjecture that D N is the global minimum.

Other symmetric points
In this subsection we shall briefly describe the other symmetric points.
• The dual symmetric points Let us first observe that we do not need to describe the dual symmetric point for the lattices A * N and D * N separately. By construction C L * = C −1 L , and the inverse of H = V ⊺ V with the ansatz (4.28) allows to describe all dual lattices. One easily extends all the results using the functional relation (4.4). As a consistency check we have evaluated numerically the function for both the lattice and its dual. The analysis of the polynomials in p ∈ p invariant under the automorphisms of the lattice proceeds in the same way for the dual lattices, and gives by construction the same number of invariant polynomials.
For example for A * N one definesp for which the traceless condition reads The action of the Weyl group is then such that S N acts by permutation of the indices and σ f acts onp as It decomposes into the vector q in the irreducible representation (N, 1) (4.80) and the orthogonal componentp ⊥ in (N − 1, 2) that has zero diagonal entriesp ⊥ ii = 0 and for i ̸ = j. One obtains then two independent quadratic polynomials in q andp ⊥ respectively, such that For large values of s, the Epstein series evaluated at the dual symmetric points D * N and A * N is larger, as expected from the sphere packing density. By the functional relation (4.4) this must be the opposite for s < N 4 , and in particular a symmetric point and its dual give the same value at s = The symmetric point A +2 5 admits the same symmetry S 6 ⊂ SL(5, Z) as the point A 5 and its dual A * 5 . This is easy to prove using the relation where γ the GL(5, Q) matrix defined such that is the change of basis (4.69). In particular C D 5 = γ ⊺ γ. One checks that for any g ∈ S 6 ⊂ SL(5, Z), γgγ −1 ∈ SL(5, Z), such that the automorphism group of A +2 5 is conjugate to the one of A * 5 in SL(5, Q) [51]. It follows that the coset p ∈ p decomposes into the two irreducible representations (5, 1) and (4, 2) of S 6 . To finds the explicit polynomial it is convenient to introduce One obtains that the condition for the Hessian of the function f to be positive definite is (4.67) for N = 5, as for the A 5 symmetric point. We have checked these conditions numerically and found that the symmetric point A +2 5 is a local minimum of the Epstein series Ep 5 s for s ≳ 2.8849. Below this value the second derivative with respect to y is negative and the saddle is unstable along 5 directions.
In the critical strip 5 4 < s < 5 2 we find that and only D 5 and D * 5 are local minima. • The E 6 , E 7 , E 8 symmetric points We shall be very brief about these cases since they are not relevant to the analysis of the type II string theory low energy effective action. One finds in these three cases that the Weyl group of E N is large enough to impose that there is a unique invariant quadratic polynomial in p. It is therefore sufficient to check that the Epstein series is negative in the critical strip to ensure that the E N symmetric point is a local minimum for all values of s. One finds indeed by numerical evaluation that the E N symmetric points are local minima for all values of s, and among all symmetric points we have checked they give the smallest value of the Epstein series Ep N s for all values of s ≥ N 4 . 7 In the critical strip N 4 < s < N 2 we find that symmetric points for d|N For N not prime one can also have symmetric points associated to irreducible bilinear form that are tensor products of lower dimensional bilinear forms. For example for N = 4 one has (4.89) More generally the density of the sphere packing of .
(4.90) 7 We have only checked symmetric points of Cartan type AN , DN , EN and their dual. This exhausts all maximal irreducible symmetry groups for N = 7, but not for N = 6, 8 [51]. In type II string theory on T d the SO(d, d) symmetric matrix H is parametrised by the Narain moduli, with the torus metric G and the Kalb-Ramond two-form B in string length. Then m is the vector of Kaluza-Klein modes and n the vector of winding numbers. For the U-duality group Spin(5, 5, Z) one can view G as the M-theory metric on T 5 and B as the Hodge dual of the three-form potential on T 5 . To define symmetric points in SO(N, N ) we start with the assumption that G is itself a symmetric point of SO(N )\GL(N ). From the previous section we find that G is then proportional to the Gram matrix C L of an even lattice L, and the relevant solution will turn out to be One finds in this case that the charges q ± q ± = 1 2 G −1 m + (B ± G)n , For any pair of elements γ ± ∈ Aut(L) such that  Its global minimum at y = 1 can be interpreted as the SU (2) self-dual radius in string theory on S 1 . The A, D, E points described above correspond more generally to the points of enhanced gauge symmetry in string theory. For N = 2 one recovers the minimum of the product of two SL(2) Epstein series at T = U = 1 2 + i √ 3 2 for the symmetric matrix H associated to the lattice L = A 2 . We will see below that for N = 3 the symmetric matrix associated to L = A 3 also reproduces the minimum of the SL(4) Epstein series at the symmetric point D 4 .
Assuming H to be determined by a lattice Gram matrix as in (4.95), one can compute the limit at large s from the theta series where the prime removes the point m = n = 0 for µ = 0 only. At large s, the leading term in Ep N,N s (H) is proportional to the smallest length of a vector in L * to the power −2s, and one obtains a minimum for the even lattice with the largest possible minimal length of a vector in L * . Although this is not exactly the same criterion as for the densest lattice sphere packing, it gives the same ADE classification (4.3) for N ≤ 8.
For small values of s we need instead to use the Fourier expansion of the Epstein series 9 where the sum is over all non-zero rank one antisymmetric integer matrices Q, the bilinear form is defined as and E SL (2) s (U Q ) is the SL(2) real analytic Eisenstein series evaluated on the SL(2) subgroup of the stabiliser of Q. 10 Note that the stabiliser of For the value s = N −2 2 , this expression simplifies drastically to π G[Q] e πi tr BQ .
(4.107) We will use this expression to evaluate numerically the Epstein series for N = 5.
We find evidence that the minimum of the Epstein series in the vector representation (4.92) admits its global minimum for all s > 0 at the symmetric point where G = 1 2 C L and G + B = 0 mod Z for the Cartan type best packing lattices (4.3).

Symmetric points as minima
In order to describe the symmetries of the polynomials in the coset derivatives we need to define a coset representative V ∈ SO(N, N ). We introduce two vielbeins basis V ± for the same metric . (4.108) It transforms under left action of k ± ∈ SO(N ) and right action of γ ∈ O(N, N, Z) as One can of course set V + = V − = V , but it is convenient to keep them different to make manifest the covariance under SO(N ) × SO(N ). The coset differential is then We now consider a point (4.95) for L of Cartan type, that we write (4.112) Using the charges q ± that transform under the two copies of the Weyl group at a symmetric point (4.96), one obtains that It follows that under an automorphism in (4.102) realised as a γ ∈ O(N, N, Z) one has where the γ ± satisfy (4.100). In particular, the γ ± can be two independent elements of the Weyl group of the Cartan type lattice L. It appears therefore convenient to expand the automorphic function of interest with respect to the pull-back momentum P * = − dG + dB * . For any invariant polynomial f H 0 (p) of p ∈ so(N, N ) ⊖ (so(N ) ⊕ so(N )), one defines the polynomialf Let us prove this for A N and D N .
• A N symmetric point For A N , the momentump transforms in the tensor product of two standard representations (N, 1) associated to the two copies of S N +1 . It is therefore in an irreducible representation of S N +1 × S N +1 and there is obviously no linear invariant. Introducing the indices i = 1 to N for one side andî = 1 to N for the other, one obtains the action of S N × S N σ ×σ :p iȷ →p σ −1 (i)σ −1 (ȷ) , (4.120) while the left σ f acts as . It is convenient to define the action of the two copies of S N ⋉ Z N 2 σ ×σ : p iȷ → p σ(i)σ(ȷ) , ϖ i × 1 : p iȷ → −p iȷ , p jȷ → p jȷ , ∀j ̸ = i , (4.124) and identically for the rightπȷ, and then take the subgroup of elements with an even number of ϖ i andπȷ. It follows directly that the only invariant quadratic polynomial is which is the Killing-Cartan form (4.119). One finds therefore that the D N symmetric point is a local minimum of any SO(N, N ) Eisenstein series in the domain of absolute convergence. We have checked numerically that Ep 5,5 s (H) is negative at the critical value s = 3 2 relevant in the string theory effective action, and the symmetric point D 5 is therefore a local minimum. It is a lower value than the A 5 symmetric point and we conjecture that it is the global minimum of the minimal Epstein series.

SO(3, 3) and SL(4)
Because of the homomorphism Spin 0 (3, 3) = SL(4) it is relevant to compare the results we have obtained for SO(N, N ) and SL(N ) in this case. It appears that the vector representation Epstein series of SO (3,3) and the Epstein series of SL(4) are related at the special s value with the identification of the SL(4) matrix and V + = V − = V and B ij = ε ijk x k in (4.108). In particular for (4.34) one has G = y 4 C A 3 , and one can check the functional relation (4.126) using (4.107) and (4.40) cos 2πx · n . We find consistently that the conjectured global minimum of the SL(4) Epstein series at the D 4 symmetric point agrees with the conjectured global minimum of the SO(3, 3) Epstein series at the A 3 symmetric point. One finds that they have the same automorphism groups because of the triality automorphism of D 4 , explaining that there is a unique invariant quadratic polynomial in this case.

Fixing the logarithmic ambiguity in eight dimensions
The different terms in the low energy expansion of the two-graviton amplitude (2.4) are not well defined individually in eight dimensions because the 1-loop box integral diverges logarithmically and so do the Epstein series defining E (0,0) (φ). To determine unambiguously the amplitude we must analyse the low energy limit of the one-loop string theory amplitude [5] and G(x) is the torus Green function. Following [12], we split the SL(2) fundamental domain into the truncated fundamental domain and the complementary region for which τ 2 > L and − 1 2 ≤ τ 1 ≤ 1 2 . At leading order one obtains where ⟲ represents the two cyclic permutations of the Mandelstam variables. To compute the two terms separately it is more convenient to introduce a dimensional regularisation d = 2 + 2ϵ. This can be achieved with the insertion of τ ϵ 2 in both terms. One can then remove the Ldependent terms that cancel out and use instead ) For the one-loop supergravity amplitude one gets the dimensional regularisation We define accordingly the renormalised 1-loop box integral in D = 8 dimensionŝ (5.8) where we introduced a renomalisation scale µ. Note thatÎ 4,µ (s, t) =Î 4,1 (s/µ 2 , t/µ 2 ).
For the perturbative Wilson coefficient one can replace τ ϵ 2 by the real analytic Eisenstein series E ϵ (τ ) without modifying the limit. One computes as in [34] that where η is the Dedekind eta function. One obtains in total where we used α ′ 3 g 2 8 = ℓ 6 P in the last step. We use the same convention as in [34] for the renormalised Eisenstein series, such that we define the SL(2) Epstein series |m + T n| e 2πi(c 0 m+c 2 n) . Note that the logarithm 4π ln(2πℓ P ) can be absorbed in the logarithm of the type IIB torus volume V 2 = (2πℓ P ) 2 T 2 g

Numerical approximations
In this section we explain the numerical checks we have carried out to establish the conjectured global minima of the Epstein series. We concentrate on the two cases of SL(5) and SO(5, 5) that constitute the main new results. The global minimum of the SL(3) Epstein series was obtained as the symmetric point A 3 for all s > 3 4 , including the renormalised value at s = 3 2 by minimal subtraction, in [36].
The moduli space SO(5)\SL(5) has 14 dimensions and SO(5) × SO(5) \SO(5, 5) 25 dimensions, and it is rather difficult to study the value of the Epstein series systematically over the whole moduli space in these two cases. Nonetheless, the Epstein series are very regular functions, that decrease somewhat monotonically from the cusps at infinity to the symmetric points in the interior.

The SL(5) Epstein series
We consider several SL(5, Z) equivalent realisations of the D 5 symmetric point H = 2 − 2 5 C D 5 with They are determined by iterated inclusions of lattices The first case is described with the ansatz (4.28) and the expansion of the Epstein series (4.40) with N = 5 as a function of y and x. We use (4.40) iteratively and truncate the sum over n i ∈ Z for N = 3 between −50 and 50, the sum over n i ∈ Z for N = 4 between −20 and 20 and the sum over n i ∈ Z for N = 5 between −10 and 10. We checked that increasing the ranges of the Fourier modes n i does not change the result within the approximation. We analyse the values of s between 5 4 and 5. For large values of s the Fourier expansion converges more slowly because the Bessel function does not decrease fast enough for the truncated expansion (4.40) to give a good approximation.
The obtained function of y and x admits only three local minima, corresponding to the symmetric points A 5 , A +2 5 and D 5 , with D 5 being the global minimum. Only D 5 is a local minimum of the function of H ∈ SO(5)\SL (5) for low values of s.
One also checks the pull-back of the function on different surfaces associated to the parametrisation  In the second case we consider the SL(N ) coset representative which defines another surface parametrised by (x, y) in the moduli space. In this case one obtains for N = 5 that the only local minimum is at y = 2 and x = 1 2 corresponding to the D 5 symmetric point.
In the third case we consider the SL(N ) coset representative with V 1 at the A N −3 +2A 1 symmetric point and x = (− 2 N −2 , . . . , 2 N −3 N −2 , 1, 1)x. One obtains again for N = 5 that the only local minimum is at y = 2 and x = 1 2 corresponding to the D 5 symmetric point.
We obtain numerically the value of the Epstein series The numerical value is stable under modification of the truncation up to fifteen digits.
that defines the leading Wilson coefficient in the string theory effective action. One difficulty is to generate an appropriate large set of rank two antisymmetric integer matrices Q ∈ ∧ 2 Z 5 . To obtain a good approximation, we must include all the M2-instanton charges Q with a Euclidean action below some threshold without including two many charges Q with a strictly greater Euclidean action. The incomplete sets of charges for a given action greater than Λ do not a priori spoil the approximation, but their inclusion increases the evaluation time of the function without providing a better approximation.
To compute a sample of charges we generate the set of rank two matrices with entries |Q ij | ≤ 6 and only keep the charges with a Euclidean action evaluated at the D 5 symmetric point bounded by Λ = 2π √ 6. The values of the action is S = π √ 1 + 4n or S = 2π √ n for all integers n ≥ 1. We have checked up to S = 2π √ 5 that the set of charges we have obtained define complete orbits of the Weyl group action W (D 5 ). They all satisfy |Q ij | ≤ 4, so we believe they are the complete set of charges for Λ = 2π √ 5. One also checks that this set of charges gives all charges of action bounded by Λ = 2π √ 6 at the A 5 symmetric point with entries |Q ij | ≤ 5. The values of the action is then S = 2π √ n or S = π √ 3 + 4n for all integers n ≥ 1. We expect therefore this set of charges to provide a good approximation of the Epstein series on the fundamental domain.
We consider two surfaces, the first parametrised by only includes a global minimum at the D 5 symmetric point at y = 1 and x = 1 2 .
The second is parametrised by (6.10) On this surface we find that the D 5 symmetric point is a global minimum at y = 2 and x = 2 5 . One also finds the local minimum at the A 5 symmetric point at y = √ 6 and x = 1 5 . The numerical approximation gives the minimum of the Epstein series To exhibit that this is the case, one writes C A N as and identifies the bilinear forms in the notation of Section 3.1. One finds from this formula that x k+1,i = 1 k+1 ∈ [0, 1 2 ] and for each Y k [n], the minimal length vectors are length one, consistently with the definition of the Grenier domain. This point is on the boundary of the Grenier domain since the inequality is saturated for all A N root, i.e.
for any j and k. For Now, solving the condition that Y 0 [n] = 1 for all roots of A N , one obtains which determines y i = x ij = 1 and fixes the One can consider similarly the symmetric point D N +1 in SO(N )\SL(N ). Starting form (4.33) and shifting n 2 → n 2 − n N +1 one obtain the equivalent representative of the D N +1 Cartan matrix where the second sum is over all pairs except n 2 n N +1 . One checks that H = 2 − 2 N +1 C ′ D N +1 is in the Grenier domain with the triangular form and the sub-components The triangular form of Y 0 exhibits that all x ij ∈ [0, 1 2 ] but x 2,N +1 = − 1 3 ∈ [− 1 2 , 1 2 ]. The subcomponents all have minimum length vectors of length one, and H is therefore in the Grenier domain. This point sits on the boundary because the inequalities Y 0 [n] ≥ 1 are saturated for for all D N +1 roots, i.e. n i = ±δ i,j , n i = δ i,j − δ i,k for (j, k) ̸ = (2, N + 1) or (N + 1, 2) , n i = ±(δ i,j − δ i,2 − δ i,N +1 ) for j ̸ = 2, N + 1 , n i = ±(δ i,j + δ i,k − δ i,2 − δ i,N +1 ) for j, k ̸ = 2, N + 1 .
In general one does not expect reducible matrices to correspond to a zero dimensional boundary of the Grenier domain. Take as an example H = (2N + 2) − 1 N +1 C A N +A 1 with One checks similarly that H is in the Grenier domain, but the saturated inequalities still allow for a dimension N boundary parametrised by x i n N +1 n i , (A.14) provided they satisfy for all k between 1 and N .
For the lattice A +2 5 , the matrix (4.35) , and the lattice A +2 5 is therefore at a dimension-zero boundary of the Grenier domain.

B Invariant polynomials at the A N symmetric point
We prove here that there are indeed only two quadratic invariant polynomials at the A N symmetric point by constructing them explicitly. We can easily see that for any polynomial F (p), its average on S N +1 defined by is an S N +1 -invariant polynomial. Therefore by calculating the ⟨p ij ⟩ S N +1 and ⟨p ijprs ⟩ S N +1 we can generate all independent S N +1 -invariant linear and quadratic polynomials. We start with the linear polynomials where the calculations are less tedious but are completely analogous to the quadratic case. We can see that 12 It is easy to see that the average on S N doesn't depend on i, j explicitly however the result will differ if i = j or i ̸ = j. Therefore there are naively only two possible linear invariant polynomials. We can easily compute the two independent averages on S N . 12 Actually it is not obvious that we don't need to take another average on SN in the end but it turns out that this is indeed not the case where i ̸ = j in the second line. Hence all that remains is to calculate ⟨p ij ⟩ Z N +1 . If we use the convention that the indices i, j ofp ij are taken modulo N + 1 we can write an arbitrary number of iterations of the transformation σ f in the simplified form for all k ≥ 0. We find that for any i, j. Hence we find that where we have used the traceless condition and where i ̸ = j in the second line. Therefore there are no linear S N +1 -invariant polynomials ofp. This means that C A N as defined in (4.30) is an extremum of any automorphic function on SO(N )\SL(N ).
We will now apply the same reasoning to show that there are only two independent quadratic S N +1 -invariant polynomials ofp. We can show that for all i, j, r, s. We can now calculate the average on S N +1 in an analogous way to the linear case. Just like the linear case we can see that the average on S N will not depend on i, j, r, s explicitly, however the result will differ depending on if some indices are equal to each other. One can see that there are 7 different averages to compute: ⟨(p ii ) 2 ⟩ S N +1 , ⟨p iipij ⟩ S N +1 , ⟨p iipjj ⟩ S N +1 , ⟨(p ij ) 2 ⟩ S N +1 , ⟨p iipjr ⟩ S N +1 , ⟨p ijpir ⟩ S N +1 and ⟨p ijprs ⟩ S N +1 where all indices are assumed to be different. We can decompose each of them on a basis of S N invariant polynomials as For N > 2 we can check that only three out of the seven averages are actually independent and they are linearly dependent with the Killing-Cartan form which vanishes once the traceless condition is imposed. This proves that there are two independent non trivial quadratic S N +1 -invariant polynomials ofp. The other non trivial independent polynomial is given by one of the averages, we give the simplest one (B.14) In the case N = 2 only the first four averages exist and they are all proportional to the Killing-Cartan form once the traceless condition is imposed. Therefore we can recover the well known result that A 2 is a local minimum (and indeed the global minimum) for the SL(2) Eisenstein series.