D6ℛ4 amplitudes in various dimensions

Abstract

Four-graviton couplings in the low energy effective action of type II string vacua compactified on tori are strongly constrained by supersymmetry and U-duality. While the R 4 and D 4 R 4 couplings are known exactly in terms of Langlands-Eisenstein series of the U-duality group, the D 6 R 4 couplings are not nearly as well understood. Exploiting the coincidence of the U-duality group in D = 6 with the T-duality group in D = 5, we propose an exact formula for the D 6 R 4 couplings in type II string theory compactified on T 4 , in terms of a genus-two modular integral plus a suitable Eisenstein series. The same modular integral computes the two-loop correction to D 6 R 4 in 5 dimensions, but here provides the non-perturbative completion of the known perturbative terms in D = 6. This proposal hinges on a systematic re-analysis of the weak coupling and large radius of the D 6 R 4 in all dimensions D ≥ 3, which fills in some gaps and resolves some inconsistencies in earlier studies.

$1/8$-BPS couplings and exceptional automorphic functions

“…is a particular regularisation prescription for genus-two modular integrals introduced in [39,40]. The ansatz (1.6) automatically satisfies the differential constraint (1.5) and reproduces the known perturbative contributions up to genus three [2]. Moreover, its decompactification predicts the full SL(5, Z)-invariant ∇ 6 R 4 coupling in D = 7, (7) 189 E SL(5)…”

“…Supersymmetry Ward identities and U-duality determine uniquely the function E (d) (0,0) in (1.2) for d ≥ 3 and E (d) (1,0) for d ≥ 5, up to an overall 1 Note that in formulae which are valid for all d such as (1.2), we use the labelling associated to E d+1 which differs from the standard labelling for d ≤ 4 -for example the E5 labelling is whereas the D5 labelling is 4 1 2 3 5 . 2 Here and elsewhere in the paper we denote by γ f (x)|γ the Poincaré sum over coset elements γ acting on the seed function f by f (x)|γ = f (γ · x), where the action of γ on x is defined by the right action on the group element g(x) as g(x)γ. A more precise definition of the y k will be given in Section 2.1.…”

“…At next-to-next-to-leading order, the coefficient E (d) (0,1) of the ∇ 6 R 4 coupling is less well understood. It satisfies a set of inhomogeneous differential equations, the simplest one being the Poisson equation [31,22,2] where ∆ E d+1 is the Laplace-Beltrami operator for the E d+1 -invariant metric on M D , and the right-hand side involves the square of the R 4 coupling, plus anomalous terms when ultraviolet logarithmic divergences appear in supergravity. As a result, the weak coupling expansion includes perturbative contributions up to genus three, 1/8-BPS instantons as well as bound states of 1/2-BPS instantons and anti-instantons.…”

“…Using the theta lift representation of ϕ KZ established in [48], we trace the relation (1.21) to a similar property (Eq. (A.16) of Appendix A) relating genus-one Siegel-Narain theta series for lattices of signature (3,2) and (2,1). Inserting (1.21) inside the genus-two 8 For the particle multiplet k = d + 1, we shall argue that Ξ Ed+1 Λd+1 (τ, φ, r) converges for Re (r) > 4, 6, 9, 29 2 for d = 4, 5, 6, 7, respectively; for the string multiplet k = 1, that Ξ modular integral in (1.6) and unfolding the integration domain, one immediately arrives at 8π R.N.…”

Worldsheet (anti)instanton bound states in type II on T2

“…The relevant Eisenstein series that appear in (1.8) are defined by (A.60) and (A.62). The 1/8 BPS couplings have also been analyzed from the worldsheet perspective in [7][8][9][10][11], and from the spacetime point of view in [12][13][14][15][16][17][18][19][20][21][22][23]. Now in (1.8) all the contributions apart from that involving F (T, T ) are given by an infinite sum of terms involving either worldsheet instantons or anti-instantons.…”