On $R^4$ threshold corrections in IIB string theory and (p,q) string instantons

TL;DR

The paper derives exact non-perturbative thresholds for $R^4$ terms in type IIB string theory compactified to eight and seven dimensions, showing these thresholds assemble into U-duality invariant non-holomorphic Eisenstein series. In eight dimensions, the threshold is the sum of an $SL(2, ext{Z})$ order-1 and an $SL(3, ext{Z})$ order-3/2 Eisenstein series, while in seven dimensions the threshold is an $SL(5, ext{Z})$ order-3/2 Eisenstein series; a regulator introduces a logarithmic term to preserve invariance. The authors also conjecture exact non-perturbative thresholds in lower dimensions and discuss their relation to M-theory, providing a unified duality-covariant framework for BPS-saturated $R^4$ corrections. They demonstrate that D-instanton and $(p,q)$-string instanton effects, together with perturbative contributions, are encoded by discrete Eisenstein series of the relevant U-duality groups. This work deepens the connection between non-perturbative string dynamics and automorphic forms across dimensions.

Abstract

We obtain the exact non-perturbative thresholds of $R^4$ terms in IIB string theory compactified to eight and seven dimensions. These thresholds are given by the perturbative tree-level and one-loop results together with the contribution of the D-instantons and of the (p,q)-string instantons. The invariance under U-duality is made manifest by rewriting the sum as a non-holomorphic modular function of the corresponding discrete U-duality group. In the eight-dimensional case, the threshold is the sum of a order-1 Eisenstein series for SL(2,Z) and a order-3/2 Eisenstein series for SL(3,Z). The seven-dimensional result is given by the order-3/2 Eisenstein series for SL(5,Z). We also conjecture formulae for the non-perturbative thresholds in lower dimensional compactifications and discuss the relation with M-theory.