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

TL;DR

This paper computes exact non-perturbative $R^4$ correction thresholds in IIB string theory compactified to 8 and 7 dimensions, incorporating instanton effects and demonstrating U-duality invariance through modular functions.

Contribution

It provides explicit formulas for non-perturbative $R^4$ thresholds in lower dimensions using Eisenstein series and discusses their relation to M-theory, extending previous perturbative results.

Findings

Thresholds expressed as Eisenstein series for U-duality groups.

Invariance under U-duality made explicit through modular functions.

Conjectured formulas for thresholds in lower 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.