Supersymmetry constraints on the R^4 multiplet in type IIB on T^2

TL;DR

This paper investigates the supersymmetry constraints on R^4 interactions in type IIB string theory compactified on T^2, deriving differential equations for moduli-dependent couplings and identifying their automorphic properties.

Contribution

It derives first and second order differential equations for moduli-dependent couplings in the R^4 multiplet, revealing their automorphic forms and Poisson equations in a compactified setting.

Findings

Couplings satisfy Laplace and Poisson equations on moduli space.

R^4 coupling depends on all moduli and exhibits specific automorphic properties.

Supersymmetry constrains the form of moduli-dependent interactions.

Abstract

We consider a class of eight derivative interactions in the effective action of type IIB string theory compactified on T^2. These 1/2 BPS interactions have moduli dependent couplings. We impose the constraints of supersymmetry to show that each of these couplings satisfy a first order differential equation on moduli space which relate it to other couplings in the same supermuliplet. These equations can be iterated to give second order differential equations for the various couplings. The couplings which only depend on the SO(2)\SL(2,R) moduli satisfy Laplace equation on moduli space, and are given by modular forms of SL(2,Z). On the other hand, the ones that only depend on the SO(3)\SL(3,R) moduli satisfy Poisson equation on moduli space, where the source terms are given by other couplings in the same supermultiplet. The couplings of the interactions which are charged under SU(2) are not automorphic forms of SL(3,Z). Among the interactions we consider, the R^4 coupling depends on all the moduli.