The automorphic NS5-brane

TL;DR

This work proposes that NS5-brane corrections to the hypermultiplet moduli space in type II string theory can be non-perturbatively captured by an SL(3,Z)–invariant automorphic form, specifically a non-holomorphic Eisenstein series with (s1,s2)=(3/2,-3/2). It introduces the extended universal hypermultiplet M_u ≅ SO(3)ackslash SL(3,R) as the universal sector and uses its SL(3,R) structure to organize perturbative, D(-1)/D5, and NS5 contributions via Abelian and non-Abelian Fourier coefficients of the Eisenstein series. The analysis yields explicit expressions for constant terms, Abelian coefficients (D(-1)/D5), and non-Abelian coefficients (NS5-bound states), along with a minimal theta series that could serve as an NS5-brane partition function, drawing connections to topological strings and quasi-conformal representations. While offering a coherent automorphic framework, the work also discusses potential divergences and the need for further development of the twistor-space data and extensions to general symmetric moduli spaces, highlighting the broader significance of automorphic methods in non-perturbative string theory corrections.

Abstract

Understanding the implications of SL(2,Z) S-duality for the hypermultiplet moduli space of type II string theories has led to much progress recently in uncovering D-instanton contributions. In this work, we suggest that the extended duality group SL(3,Z), which includes both S-duality and Ehlers symmetry, may determine the contributions of D5 and NS5-branes. We support this claim by automorphizing the perturbative corrections to the "extended universal hypermultiplet", a five-dimensional universal SL(3,R)/SO(3) subspace which includes the string coupling, overall volume, Ramond zero-form and six-form and NS axion. Using the non-Abelian Fourier expansion of the Eisenstein series attached to the principal series of SL(3,R), first worked out by Vinogradov and Takhtajan 30 years ago, we extract the contributions of D(-1)-D5 and NS5-brane instantons, corresponding to the Abelian and non-Abelian coefficients, respectively. In particular, the contributions of k NS5-branes can be summarized into a vector of wave functions Ψ_{k,l}, l=0... k-1, as expected on general grounds. We also point out that for more general models with a symmetric moduli space G/K, the minimal theta series of G generates an infinite series of exponential corrections of the form required for "small" D(-1)-D1-D3-D5-NS5 instanton bound states. As a mathematical spin-off, we make contact with earlier results in the literature about the spherical vectors for the principal series of SL(3,R) and for minimal representations.