Double-Janus Linear Sigma Models and Generalized Reciprocity for Gauss Sums

TL;DR

This paper explores the supersymmetric partition function of a 2d linear sigma model with a torus target space, revealing new identities for quadratic Gauss sums related to duality and geometry, and connecting Berry phases to supersymmetric configurations.

Contribution

It introduces novel identities for quadratic Gauss sums derived from supersymmetric partition functions with varying complex structures and duality twists, generalizing known mathematical relations.

Findings

Derived identities relating quadratic Gauss sums.

Connected Berry phases to supersymmetric Janus configurations.

Demonstrated the role of complex structure variation in supersymmetry.

Abstract

We study the supersymmetric partition function of a 2d linear $\sigma$-model whose target space is a torus with a complex structure that varies along one worldsheet direction and a K\"ahler modulus that varies along the other. This setup is inspired by the dimensional reduction of a Janus configuration of 4d $\mathcal{N}=4$ $U(1)$ Super-Yang-Mills theory compactified on a mapping torus ($T^2$ fibered over $S^1$) times a circle with an $SL(2,\mathbb{Z})$ duality wall inserted on $S^1$, but our setup has minimal supersymmetry. The partition function depends on two independent elements of $SL(2,\mathbb{Z})$, one describing the duality twist, and the other describing the geometry of the mapping torus. It is topological and can be written as a multivariate quadratic Gauss sum. By calculating the partition function in two different ways, we obtain identities relating different quadratic Gauss sums, generalizing the {\it Landsberg-Schaar} relation. These identities are a subset of a collection of identities discovered by F. Deloup. Each identity contains a phase which is an eighth root of unity, and we show how it arises as a Berry phase in the supersymmetric Janus-like configuration. Supersymmetry requires the complex structure to vary along a semicircle in the upper half-plane, as shown by Gaiotto and Witten in a related context, and that semicircle plays an important role in reproducing the correct Berry phase.