Double-Janus linear sigma models and generalized reciprocity for Gauss sums

A bstract We study the supersymmetric partition function of a 2d linear σ -model whose target space is a torus with a complex structure that varies along one worldsheet direction and a Kähler modulus that varies along the other. This setup is inspired by the dimensional reduction of a Janus configuration of 4d 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 , ℤ) duality wall inserted on S 1 , but our setup has minimal supersymmetry. The partition function depends on two independent elements of SL(2 , 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 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.