Abstract
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 mathcal{N} = 4 U(1) Super-Yang-Mills theory compactified on a mapping torus (T2 fibered over S1) times a circle with an SL(2, ℤ) duality wall inserted on S1, 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.
Highlights
In these notes, we study the torus partition function of a supersymmetric 2d linear σmodel with T 2 target space whose complex structure varies along one of the worldsheet directions and whose Kähler modulus varies along the other direction
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 S1) times a circle with an SL(2, Z) duality wall inserted on S1, but our setup has minimal supersymmetry
Summary
We study the torus partition function of a supersymmetric 2d linear σmodel with T 2 target space (a free theory) whose complex structure varies along one of the worldsheet directions (parametrized by 0 ≤ σ1 < 1) and whose Kähler modulus varies along the other direction (parametrized by 0 ≤ σ2 < 1). The setup for the present paper is derived from an abelian double-Janus configuration that is a prelude to the study of a nonabelian theory It can be obtained as a limit of a Gaiotto-Witten Janus configuration by compactifying on a small torus and allowing its complex structure parameter to vary as a function of time. We use this dual formulation to determine the precise normalization of the partition function.
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