  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.
