Generalized theta series and monodromy of Casimir connection. The case of rank 1

TL;DR

This paper investigates the monodromy of the sl(2) Casimir connection, revealing that its trace relates to classical special functions like the Jacobi theta constant and Appell-Lerch sums, thus connecting monodromy with special functions.

Contribution

It establishes a novel link between the monodromy of the sl(2) Casimir connection and classical special functions, providing explicit formulas for the trace of the monodromy operator.

Findings

Trace of monodromy operator equals Jacobi theta constant.

Trace relates to partial Appell-Lerch sums.

Connects monodromy with special functions in rank 1 case.

Abstract

The monodromy of the $\mfsl(2)$ Casimir connection is considered. It is shown that the trace of the monodromy operator over the appropriate space of flat sections gives rise to the Jacobi theta constant and to the partial Appell-Lerch sums.