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

TL;DR

This work addresses the monodromy of the deformed $\mathfrak{sl}(2)$ Casimir connection and computes traces of its monodromy on spaces of flat sections associated with Verma modules. It demonstrates that these traces reproduce modular theta-type objects, specifically the Jacobi theta-constant and partial theta functions, and that partial Appell-Lerch sums arise from traces over tensor products of representations. The paper derives explicit trace formulas and decompositions, notably $\mathsf{M}_0 \otimes \mathsf{M}_0 = \bigoplus_{i \ge 0} \mathsf{M}_{-2i}$, linking monodromy to $q$-series and proposing a conjectural relation for these structures, while also introducing a deformation $\nabla(\phi)$ and its modular behavior. These results illuminate deep connections between representation-theoretic monodromy and modular-type special functions, with potential extensions to higher rank algebras in future work.

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.

Theorems & Definitions (8)

• Theorem 2.1
• proof
• Theorem 3.1
• proof
• Definition 4.1
• Theorem 4.1
• proof
• Conjecture 4.1