Generalized theta series and monodromy of a Casimir connection. Case of rank 1

The monodromy of the \(\mathfrak{sl}(2)\) Casimir connection is considered. It is shown that the trace of the monodromy operator over an appropriate space of flat sections gives the Jacobi theta constant and incomplete theta functions. A definition of new objects, namely, incomplete Appell–Lerch sums, is given, and their connection with the trace of the monodromy operator is revealed.