Modular $q$-holonomic modules

TL;DR

This paper introduces modular $q$-holonomic modules with enhanced analyticity, demonstrating their modularity in key $q$-hypergeometric and Chern--Simons related modules, and offers a new approach to solving related $q$-difference equations.

Contribution

It defines modular $q$-holonomic modules with improved properties and shows their applicability to important $q$-hypergeometric and Chern--Simons modules, linking structural properties and solution methods.

Findings

The fundamental matrices form a cocycle with better analyticity.

The generalised $q$-hypergeometric equation is modular.

Key $q$-holonomic modules in Chern--Simons theory are modular.

Abstract

We introduce the notion of modular $q$-holonomic modules whose fundamental matrices define a cocycle with improved analyticity properties and show that the generalised $q$-hypergeometric equation, as well as three key $q$-holonomic modules of complex Chern--Simons theory are modular. This notion explains conceptually recent structural properties of quantum invariants of knots and 3-manifolds, and of exact and perturbative Chern--Simons theory, and in addition provides an effective method to solve the corresponding linear $q$-difference equations. An alternative title of our paper, emphasising the equations rather than the modules, is: Modular linear $q$-difference equations.