Modular $q$-holonomic modules

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 [22,23,24,33], 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 Contents 1. Introduction 2 1.1. Summary 2 1.2. Motivation 3 1.3. Definition and properties 3 1.4. The generalised q-hypergeometric equation is modular 5 1.5. Modular q-holonomic modules in Chern-Simons theory 7 2. A review of linear q-difference equations 12 2.1. Preliminaries 12 2.2. An algorithm for a fundamental matrix 13 2.3. The q-Borel and the q-Laplace transforms 14 2.4. The slash operator and proof of Theorem 1.2 17 2.5. Duality 20 2.6. Categorical aspects 23 3. Heine's q-hypergeometric functions 24 3.1. Solutions 24 3.2. Self duality 25 3.3. State integral 28 3.4. Resonance 29 4. Proof of Theorems 1.5 and 1.6 30 4.1. The q-Pochhammer symbol 30 4.2. The Appell-Lerch sums 32

A Generalization of Zwegers' μ-Function According to the q -Hermite-Weber Difference Equation

“…As a corollary, we obtain an interpretation of µ(u, v) = µ(u, v; 1) that the original µ-function is regarded as the image of the q-Borel and q-Laplace transformations of the fundamental solution f 0 of the q-Hermite-Weber equation (very recently S. Garoufalidis and C. Wheeler [7] pointed out this fact independently of us). This interpretation gives some answers to our questions mentioned before.…”

“…During the preparation of this paper, we have been informed by Hikami and Matsusaka of a recent very interesting paper [7]. Some of the examples of the paper [7] overlap with our results on the µ-function, though the one parameter generalization µ(u, v; α) is not considered there (e.g., see equation (1.18) and [7, Section 4.2]).…”

“…We also wish to thank Professor Yosuke Ohyama (Tokushima University) for his valuable suggestions on q-special functions, including in the unpublished proof of Lemma 2.2 [17]. We are also indebted to Professor Kazuhiro Hikami (Kyushu University) for his information for [7] and quantum invariants. Professor Toshiki Matsusaka (Kyushu University) also provides information on [7] and his note [15].…”

“…We are also indebted to Professor Kazuhiro Hikami (Kyushu University) for his information for [7] and quantum invariants. Professor Toshiki Matsusaka (Kyushu University) also provides information on [7] and his note [15]. Some pieces of information on the q-Appell hypergeometric function Φ (1) and its q-difference equations are provided by Dr. T. Nobukawa.…”

Knots and Their Related $q$-Series

“…Related aspects of this work appeared in [13,14,21,27,36,37]. Modular linear q-difference equations were introduced in [29]. An extension of the matrix-valued q-series to a matrix of one additional row and column that sees the trivial PSL 2 (C)-representation was given in [24].…”