We study the Masur-Veech volumes MV g,n of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus g with n punctures. We show that the volumes MV g,n are the constant terms of a family of polynomials in n variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of [11] proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in [3]. We also obtain an expression of the area Siegel-Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur-Veech volumes, and thus of area Siegel-Veech constants, for low g and n, which leads us to propose conjectural formulas for low g but all n.

We study the Masur-Veech volumes MV g,n of the principal stratum of the moduli space of quadratic differentials of unit area on curves of genus g with n punctures. We show that the volumes MV g,n are the constant terms of a family of polynomials in n variables governed by the topological recursion/Virasoro constraints. This is equivalent to a formula giving these polynomials as a sum over stable graphs, and retrieves a result of [17] proved by combinatorial arguments. Our method is different: it relies on the geometric recursion and its application to statistics of hyperbolic lengths of multicurves developed in [3]. We also obtain an expression of the area Siegel-Veech constants in terms of hyperbolic geometry. The topological recursion allows numerical computations of Masur-Veech volumes, and thus of area Siegel-Veech constants, for low g and n, which leads us to propose conjectural formulas for low g but all n. We also relate our polynomials to the asymptotic counting of square-tiled surfaces with large boundaries.

Some years ago, it was conjectured by the first author that the Chern-Simons perturbation theory of a 3-manifold at the trivial flat connection is a resurgent power series. We describe completely the resurgent structure of the above series (including the location of the singularities and their Stokes constants) in the case of a hyperbolic knot complement in terms of an extended square matrix of (x, q)-series whose rows are indexed by the boundary parabolic SL 2 (C)-flat connections, including the trivial one. We use our extended matrix to describe the Stokes constants of the above series, to define explicitly their Borel transform and to identify it with state-integrals. Along the way, we use our matrix to give an analytic extension of the Kashaev invariant and of the colored Jones polynomial and to complete the matrix valued holomorphic quantum modular forms as well as to give an exact version of the refined quantum modularity conjecture of Zagier and the first author. Finally, our matrix provides an extension of the 3D-index in a sector of the trivial flat connection. We illustrate our definitions, theorems, numerical calculations and conjectures with the two simplest hyperbolic knots.Contents 2.1. A 2 × 2 matrix of q-series 2.2. A 3 × 3 matrix of q-series 2.3. The Φ (σ0) (τ ) asymptotic series 2.4. Borel resummation and Stokes constants 2.5. The Andersen-Kashaev state-integral 2.6. A new state-integral 2.7. A 3 × 3 matrix of state-integrals 3. The x-variable 3.1. The Φ (σ0) (x, τ ) series 3.2. A 3 × 3 matrix of (x, q)-series 3.3. Borel resummation and Stokes constants

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