Resurgence of Chern–Simons Theory at the Trivial Flat Connection

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 (<i>x</i>, <i>q</i>)-series whose rows are indexed by the boundary parabolic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow><mml:msub><mml:mtext>SL</mml:mtext> <mml:mn>2</mml:mn></mml:msub> <mml:mrow><mml:mo>(</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo></mml:mrow> </mml:mrow> </mml:math> -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.