Chern-Simons理论在平凡平坦连接处的 resurgence
Resurgence of Chern-Simons theory at the trivial flat connection
AI总结：
本文研究了Chern-Simons理论在平凡平坦连接处的 resurgence 结构，通过扩展的(x,q)级数矩阵描述了超曲面结补的奇异性和 Stokes 常数，并扩展了Kashaev不变量和着色Jones多项式。
摘要：
数年前，第一作者推测了3流形在平凡平坦连接处的Chern-Simons扰动理论是一个resurgent幂级数。我们完全描述了该级数的resurgent结构（包括奇点的位置及其Stokes常数），在超曲面结补的情况下，通过一个扩展的(x,q)级数矩阵来描述，其行由边界抛物线SL₂(C)平坦连接索引，包括平凡连接。我们利用该矩阵描述上述级数的Stokes常数，定义其Borel变换并将其与状态积分联系起来。在过程中，我们利用该矩阵对Kashaev不变量和着色Jones多项式进行解析延拓，并完善了矩阵值的全纯量子模形式，以及给出了Zagier和第一作者的精炼量子模性猜想的精确版本。最后，该矩阵扩展了3D索引在平凡平坦连接的某个扇区。我们用两个最简单的超曲面结来说明我们的定义、定理、数值计算和猜想。
英文摘要：
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 $\text{SL}_2(\mathbb{C})$-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.
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