论文标题
孔雀模式和复杂的Chern-Simons理论的复兴
Peacock patterns and resurgence in complex Chern-Simons theory
论文作者
论文摘要
复杂的Chern-Simons理论在具有圆环边界的3个manifold上的分区函数还原为有限的尺寸状态融合，这是复杂的切割平面中复杂的普朗克常数$τ$的全态函数，并且是复杂参数$ u $的整个函数。这产生了一个分歧扰动形式功率系列的矢量，其stokes射线在复杂平面中形成了类似孔雀的模式。 我们猜想这些扰动序列是复兴的，它们的跨系列涉及两个非扰动变量，它们的Stokes自动形态满足了独特的分解属性，并且根据（双重）线性$ q $ -Difference方程的基本矩阵解决方案明确给出了它。我们进一步猜测，Stokes自动形态矩阵的条目是Dimofte-Gaiotto-Gukov的3D指数。我们提供有关$ Q $ - 差异方程式及其基本解决方案的属性的证明，并说明了我们对Stokes矩阵的猜想，该矩阵具有数值计算，其中两个最简单的双曲线$ 4_1 $和$ 5_2 $结。
The partition function of complex Chern-Simons theory on a 3-manifold with torus boundary reduces to a finite dimensional state-integral which is a holomorphic function of a complexified Planck's constant $τ$ in the complex cut plane and an entire function of a complex parameter $u$. This gives rise to a vector of factorially divergent perturbative formal power series whose Stokes rays form a peacock-like pattern in the complex plane. We conjecture that these perturbative series are resurgent, their trans-series involve two non-perturbative variables, their Stokes automorphism satisfies a unique factorization property and that it is given explicitly in terms of a fundamental matrix solution to a (dual) linear $q$-difference equation. We further conjecture that entries of the Stokes automorphism matrix are the 3D-indices of Dimofte-Gaiotto-Gukov. We provide proofs of our statements regarding the $q$-difference equations and their properties of their fundamental solutions and illustrate our conjectures regarding the Stokes matrices with numerical calculations for the two simplest hyperbolic $4_1$ and $5_2$ knots.
