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- Title
Peacock patterns and resurgence in complex Chern–Simons theory.
- Authors
Garoufalidis, Stavros; Gu, Jie; Mariño, Marcos
- Abstract
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 a distinguished entry of the Stokes automorphism matrix is the 3D-index 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.
- Subjects
CHERN-Simons gauge theory; PLANCK'S constant; PEAFOWL; INTEGRAL functions; HOLOMORPHIC functions; POWER series; FACTORIZATION; PARTITION functions
- Publication
Research in the Mathematical Sciences, 2023, Vol 10, Issue 3, p1
- ISSN
2522-0144
- Publication type
Article
- DOI
10.1007/s40687-023-00391-1