Peacock patterns and resurgence in complex Chern-Simons theory. (English) Zbl 07715597
Summary: 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 \(\tau\) 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 \(\mathbf{4}_1\) and \(\mathbf{5}_2\) knots.
MSC:
| 57K16 | Finite-type and quantum invariants, topological quantum field theories (TQFT) |
| 81T45 | Topological field theories in quantum mechanics |
Keywords:
Chern-Simons theory; holomorphic blocks; state-integrals; knots; 3-manifolds; resurgence; perturbative series; Borel resummation; Stokes automorphisms; Stokes constants; \(q\)-holonomic modules; \(q\)-difference equations; DT-invariants; BPS states; peacocks; meromorphic quantum Jacobi formsSoftware:
HolonomicFunctionsReferences:
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