Peacock patterns and new integer invariants in topological string theory

TL;DR

This paper explores the factorial divergence in topological string theory near the conifold point, introduces integer invariants called Stokes constants, and confirms their integrality through calculations in toric examples using the TS/ST correspondence.

Contribution

It introduces the concept of peacock patterns and conjectures Stokes constants as integer invariants, providing explicit calculations and generating functions in toric cases.

Findings

Confirmed Stokes constants are integer invariants in some toric examples.

Established a connection between topological string theory and complex Chern-Simons theory.

Provided explicit q-series generating functions for new invariants.

Abstract

Topological string theory near the conifold point of a Calabi-Yau threefold gives rise to factorially divergent power series which encode the all-genus enumerative information. These series lead to infinite towers of singularities in their Borel plane (also known as "peacock patterns"), and we conjecture that the corresponding Stokes constants are integer invariants of the Calabi-Yau threefold. We calculate these Stokes constants in some toric examples, confirming our conjecture and providing in some cases explicit generating functions for the new integer invariants, in the form of q-series. Our calculations in the toric case rely on the TS/ST correspondence, which promotes the asymptotic series near the conifold point to spectral traces of operators, and makes it easier to identify the Stokes data. The resulting mathematical structure turns out to be very similar to the one of complex Chern-Simons theory. In particular, spectral traces correspond to state integral invariants and factorize in holomorphic/anti-holomorphic blocks.