Conformal TBA for resolved conifolds

TL;DR

This paper investigates the Riemann-Hilbert problem related to Donaldson-Thomas invariants for resolved conifolds, proposing new prescriptions to handle divergences and connecting solutions to topological string theory.

Contribution

It introduces alternative well-behaved solutions to the TBA equations for resolved conifolds and links asymptotic expansions to topological string partition functions.

Findings

Reproduces existing solutions using new prescriptions.

Identifies a better-behaved solution in a specific limit.

Connects asymptotic expansion of the τ function to genus expansion.

Abstract

We revisit the Riemann-Hilbert problem determined by Donaldson-Thomas invariants for the resolved conifold and for other small crepant resolutions. While this problem can be recast as a system of TBA-type equations in the conformal limit, solutions are ill-defined due to divergences in the sum over infinite trajectories in the spectrum of D2-D0-brane bound states. We explore various prescriptions to make the sum well-defined, show that one of them reproduces the existing solution in the literature, and identify an alternative solution which is better behaved in a certain limit. Furthermore, we show that a suitable asymptotic expansion of the $\tau$ function reproduces the genus expansion of the topological string partition function for any small crepant resolution. As a by-product, we conjecture new integral representations for the triple sine function, similar to Woronowicz' integral representation for Faddeev's quantum dilogarithm.