Conformal TBA for Resolved Conifolds

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.