Conformal TBA for Resolved Conifolds

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.
Comment: 25+14 pages; added proof for integral representation of the double sine function and a similar conjecture for the triple sine; version accepted for publication in Annales Henri Poincar\'e