Twist operator correlator revisited and tau function on Hurwitz space

Abstract: Correlation function of twist operators is a natural quantity of interest in two-dimensional conformal field theory (2d CFT) and finds relevance in various physical contexts. For computing twist operator correlators associated with generic branched covers of genus zero and one, we present a generalization of the conventional stress-tensor method to encompass generic 2d CFTs without relying on any free field realization. This is achieved by employing a generalization of the argument of Calabrese-Cardy in the cyclic genus zero case. The generalized stress-tensor method reveals a compelling relation between the twist operator correlator and the tau function on Hurwitz space, the moduli space of branched covers, of Kokotov-Korotkin. This stems from the close relation between stress-tensor one-point function and Bergman projective connection of branched cover. The tau function on Hurwitz space is in turn related to the more general isomonodromic tau function, and this chain of correspondence thus relates the twist operator correlator to a canonical algebro-geometric object and endows it with an integrable system interpretation. Conversely, the tau function on Hurwitz space essentially admits a CFT interpretation as the holomorphic part of the twist operator correlator of $c=1$ free boson.