Modular transformations of tau functions and conformal blocks on the torus

Abstract

The connection problem for isomonodromic tau functions on the one-punctured torus concerns the ratio between the tau function and its modular transform, associated to dual pants decompositions of the torus. In this paper, we study the modular transformations of the tau function and consequently derive the connection constant. Moreover, through the relation with two-dimensional Conformal Field Theory, we also obtain an exact closed formula for the $c=1$ Virasoro modular kernel, whose expression was previously unknown, and relate it to the $c\rightarrow\infty$ (semiclassical) modular kernel and $SL_2(\mathbb{C})$ complex Chern-Simons amplitudes. Finally, we prove that the connection constant and the two, $c=1$ and $c\to \infty$, modular kernels are generating functions of canonical transformations on the character variety of the one-punctured torus. Our results are also relevant for the $\mathcal{N}=2^*$ gauge theory.