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Abstract |
We will discuss recent analytic progress on the "non-rational" Virasoro Crossing Kernels that implement modular and crossing transformations on torus 1-point and sphere 4-point Virasoro blocks, respectively, in 2d CFTs. These kernels are realized as solutions to special difference equations arising, in turn, from the "non-rational" versions of the Moore-Seiberg polynomial relations. Using techniques from quantum topology and the so-called `state integrals', we will show that at rational values of the central charge in the regime (-\infty,1]U[25,+\infty), and generic Liouville momenta, there are new solutions to these difference equations that have the following features: (i) are non-meromoprhic as functions of the Liouville momenta, (ii) are non-reflection symmetric in the momenta, (iii) take a "semi-classical and one-loop" exact form. The last feature will lead us to propose a "quantum modularity" conjecture for these kernels as functions of the central charge. If there is time, we will discuss how these results lead us to prove analytically for the first time that Liouville theory with c < = 1 is modular covariant and crossing symmetric. Based on https://arxiv.org/pdf/2512.03172. |
