QM Research Seminar: Abelianization of Virasoro conformal blocks at $c=1$

Speaker: Qianyu Hao (University of Geneva)


Abstract:

Conformal blocks are essential objects to study in the 2d CFTs. They depend on the data of a vertex algebra $\CV$, a punctured Riemann surface $C$, and possible decorations inserted at the punctures. The Virasoro conformal blocks are very interesting since they have many connections to other areas of math and physics. I will describe a new way to construct Virasoro conformal blocks at $c=1$. This is closely related to the idea of nonabelianization in the study of $SL(N,\mathbb{C})$ connections by using $GL(1,\mathbb{C})$ connection in the work of Gaiotto-Moore-Neitzke and Hollands-Neitzke. I will talk about our work on relating the $c=1$ Virasoro conformal blocks on $C$ to the "abelian" Heisenberg conformal blocks on a branched double cover of $C$. The main new idea in our work is the use of the spectral network on the surface $C$. The nonabelianization construction enables us to study the harder to get Virasoro conformal blocks using the simpler abelian objects. This is joint work in progress with Andrew Neitzke.