Conformal blocks, parenthesized braid operad, and $c=1/2$ Virasoro vertex operator algebra

Abstract

We review the construction of a pseudo-braided category structure on the $C_1$-cofinite module category of a vertex operator algebra using conformal blocks and analytic continuation along paths in configuration spaces. In the rational $C_2$-cofinite case, the pseudo-braided category is represented by tensor products and becomes a balanced braided tensor category. We then compute all four-point conformal blocks of the Virasoro vertex operator algebra of central charge $1/2$ in terms of hypergeometric functions. We explain how analytic continuation of these blocks determines the braiding and associator, and identify the resulting module category with the Tambara--Yamagami category over ${\mathbb{Z}}_2$ as a balanced braided tensor category.

Contribution to the proceedings of the 2nd AMS-UMI International Joint Meeting, Special Session "New Developments in Infinite-Dimensional Lie Algebras, Vertex Operator Algebras and the Monster", Palermo, 2024