Conformal field theory from automorphic forms on a quantum group

Abstract

We reformulate two-dimensional conformal field theory (2d CFT) in the language of noncommutative geometry. To this end, we introduce automorphic forms for the von Neumann algebraic quantum group $\mathrm{SL}^+_b(2,\mathbb{R})$. A quantum measure space is a von Neumann algebra equipped with a weight. We study such spaces carrying a weight-preserving action of $\mathrm{SL}^+_b(2,\mathbb{R})$. We argue that every unitary bosonic 2d CFT with central charge $c=1+6(b+b^{-1})^2>1$ is equivalent to a quantum measure space with an action of $\mathrm{SL}^+_b(2,\mathbb{R})\times\mathrm{SL}^+_b(2,\mathbb{R})$. In this dictionary, Virasoro primary operators are quantum automorphic forms and their scaling dimensions correspond to the spectrum of a quantum Laplace operator. The Virasoro crossing equations express associativity of multiplication on the quantum measure space. For compact theories, the algebra is a type III$_1$ factor. Liouville CFT is realized as $\mathrm{SL}^+_b(2,\mathbb{R})$ acting on itself by left and right translations. Virasoro boundary conditions correspond to quantum measure spaces with an action of a single copy of $\mathrm{SL}^+_b(2,\mathbb{R})$. When the boundary spectrum is discrete, these are noncommutative counterparts of compact hyperbolic surfaces. The formulation accommodates models both with and without extended chiral algebras. It bridges the Virasoro conformal bootstrap with categorical and von Neumann algebraic approaches to 2d CFT.