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 SL+b(2,ℝ). A quantum measure space is a von Neumann algebra equipped with a weight. We study such spaces carrying a weight-preserving action of SL+b(2,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 SL+b(2,ℝ)×SL+b(2,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 III1 factor. Liouville CFT is realized as SL+b(2,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 SL+b(2,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.
Related
- Construction of vector-valued adelic automorphic forms
- Haagerup conformal field theory at central charge eight
- Vertex operator algebra bundles on Riemann surfaces of higher genus and automorphic forms for Fuchsian groups
- Harder's conjecture and Hermitian automorphic forms
- K-Moduli Wall Crossing and Automorphic Forms for the Moduli Space of Rational Elliptic Surfaces
BibTeX
@misc{mazac2026conformal,
title = {{Conformal field theory from automorphic forms on a quantum group}},
author = {Dalimil Mazac},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.06907},
eprint = {2611.06907},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.06907},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Conformal field theory from automorphic forms on a quantum group
AU - Dalimil Mazac
PY - 2026
DA - 2026/11/04
PB - arXiv
DO - 10.48550/arxiv.2611.06907
UR - https://arxiv.org/abs/2611.06907
AB - 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 SL^+_b(2,ℝ). A quantum measure space is a von Neumann algebra equipped with a weight. We study such spaces carrying a weight-preserving action of SL^+_b(2,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 SL^+_b(2,ℝ)×SL^+_b(2,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 SL^+_b(2,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 SL^+_b(2,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.
ER -