Quantum groups and Nichols algebras acting on conformal field theories

Abstract

We prove that certain screening operators in conformal field theory obey the algebra relations of a corresponding Nichols algebra with diagonal braiding. Our result proves in particular a long-standing expectation that the Borel parts of small quantum groups appear as the algebra of screening operators. The proof is based on a novel, intimate relation between Hopf algebras, vertex algebras and a class of multivalued analytic special functions, which are generalizations of Selberg integrals. We prove that the zeroes of these special functions correspond to the algebra relations of the respective Nichols algebra, by proving an analytical quantum symmetrizer formula for the functions. Moreover, certain poles of the functions encode module extensions and a Weyl group action. At other poles, the quantum {symmetrizer} formula fails and the screening operators generate an extension of the Nichols algebra. The intended application of our result is the conjectural logarithmic Kazhdan-Lusztig correspondence. More generally, our result seems to suggest that non-local screening operators in an arbitrary vertex algebra should be described by appropriate Nichols algebras, just as local screening operators can be described by Lie algebras.