Quantum Group as Semi-infinite Cohomology

TL;DR

This paper constructs the quantum group SL_q(2) as semi-infinite cohomology of the Virasoro algebra using braided vertex operator algebras, linking it to string theory and Liouville models.

Contribution

It introduces a novel cohomological construction of quantum groups via braided VOAs with complementary central charges.

Findings

Explicit generators of SL_q(2) identified

Relations among generators verified

Connections to Liouville and minimal string theory discussed

Abstract

We obtain the quantum group $SL_q(2)$ as semi-infinite cohomology of the Virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges $c+\bar{c}=26$. Each braided VOA is constructed from the free Fock space realization of the Virasoro algebra with an additional q-deformed harmonic oscillator degree of freedom. The braided VOA structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology. We explicitly provide the four cohomology classes that serve as the generators of $SL_q(2)$ and verify their relations. We also discuss the possible extensions of our construction and its connection to the Liouville model and minimal string theory.