Quantum Group Structure and Local Fields in the Algebraic Approach to 2D Gravity

TL;DR

This paper reviews the operator approach to 2D gravity, focusing on local observables, quantum group structures, and the formulation of a quantum tau function within the algebraic framework.

Contribution

It introduces a quantum group-based algebraic construction of local fields and observables in 2D gravity, including a noncommutative quantum tau function.

Findings

Construction of local Liouville exponentials and fields

Identification of double quantum group structure

Definition of a quantum tau function

Abstract

This review contains a summary of work by J.-L. Gervais and the author on the operator approach to 2d gravity. Special emphasis is placed on the construction of local observables -the Liouville exponentials and the Liouville field itself - and the underlying algebra of chiral vertex operators. The double quantum group structure arising from the presence of two screening charges is discussed and the generalized algebra and field operators are derived. In the last part, we show that our construction gives rise to a natural definition of a quantum tau function, which is a noncommutative version of the classical group-theoretic representation of the Liouville fields by Leznov and Saveliev.