Hidden $U_q(sl(2))\otimes U_q(sl(2))$ Quantum Group Symmetry in Two Dimensional Gravity

TL;DR

This paper reveals a hidden $U_q(sl(2)) imes U_q(sl(2))$ quantum group symmetry in 2D gravity, demonstrating its properties and structure within the Coulomb gas vertex operator framework, including at roots of unity.

Contribution

The paper establishes for the first time the quantum group symmetry in 2D gravity using Coulomb gas operators, revealing a novel Hopf-like structure and detailed symmetry properties.

Findings

Identifies a hidden $U_q(sl(2)) imes U_q(sl(2))$ symmetry in 2D gravity.

Defines the co-product with a matching condition based on Hilbert space structure.

Shows the symmetry truncates correctly at roots of unity.

Abstract

In a previous paper, we proposed a construction of $U_q(sl(2))$ quantum group symmetry generators for 2d gravity, where we took the chiral vertex operators of the theory to be the quantum group covariant ones established in earlier works. The basic idea was that the covariant fields in the spin $1/2$ representation themselves can be viewed as generators, as they act, by braiding, on the other fields exactly in the required way. Here we transform this construction to the more conventional description of 2d gravity in terms of Bloch wave/Coulomb gas vertex operators, thereby establishing for the first time its quantum group symmetry properties. A $U_q(sl(2))\otimes U_q(sl(2))$ symmetry of a novel type emerges: The two Cartan-generator eigenvalues are specified by the choice of matrix element (bra/ket Verma-modules); the two Casimir eigenvalues are equal and specified by the Virasoro weight of the vertex operator considered; the co-product is defined with a matching condition dictated by the Hilbert space structure of the operator product. This hidden symmetry possesses a novel Hopf like structure compatible with these conditions. At roots of unity it gives the right truncation. Its (non linear) connection with the $U_q(sl(2))$ previously discussed is disentangled.