Continous Spins in 2D Gravity: Chiral Vertex Operators and Local Fields

TL;DR

This paper develops a framework for continuous-spin Liouville fields in 2D gravity, constructing their operators, analyzing their quantum group structure, and verifying their fundamental equations and braiding properties.

Contribution

It introduces a novel operator approach for continuous powers of the Liouville field and explores their quantum group symmetries and braiding relations.

Findings

Constructed exponential operators with continuous powers of the Liouville field.

Verified the canonical commutation relations and quantum Liouville equations.

Derived the braiding relations consistent with an extended quantum group structure.

Abstract

We construct the exponentials of the Liouville field with continuous powers within the operator approach. Their chiral decomposition is realized using the explicit Coulomb-gas operators we introduced earlier. {}From the quantum-group viewpoint, they are related to semi-infinite highest or lowest weight representations with continuous spins. The Liouville field itself is defined, and the canonical commutation relations verified, as well as the validity of the quantum Liouville field equations. In a second part, both screening charges are considered. The braiding of the chiral components is derived and shown to agree with the ansatz of a parallel paper of J.-L. G. and Roussel: for continuous spins the quantum group structure $U_q(sl(2)) \odot U_{\qhat}(sl(2))$ is a non trivial extension of $U_q(sl(2))$ and $U_{\qhat}(sl(2))$. We construct the corresponding generalized exponentials and the generalized Liouville field.