Nonlinear $\hat{W}_{\infty}$ Current Algebra in the SL(2,R)/U(1) Coset Model

TL;DR

This paper introduces a free-field realization of the nonlinear $ olinebreak ext{ extasciitilde}W_{ ext{ extasciitilde} ext{infinity}}$ algebra in the classical $SL(2,R)/U(1)$ coset model, revealing a quantum deformation that leads to an infinite set of conserved charges.

Contribution

It provides a new free-field representation and generating function for the $ olinebreak ext{ extasciitilde}W_{ ext{ extasciitilde} ext{infinity}}$ algebra in the $SL(2,R)/U(1)$ model, connecting classical and quantum structures.

Findings

Identification of a classical $ olinebreak ext{ extasciitilde}W_{ ext{ extasciitilde} ext{infinity}}$ algebra in the coset model

Derivation of a quantum deformation as a hidden current algebra

Existence of infinitely many conserved charges related to $W$-hair in 2D black holes

Abstract

Previously we have established that the second Hamiltonian structure of the KP hierarchy is a nonlinear deformation, called $\hat{W}_{\infty}$, of the linear, centerless $W_{\infty}$ algebra. In this letter we present a free-field realization for all generators of $\hat{W}_{\infty}$ in terms of two scalars as well as an elegant generating function for the $\hat{W}_{\infty}$ currents in the classical conformal $SL(2,R)/U(1)$ coset model. After quantization, a quantum deformation of $\hat{W}_{\infty}$ appears as the hidden current algebra in this model. The $\hat{W}_{\infty}$ current algebra results in an infinite set of commuting conserved charges, which might give rise to $W$-hair for the 2d black hole arising in the corresponding string theory at level $k=9/4$.