Hamiltonian Reduction of $SL(2)$-theories at the Level of Correlators

TL;DR

This paper provides a direct proof that correlators of $SL(2)$ current algebra reduce to minimal model correlators at the level of explicit calculations, using the Wakimoto free field construction.

Contribution

It offers a simple, direct proof of the Hamiltonian reduction of $SL(2)$ correlators to minimal models, filling a gap in the existing literature.

Findings

Explicit $SL(2)$ correlators reduce to minimal model correlators when $x_n=z_n$

Verification of reduction using Wakimoto free field construction

Calculation of the constant factor in the reduction

Abstract

Since the work of Bershadsky and Ooguri and Feigin and Frenkel it is well known that correlators of $SL(2)$ current algebra for admissible representations should reduce to correlators for conformal minimal models. A precise proposal for this relation has been given at the level of correlators: When $SL(2)$ primary fields are expressed as $\phi_j(z_n,x_n)$ with $x_n$ being a variable to keep track of the $SL(2)$ representation multiplet (possibly infinitely dimensional for admissible representations), then the minimal model correlator is supposed to be obtained simply by putting all $x_n=z_n$. Although strong support for this has been presented, to the best of our understanding a direct, simple proof seems to be missing so in this paper we present one based on the free field Wakimoto construction and our previous study of that in the present context. We further verify that the explicit $SL(2)$ correlators we have published in a recent preprint reduce in the above way, up to a constant which we also calculate. We further discuss the relation to more standard formulations of hamiltonian reduction.