Conformal Current Algebra in Two Dimensions

TL;DR

The paper constructs a conformal, non-chiral current algebra in two dimensions and realizes it in supergroup sigma-models with vanishing Killing form, with or without a Wess-Zumino term. It derives the algebra by both generic OPE constraints and exact correlator methods, showing that a Sugawara stress tensor reproduces a conformal structure with central charge given by the superdimension $sdim G$, and identifies a non-chiral Kac-Moody framework along with an infinite set of commuting operators. Explicit current-algebra coefficients $c_1,c_2,c_3,c_4,g$ are obtained from exact two- and three-point functions and corroborated by conformal perturbation theory about the WZW point, with resummation confirming all-orders results. The findings have implications for quantum integrability and AdS/CFT contexts, notably in PSU(n|n) models and worldsheet theories on supersymmetric $AdS$ backgrounds, offering a principled route to analyze spectra and integrability in these theories.

Abstract

We construct a non-chiral current algebra in two dimensions consistent with conformal invariance. We show that the conformal current algebra is realized in non-linear sigma-models on supergroup manifolds with vanishing dual Coxeter number, with or without a Wess-Zumino term. The current algebra is computed using two distinct methods. First we exploit special algebraic properties of supergroups to compute the exact two- and three-point functions of the currents and from them we infer the current algebra. The algebra is also calculated by using conformal perturbation theory about the Wess-Zumino-Witten point and resumming the perturbation series. We also prove that these models realize a non-chiral Kac-Moody algebra and construct an infinite set of commuting operators that is closed under the action of the Kac-Moody generators. The supergroup models that we consider include models with applications to statistical mechanics, condensed matter and string theory. In particular, our results may help to systematically solve and clarify the quantum integrability of PSU(n|n) models and their cosets, which appear prominently in string worldsheet models on anti-deSitter spaces.