On the tensionless limit of gauged WZW models

TL;DR

The paper investigates the tensionless limit of gauged WZW models at the quantum level, realized when the Kac‑Moody level reaches the dual Coxeter number $g^{\vee}$ and the Virasoro central charge diverges. It analyzes both world‑sheet and target‑space pictures, showing gravity decouples via Liouville dynamics and uncovering a rich higher‑spin symmetry on the world sheet, realized concretely in the coset $SL(2,\mathbb{R})_k/U(1)$ with the tensionless point $k=2$ yielding a truncated $W_{\infty}$, while the Virasoro sector abelianizes. The authors develop a parafermion‑based construction of the world‑sheet symmetry $\hat{W}_{\infty}(k)$, connect its large‑$k$ limit to $W_{\infty}$ and its tensionless limit to the linear algebra $W_{\infty}^{(3)}$, and perform a BRST analysis to fix higher‑spin central charges, confirming anomaly cancellation at a critical value $c=2$. They extend the framework to higher‑dimensional cosets and discuss null gauging, suggesting a topological character for tensionless strings and highlighting connections to non‑commutative geometry, with open questions about the target‑space role of these symmetries and their full non‑perturbative formulation.

Abstract

The tensionless limit of gauged WZW models arises when the level of the underlying Kac-Moody algebra assumes its critical value, equal to the dual Coxeter number, in which case the central charge of the Virasoro algebra becomes infinite. We examine this limit from the world-sheet and target space viewpoint and show that gravity decouples naturally from the spectrum. Using the two-dimensional black-hole coset SL(2,R)_k/U(1) as illustrative example, we find for k=2 that the world-sheet symmetry is described by a truncated version of W_{\infty} generated by chiral fields with integer spin s \geq 3, whereas the Virasoro algebra becomes abelian and it can be consistently factored out. The geometry of target space looks like an infinitely curved hyperboloid, which invalidates the effective field theory description and conformal invariance can no longer be used to yield reliable space-time interpretation. We also compare our results with the null gauging of WZW models, which correspond to infinite boost in target space and they describe the Liouville mode that decouples in the tensionless limit. A formal BRST analysis of the world-sheet symmetry suggests that the central charge of all higher spin generators should be fixed to a critical value, which is not seen by the contracted Virasoro symmetry. Generalizations to higher dimensional coset models are also briefly discussed in the tensionless limit, where similar observations are made.