On the string theory of a single NS5-brane

TL;DR

The paper provides a complete worldsheet construction for string theory on a single NS5-brane background by using a gauged WZW coset on ${\cal G}/{\cal H}$ with a hybrid formalism and a novel constraint ${Q_{\rm con}}$ that renders the $n_5=1$ theory unitary-like within the BRST cohomology. It establishes an explicit $n_5=1$ hybrid formulation, builds a precise RNS$\leftrightarrow$hybrid map to connect with null gauging, and constructs a nilpotent BRST operator that preserves the ${\cal N}=(4,4)$ spacetime supersymmetry while decoupling the long-string continuum via a null-shortening condition. It further analyzes the representation theory, showing that only short ${\mathfrak{psu}(1,1|2)}_1$ representations survive, in agreement with the dual symmetric product orbifold and its single-trace $T\bar T$ deformation, and discusses RR moduli and their role as sources of the constraint. The work thus provides a solvable, holographically interpretable model for little string theory on a single NS5-brane, with a transparent bulk-dual perspective to a $T\bar T$-deformed CFT and a clear link between free-field realizations, null gauging, and BRST structure.

Abstract

We formulate a worldsheet description of string theory in the background of a single decoupled NS5-brane, which is a particularly simple example of ``little string'' holography. The worldsheet theory involves a gauged Wess-Zumino-Witten model for the group coset $\mathcal{G}/\mathcal{H}$, where $\mathcal{G} = {PSU(1,1|2)} \times \mathbb R^{5,1}$ and $\mathcal{H}=U(1) \times U(1)$. A variant of the embedding of $\mathcal{H}$ into $\mathcal{G}$ describes a background that interpolates between the fivebrane throat in the UV and $AdS_3$ in the IR, which is dual to the (single trace) $T\bar T$-deformed symmetric product orbifold $(\mathbb T^4)^N/S_N$. Well-known difficulties with a worldsheet theory for $n_5=1$ fivebranes are bypassed by the incorporation of an extra constraint on the dynamics. This constraint arises naturally from consideration of the effects of deforming the theory by one of the four Ramond-Ramond moduli of the spacetime background, which for $n_5=1$ imposes the constraint that our analysis has identified. Along the way, we construct a map between the supergroup formulation of the $n_5=1$ worldsheet theory and the more conventional RNS worldsheet formalism.