NS5-Branes, T-Duality and Worldsheet Instantons

TL;DR

This paper resolves a long-standing puzzle about T-duality between ALF Taub–NUT backgrounds and localized NS5-branes by showing that worldsheet instantons provide the missing localization mechanism. Using an ${\cal N}=(4,4)$ gauged linear sigma model, Tong demonstrates that T-duality maps the ALF geometry to a smeared NS5-brane, but instanton corrections from Nielsen–Olesen vortices break the isometry and localize the NS5-brane at a point; the $k$-instanton contributions are governed by the weighted Euler form over the $k$-vortex moduli space. The corrections take the form $H_{inst}(r,\theta)=\frac{1}{g^{2}}+\frac{1}{2r}\left(1+\sum_{k\ge1}\sum_{\pm} m_k e^{-k r\pm i k\theta}\right)$ with $m_k=(2k)^{-3}\nu(\tilde{M}_{k})$, and the $k=1$ sector yields a precise prediction for the exponential decay coefficient $l_1$ of the vortex tail, $l_1=8^{1/4}$, in agreement with numerical vortex studies. The work thus links worldsheet instanton physics, vortex moduli-space geometry, and T-duality in a concrete, testable framework, clarifying how localization arises in string dualities.

Abstract

The equivalence of NS5-branes and ALF spaces under T-duality is well known. However, a naive application of T-duality transforms the ALF space into a smeared NS5-brane, de-localized on the dual, transverse, circle. In this paper we re-examine this duality, starting from a two-dimensional N=(4,4) gauged linear sigma model describing Taub-NUT space. After dualizing the circle fiber, we find that the smeared NS5-brane target space metric receives corrections from multi-worldsheet instantons. These instantons are identified as Nielsen-Olesen vortices. We show that their effect is to break the isometry of the target space, localizing the NS5-brane at a point. The contribution from the k-instanton sector is shown to be proportional to the weighted integral of the Euler form over the k-vortex moduli space. The duality also predicts the, previously unknown, asymptotic exponential decay coefficient of the BPS vortex solution.