Non-invertible symmetries of two-dimensional Non-Linear Sigma Models

TL;DR

This work constructs non-invertible duality defects in two-dimensional non-linear sigma models (NLSMs) with Wess-Zumino terms by half-space gauging of abelian isometries, exploiting T-duality to relate a theory to a potentially equivalent dual description. The method uses a doubled gauged NLSM framework and yields a boundary BF-type theory whose topological nature enforces Tambara-Yamagami fusion rules for the defect, provided self-duality constraints are satisfied. The authors illustrate the construction across a range of target spaces, including S^3 (Hopf fibration) and S^2×S^1 with H-flux, lens spaces, tori, nilfolds, and SU(N)$_ ext{κ}$ WZW models, showing how discrete gauging with a $Z_p$ subgroup enforces duality with a precise matching of Chern classes and torus moduli. A key outcome is that conformal (and rational) properties are not essential; non-invertible defects can exist in non-conformal NLSMs, with WZW models providing particularly rich examples and a bridge to Verlinde structures in special cases.

Abstract

Global symmetries can be generalised to transformations generated by topological operators, including cases in which the topological operator does not have an inverse. A family of such topological operators are intimately related to dualities via the procedure of half-space gauging. In this work we discuss the construction of non-invertible defects based on T-duality in two dimensions, generalising the well-known case of the free compact boson to any Non-Linear Sigma Model with Wess-Zumino term which is T-dualisable. This requires that the target space has an isometry with compact orbits that acts without fixed points. Our approach allows us to include target spaces without non-trivial 1-cycles, does not require the NLSM to be conformal, and when it is conformal it does not need to be rational; moreover, it highlights the microscopic origin of the topological terms that are responsible for the non-invertibility of the defect. An interesting class of examples are Wess-Zumino-Witten models, which are self-dual under a discrete gauging of a subgroup of the isometry symmetry and so host a topological defect line with Tambara-Yamagami fusion. Along the way, we discuss how the usual 0-form symmetries match across T-dual models in target spaces without 1-cycles, and how global obstructions can prevent locally conserved currents from giving rise to topological operators.

Figures (3)

• Figure 1: Idea of the construction of a half-space gauging or self-duality defect. We divide $W$ into two regions $\Gamma_-$ and $\Gamma_+$. If the original theory $\mathcal{T}$ has a global symmetry $G$, we can gauge it in $\Gamma_+$, and impose boundary conditions on the gauge fields such that the interface $\gamma$ is topological. If the gauged theory $\mathcal{T}/G$ is then dual to the original theory $\mathcal{T}$, the construction defines a topological operator in $\mathcal{T}$.
• Figure 2: A valid choice of defect locus $\gamma$ for a worldsheet of genus 2. The defect $\gamma$ must be chosen to lie on a codimension-1 cycle which splits the worldsheet into two parts, $\Gamma_+$ and $\Gamma_-$.
• Figure 3: Geometry of a NLSM on a worldsheet $\Gamma_+$ with boundary, $\Phi_+:(\Gamma_+,\gamma)\to(M,\Phi_+(\gamma)=\Phi_-(\gamma))$. There exists a disc $\Delta$ with image $\Phi_+(\Delta)$, and a 3-volume $V_+$, that satisfy $\partial V_+=\Gamma_+ \cup \Delta$ such that $\partial \Gamma_+ = - \partial \Delta = \gamma$.