Representation Theory of Solitons

TL;DR

The paper develops a representation-theoretic framework for solitons in 2d QFTs with finite non-invertible symmetry, grounded in the strip algebra Str_C(M), a $C^*$-weak Hopf algebra. By linking Str_C(M) representations to the dual category C_M^* via the symmetry TQFT, it provides a quiver-based method to classify multiplets of particles and solitons that arise when boundary conditions are present, and demonstrates how degeneracies and selection rules are organized by non-invertible symmetry. Through explicit examples—group and non-invertible cases such as Fibonacci, Z2 Tambara-Yamagami, and su(2)_k_WZW deformations—the framework reproduces known spectra and predicts new multiplet structures and scattering constraints. The results give a unifying open-boundary perspective on non-invertible symmetries, applicable to particle physics, conformal field theory, and condensed matter, and supply practical tools for computing selection rules and degeneracies in the presence of boundaries.

Abstract

Solitons in two-dimensional quantum field theory exhibit patterns of degeneracies and associated selection rules on scattering amplitudes. We develop a representation theory that captures these intriguing features of solitons. This representation theory is based on an algebra we refer to as the "strip algebra", $\textrm{Str}_{\mathcal{C}}(\mathcal{M})$, which is defined in terms of the non-invertible symmetry, $\mathcal{C},$ a fusion category, and its action on boundary conditions encoded by a module category, $\mathcal{M}$. The strip algebra is a $C^*$-weak Hopf algebra, a fact which can be elegantly deduced by quantizing the three-dimensional Drinfeld center TQFT, $\mathcal{Z}(\mathcal{C}),$ on a spatial manifold with corners. These structures imply that the representation category of the strip algebra is also a unitary fusion category which we identify with a dual category $\mathcal{C}_{\mathcal{M}}^{*}.$ We present a straightforward method for analyzing these representations in terms of quiver diagrams where nodes are vacua and arrows are solitons and provide examples demonstrating how the representation theory reproduces known degeneracies and selection rules of soliton scattering. Our analysis provides the general framework for analyzing non-invertible symmetry on manifolds with boundary and applies both to the case of boundaries at infinity, relevant to particle physics, and boundaries at finite distance, relevant in conformal field theory or condensed matter systems.