Representation Theory of Solitons

TL;DR

This paper develops a representation theory for solitons in 2D quantum field theory using the novel strip algebra, capturing degeneracies and selection rules, and provides a framework applicable to various physical systems with boundaries.

Contribution

It introduces the strip algebra as a $C^*$-weak Hopf algebra derived from the Drinfeld center TQFT, offering a new method to analyze soliton degeneracies and selection rules.

Findings

Representation category is a unitary fusion category.

Method using quiver diagrams simplifies analysis.

Reproduces known soliton degeneracies and rules.

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.