Weak Hopf tube algebra for domain walls between 2d gapped phases of Turaev-Viro TQFTs

TL;DR

This paper develops a mathematical framework for understanding domain walls in 2D topological phases using weak Hopf algebras, classifying excitations and defects via representation theory.

Contribution

It constructs domain wall tube algebras with weak Hopf algebra structures and relates them to Drinfeld doubles, advancing the algebraic understanding of domain walls in TQFTs.

Findings

Domain wall tube algebras are equipped with $C^*$ weak Hopf algebra structures.

Topological excitations on domain walls correspond to representations of these algebras.

The paper establishes an isomorphism between domain wall tube algebras and Drinfeld quantum doubles.

Abstract

We investigate domain walls between 2d gapped phases of Turaev-Viro type topological quantum field theories (TQFTs) by constructing domain wall tube algebras. We begin by analyzing the domain wall tube algebra associated with bimodule categories, and then extend the construction to multimodule categories over $N$ base fusion categories. We prove that the resulting tube algebra is naturally equipped with a $C^*$ weak Hopf algebra structure. We show that topological excitations localized on domain walls are classified by representations of the corresponding domain wall tube algebra, in the sense that the functor category of bimodules admits a fusion-preserving embedding into the representation category of the domain wall tube algebra. We further establish the folding trick and Morita theory in this context. Then, most crucially, we provide a rigorous construction of the Drinfeld quantum double from weak Hopf boundary tube algebras using a skew-pairing, and establish an isomorphism between domain wall tube algebra and Drinfeld quantum double of boundary tube algebras. Motivated by the correspondence between the domain wall tube algebra and the quantum double of the boundary tube algebras, we introduce the notion of an $N$-tuple algebra and demonstrate how it arises in the multimodule domain wall setting. Finally, we consider defects between two domain walls, showing that such defects can be characterized by representations of a domain wall defect tube algebra. We briefly outline how these defects can be systematically treated within this representation-theoretic framework.