Anyonic Chains -- $\alpha$-Induction -- CFT -- Defects -- Subfactors

TL;DR

This paper explores the connection between anyonic spin-chains, fusion categories, and conformal field theories, constructing a novel algebra of matrix-product-operators that models defects in 1+1D CFTs using subfactor theory.

Contribution

It introduces a new algebra of MPOs on anyonic chains that is isomorphic to the defect algebra of 1+1D CFTs, linking lattice models with conformal defects via subfactor theory.

Findings

Constructed a novel algebra of MPOs on bi-partite anyonic chains.

Showed the algebra is isomorphic to the defect algebra in 1+1D CFTs.

Conjectured the relation between central projections and irreducible defects.

Abstract

Given a unitary fusion category, one can define the Hilbert space of a so-called ``anyonic spin-chain'' and nearest neighbor Hamiltonians providing a real-time evolution. There is considerable evidence that suitable scaling limits of such systems can lead to $1+1$-dimensional conformal field theories (CFTs), and in fact, can be used potentially to construct novel classes of CFTs. Besides the Hamiltonians and their densities, the spin chain is known to carry an algebra of symmetry operators commuting with the Hamiltonian, and these operators have an interesting representation as matrix-product-operators (MPOs). On the other hand, fusion categories are well-known to arise from a von Neumann algebra-subfactor pair. In this work, we investigate some interesting consequences of such structures for the corresponding anyonic spin-chain model. One of our main results is the construction of a novel algebra of MPOs acting on a bi-partite anyonic chain. We show that this algebra is precisely isomorphic to the defect algebra of $1+1$ CFTs as constructed by Fr\" ohlich et al. and Bischoff et al., even though the model is defined on a finite lattice. We thus conjecture that its central projections are associated with the irreducible vertical (transparent) defects in the scaling limit of the model. Our results partly rely on the observation that MPOs are closely related to the so-called ``double triangle algebra'' arising in subfactor theory. In our subsequent constructions, we use insights into the structure of the double triangle algebra by B\" ockenhauer et al. based on the braided structure of the categories and on $\alpha$-induction. The introductory section of this paper to subfactors and fusion categories has the character of a review.