Boson Condensation in Topologically Ordered Quantum Liquids

TL;DR

This paper introduces a new, computationally efficient formalism for boson condensation in topologically ordered quantum liquids, linking it to chiral algebra extensions and matrix factorizations, and demonstrating its effectiveness through various theorems and examples.

Contribution

It provides a novel, practical approach to analyze boson condensation in TQFTs, simplifying calculations and establishing new theoretical links.

Findings

Efficient methods for computing condensed theory fusion algebra and S matrices.

Proved theorems relating boson condensation to chiral algebra extensions.

Reproduced known results such as noncondensability of Fibonacci layers.

Abstract

Boson condensation in topological quantum field theories (TQFT) has been previously investigated through the formalism of Frobenius algebras and the use of vertex lifting coefficients. While general, this formalism is physically opaque and computationally arduous: analyses of TQFT condensation are practically performed on a case by case basis and for very simple theories only, mostly not using the Frobenius algebra formalism. In this paper we provide a new way of treating boson condensation that is computationally efficient. With a minimal set of physical assumptions, such as commutativity of lifting and the definition of confined particles, we can prove a number of theorems linking Boson condensation in TQFT with chiral algebra extensions, and with the factorization of completely positive matrices over the nonnegative integers. We present numerically efficient ways of obtaining a condensed theory fusion algebra and S matrices; and we then use our formalism to prove several theorems for the S and T matrices of simple current condensation and of theories which upon condensation result in a low number of confined particles. We also show that our formalism easily reproduces results existent in the mathematical literature such as the noncondensability of 5 and 10 layers of the Fibonacci TQFT.