Alterfold Theory and Topological Modular Invariance

TL;DR

This paper introduces a topological framework for alterfold topological quantum field theory, providing new proofs, generalizations, and invariance results related to modular fusion categories and Morita contexts.

Contribution

It offers a topological perspective that simplifies proofs, generalizes results on $\alpha$-induction, and introduces double $\alpha$-induction with higher-genus invariance in modular categories.

Findings

Streamlined proofs of $\alpha$-induction results.

Introduction of double $\alpha$-induction and its invariance properties.

A new integral identity for modular invariance across Morita contexts.

Abstract

We propose a topological paradigm in alterfold topological quantum field theory to explore various concepts, including modular invariants, $\alpha$-induction and connections in Morita contexts within a modular fusion category of non-zero global dimension over an arbitrary field. Using our topological perspective, we provide streamlined quick proofs and broad generalizations of a wide range of results. These include all theoretical findings by B\"{o}ckenhauer, Evans, and Kawahigashi on $\alpha$-induction. Additionally, we introduce the concept of double $\alpha$-induction for pairs of Morita contexts and define its higher-genus $Z$-transformation, which remains invariant under the action of the mapping class group. Finally, we establish a novel integral identity for modular invariance across multiple Morita contexts, unifying several known identities as special cases.