Fully-Dualizable and Invertible $\mathcal{E}_n$-Algebras

TL;DR

This paper proves a conjecture characterizing fully-dualizable and invertible $ ext{E}_n$-algebras, which are crucial for understanding $(n+1)$-dimensional topological quantum field theories and their invertibility.

Contribution

It provides a proof of a conjecture that characterizes fully-dualizable and invertible $ ext{E}_n$-algebras within higher Morita categories, advancing the understanding of TQFTs.

Findings

Characterization of fully-dualizable $ ext{E}_n$-algebras

Identification of invertible $ ext{E}_n$-algebras

Connection to $(n+1)$-dimensional TQFTs

Abstract

We prove a conjecture of Brochier, Jordan, Safronov, and Snyder [BJSS21], first formulated by Lurie [Lur09b], characterizing fully-dualizable and invertible $\mathcal{E}_n$-algebras viewed as objects in the higher Morita categories $\mathbf{Mor}_n(\mathcal{V})$ [Lur09b, Sch14, Hau17b, Hau23]. In other words, we characterize those $\mathcal{E}_n$-algebras which give rise to $(n + 1)$-dimensional topological quantum field theories (TQFT), and those which give rise to invertible theories.