Bisognano-Wichmann property for rigid categorical extensions and non-local extensions of conformal nets

TL;DR

This paper proves a Bisognano-Wichmann theorem for categorical extensions and non-local extensions of conformal nets, advancing the understanding of modular theory in algebraic quantum field theory.

Contribution

It establishes a Bisognano-Wichmann property for categorical extensions and non-local extensions of conformal nets using braided tensor categories and $C^*$-Frobenius algebras.

Findings

Proves Bisognano-Wichmann theorem for categorical extensions of conformal nets.

Establishes a modified Bisognano-Wichmann theorem for non-local extensions.

Analyzes the relation between modular operators and unbounded operators in the extension category.

Abstract

Given an (irreducible) Mobius covariant net $\mathcal A$, we prove a Bisognano-Wichmann theorem for its categorical extension $\mathscr E^{\textrm{d}}$ associated to the braided $C^*$-tensor category $\textrm{Rep}^{\textrm{d}}(\mathcal A)$ of dualizable (more precisely "dualized") Mobius covariant $\mathcal A$-modules. As a closely related result, we prove a (modified) Bisognano-Wichmann theorem for any (possibly) non-local extension of $\mathcal A$ obtained by a $C^*$-Frobenius algebra $Q$ in $\textrm{Rep}^{\textrm{d}}(\mathcal A)$. As an application, we discuss the relation between the domains of modular operators and the preclosedness of certain unbounded operators in $\mathscr E^{\textrm{d}}$.