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

Abstract

Given an (irreducible) Mobius covariant net $\mathcal{A}$, we prove a Bisognano-Wichmann theorem for its categorical extension ${\mathcal{E}}^{\text{d}}$ associated to the braided $C^{\ast }$-tensor category ${\text{Rep}}^{\text{d}}(\mathcal{A})$ 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 $C^{\ast }$-Frobenius algebra $Q$ in ${\text{Rep}}^{\text{d}}(\mathcal{A})$. As an application, we discuss the relation between the domains of modular operators and the preclosedness of certain unbounded operators in ${\mathcal{E}}^{\text{d}}$.

44 pages, to appear in Annales Henri Poincaré. A new section added (Sec. 3), notation changed, Sec. 5 (previously Sec. 4) rewritten, the proofs of the main results of Sec. 6 and Sec. 7 are rewritten

References in corpus (3)

• Structure and Classification of Superconformal Nets
• Categorical extensions of conformal nets
• Unitarity of the modular tensor categories associated to unitary vertex operator algebras, I

Cited by in corpus (2)

• Covariant homogeneous nets of standard subspaces
• Unbounded field operators in categorical extensions of conformal nets