Boundary conformal fields and Tomita--Takesaki theory

TL;DR

This paper explores the connection between boundary conformal field theory, Ishibashi states, and Tomita--Takesaki theory, providing a geometric perspective and addressing normalisable and non-normalisable states.

Contribution

It introduces a geometric approach to Ishibashi states and links them with Tomita--Takesaki theory, extending the operator state correspondence in boundary conformal field theory.

Findings

Normalisable Ishibashi states are cyclic separating states.

Tomita--Takesaki theory offers an alternative for non-normalisable states.

The approach facilitates extensions of the state-operator correspondence.

Abstract

Motivated by formal similarities between the continuum limit of the Ising model and the Unruh effect, this paper connects the notion of an Ishibashi state in boundary conformal field theory with the Tomita--Takesaki theory for operator algebras. A geometrical approach to the definition of Ishibashi states is presented, and it is shownthat, when normalisable the Ishibashi states are cyclic separating states, justifying the operator state correspondence. When the states are not normalisable Tomita--Takesaki theory offers an alternative approach based on left Hilbert algebras, opening the way to extensions of our construction and the state-operator correspondence.