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Relative Entropy in CFT

Roberto Longo, Feng Xu

TL;DR

The paper provides a rigorous mathematical treatment of entanglement entropies in two-dimensional chiral CFTs by computing Araki's relative entropy—via Lieb's convexity and singular integral methods—for free-fermion nets. It delivers an explicit mutual information formula for disjoint intervals, $F(A,B)= -\frac{r}{6}\ln \eta$, and extends the results to chiral CFTs embedded into free fermions and their finite-index extensions, revealing a deep link to subfactor theory. A key finding is the observed violation of duality in finite-index cases, with the deviation governed by the global dimension, connecting to topological entanglement entropy concepts. The work also develops a Kosaki-formula approach to limits of relative entropy and provides rich examples (orbifolds, conformal inclusions) to illustrate the framework and its breadth.

Abstract

By using Araki's relative entropy, Lieb's convexity and the theory of singular integrals, we compute the mutual information associated with free fermions, and we deduce many results about entropies for chiral CFT's which are embedded into free fermions, and their extensions. Such relative entropies in CFT are here computed explicitly for the first time in a mathematical rigorous way. Our results agree with previous computations by physicists based on heuristic arguments; in addition we uncover a surprising connection with the theory of subfactors, in particular by showing that a certain duality, which is argued to be true on physical grounds, is in fact violated if the global dimension of the conformal net is greater than $1.$

Relative Entropy in CFT

TL;DR

The paper provides a rigorous mathematical treatment of entanglement entropies in two-dimensional chiral CFTs by computing Araki's relative entropy—via Lieb's convexity and singular integral methods—for free-fermion nets. It delivers an explicit mutual information formula for disjoint intervals, , and extends the results to chiral CFTs embedded into free fermions and their finite-index extensions, revealing a deep link to subfactor theory. A key finding is the observed violation of duality in finite-index cases, with the deviation governed by the global dimension, connecting to topological entanglement entropy concepts. The work also develops a Kosaki-formula approach to limits of relative entropy and provides rich examples (orbifolds, conformal inclusions) to illustrate the framework and its breadth.

Abstract

By using Araki's relative entropy, Lieb's convexity and the theory of singular integrals, we compute the mutual information associated with free fermions, and we deduce many results about entropies for chiral CFT's which are embedded into free fermions, and their extensions. Such relative entropies in CFT are here computed explicitly for the first time in a mathematical rigorous way. Our results agree with previous computations by physicists based on heuristic arguments; in addition we uncover a surprising connection with the theory of subfactors, in particular by showing that a certain duality, which is argued to be true on physical grounds, is in fact violated if the global dimension of the conformal net is greater than

Paper Structure

This paper contains 23 sections, 25 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Entropy and relative entropy
  4. Graded nets and subnets
  5. Mutual information in the case of free fermions
  6. Basic representation of $LU_r$ and free fermion net
  7. Mutual information for free fermions
  8. Computation of mutual information in finite dimensional case
  9. Inequality from operator convexity
  10. Reversed inequality from Lieb's joint convexity
  11. Regularized Kernel for one free fermion case
  12. The proof of Theorem \ref{['main1']}
  13. Subnets of free fermion nets and their finite index extensions
  14. Formal properties of entropy for free fermion nets and their subnets
  15. Structure of singularities in the finite index case
  16. ...and 8 more sections

Key Result

Theorem 2.2

(1) Let $M$ be a von Neumann algebra and $M_1$ a von Neumann subalgebra of M. Assume that there exists a faithful normal conditional expectation $E$ of $M$onto $M_1$. If $\psi$ and $\omega$ are states of $M_1$ and $M$, respectively, then $S(\omega, \psi\cdot E) = S(\omega\!\restriction\! M_1, \psi)

Theorems & Definitions (33)

  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Corollary 2.7
  • Definition 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 23 more