Modular invariance as completeness

TL;DR

This paper explores the significance of modular invariance in 2D conformal quantum field theories, linking it to the concept of completeness via Haag duality, and introduces an index to measure deviations from this invariance.

Contribution

It provides a mathematical and physical analysis of modular invariance as a criterion for the completeness of 2D CFTs, including methods to compute related indices from modular data.

Findings

Modular invariance relates to the completeness of 2D CFTs.

An index measuring deviations from modular invariance is introduced.

The index can be computed from modular transformation matrices.

Abstract

We review the physical meaning of modular invariance for unitary conformal quantum field theories in d=2. For QFT models, while T invariance is necessary for locality, S invariance is not mandatory. S invariance is a form of completeness of the theory that has a precise meaning as Haag duality for arbitrary multi-interval regions. We present a mathematical proof as well as derive this result from a physical standpoint using Renyi entropies and the replica trick. For rational CFT's, the failure of modular invariance or Haag duality can be measured by an index, related to the quantum dimensions of the model. We show how to compute this index from the modular transformation matrices. The index also appears in a limit of the Renyi mutual informations. Cases of infinite index are briefly discussed. Part of the argument can be extended to higher dimensions, where the lack of completeness can also be diagnosed using the CFT data through the thermal partition function and measured by an index.