Modular invariance as completeness
Valentín Benedetti, Horacio Casini, Yasuyuki Kawahigashi, Roberto Longo, Javier M. Magán
Abstract
We review the physical meaning of modular invariance for unitary conformal quantum field theories in <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mrow> <a:mi>d</a:mi> <a:mo>=</a:mo> <a:mn>2</a:mn> </a:mrow> </a:math> . For quantum field theory models, while <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>T</c:mi> </c:math> invariance is necessary for locality, <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>S</e:mi> </e:math> invariance is not mandatory. <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mrow> <g:mi>S</g:mi> </g:mrow> </g:math> 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 conformal field theories (CFTs), 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 information. 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. Published by the American Physical Society 2024