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arXiv:2208.05495 · Journal of High Energy Physics

Asymptotic density of states in 2d CFTs with non-invertible symmetries

Ying-Hsuan Lin, Masaki Okada, Sahand Seifnashri, Yuji Tachikawa

Published Mar 14, 2023• 128 citations
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Abstract

A bstract It is known that the asymptotic density of states of a 2d CFT in an irreducible representation ρ of a finite symmetry group G is proportional to (dim ρ ) 2 . We show how this statement can be generalized when the symmetry can be non-invertible and is described by a fusion category $$ \mathcal{C} $$ C . Along the way, we explain what plays the role of a representation of a group in the case of a fusion category symmetry; the answer to this question is already available in the broader mathematical physics literature but not yet widely known in hep-th. This understanding immediately implies a selection rule on the correlation functions, and also allows us to derive the asymptotic density.

DOI
10.1007/jhep03(2023)094

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On the (Fib⊠Fib)⋊S2(\text{Fib} \boxtimes \text{Fib}) \rtimes S_2(Fib⊠Fib)⋊S2​ fusion category
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A bstract We derive Cardy-like formulas for the growth of operators in different sectors of unitary 2 dimensional CFT in the presence of topological defect lines by putting an upper and lower bound on the number of states with scaling dimension in the interval [∆ − δ, ∆ + δ ] for large ∆ at fixed δ . Consequently we prove that given any unitary modular invariant 2D CFT symmetric under finite global symmetry G (acting faithfully), all the irreducible representations of G appear in the spectra of the untwisted sector; the growth of states is Cardy like and proportional to the “square” of the dimension of the irrep. In the Schwarzian limit, the result matches onto that of JT gravity with a bulk gauge theory. If the symmetry is non-anomalous, the result applies to any sector twisted by a group element. For c > 1, the statements are true for Virasoro primaries. Furthermore, the results are applicable to large c CFTs. We also extend our results for the continuous U(1) group.
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