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arXiv:1907.08149

The Cardy Formula from Goldstone Bosons

Edi Halyo

Published Jul 18, 2019• 1 citation
arXivHTMLPDF

Abstract

Two dimensional conformal field theories, can be described by their pseudo Goldstone bosons when the conformal symmetry is spontaneously and anomalously broken. We show that the Schwarzian action of these bosons leads to the Cardy formula without using modular invariance. As a result, the Cardy formula applies to conformal field theories on a cylinder and chiral theories in one dimension. This also explains why the Cardy--Verlinde formula for theories on $S^1 \times S^{d-2}$ can be written in the form of the Cardy formula of an effective two dimensional theory.

DOI
10.48550/arxiv.1907.08149

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We review and extend evidence for the validity of a generalized Verlinde formula, in particular, nonrational conformal field theories. We identify a subset of representations of the chiral algebra in nonrational conformal field theories that give rise to an analogue of the relation between modular SSS-matrices and fusion coefficients in rational conformal field theories. To that end we review and extend the Cardy-type brane calculations in bosonic and supersymmetric Liouville theory (and its duals) as well as in H3+{H}_{3}^{+}H3+​. We analyze the three-point functions of Liouville theory and of H3+{H}_{3}^{+}H3+​ in detail to directly identify the fusion coefficients from the operator product expansion. Moreover, we check the validity of a proposed generic formula for localized brane one-point functions in nonrational conformal field theories.
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, Δ + δ] for large Δ 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.
We study a relationship between conformally invariant boundary conditions and anomalies of conformal field theories (CFTs) in 1+11+11+1 dimensions. For a given CFT with a global symmetry, we consider symmetric ``gapping potentials'', which are relevant perturbations to the CFT. If a gapping potential is introduced only in a subregion of the system, it provides a certain boundary condition to the CFT. From this equivalence, if there exists a Cardy boundary state, which is invariant under a symmetry, then the CFT can be gapped with a unique ground state by adding the corresponding gapping potential. This means that the symmetry of the CFT is anomaly free. Using this approach, we systematically deduce the anomaly-free conditions for various types of CFTs with several different symmetries. They include the free compact boson theory, Wess-Zumino-Witten models, and unitary minimal models. When the symmetry of the CFT is anomalous, it implies a Lieb-Schultz-Mattis type ingappability of the system. Our results are consistent with, where available, known results in the literature. Moreover, we extend the discussion to other symmetries including spin groups and generalized time-reversal symmetries.
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