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Bounds for state degeneracies in 2D conformal field theory

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Abstract

In this note we explore the application of modular invariance in 2-dimensional CFT to derive universal bounds for quantities describing certain state degeneracies, such as the thermodynamic entropy, or the number of marginal operators. We show that the entropy at inverse temperature 2π satisfies a universal lower bound, and we enumerate the principal obstacles to deriving upper bounds on entropies or quantum mechanical degeneracies for completely general CFT. We then restrict our attention to infrared-stable CFT with moderately low central charge, in addition to the usual assumptions of modular invariance, unitarity and discrete operator spectrum. For CFT in the range c L + c R < 48 with no relevant operators, we are able to prove an upper bound on the thermodynamic entropy at inverse temperature 2π. Under the same conditions we also prove that a CFT can have no more than \( \left( {\frac{{{c_{\text{L}}} + {c_{\text{R}}}}}{{48 - {c_{\text{L}}} - {c_{\text{R}}}}}} \right) \cdot \exp \left\{ { + 4\pi } \right\} - 2 \) marginal deformations.

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Authors and Affiliations

  1. Institute for the Physics and Mathematics of the Universe, Todai Institutes for Advanced Study (TODIAS), The University of Tokyo, Kashiwa, Chiba, 277-8582, Japan

    Simeon Hellerman & Cornelius Schmidt-Colinet

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  1. Simeon Hellerman
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  2. Cornelius Schmidt-Colinet
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Correspondence to Simeon Hellerman.

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ArXiv ePrint: 1007.0756

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Hellerman, S., Schmidt-Colinet, C. Bounds for state degeneracies in 2D conformal field theory. J. High Energ. Phys. 2011, 127 (2011). https://doi.org/10.1007/JHEP08(2011)127

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