Bounds for State Degeneracies in 2D Conformal Field Theory

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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 2 pi satisfies a universal lower bound, and we enumerate the principal obstacles to deriving upper bounds on entropies or quantum mechanical degeneracies for fully general CFTs.
• 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_left + c_right < 48 with no relevant operators, we are able to prove an upper bound on the thermodynamic entropy at inverse temperature 2 pi.