Bounds for State Degeneracies in 2D Conformal Field Theory

TL;DR

This paper uses modular invariance to establish universal bounds on state degeneracies and entropy in 2D conformal field theories, especially for theories with low central charge and no relevant operators.

Contribution

It derives the first universal lower bound on entropy at inverse temperature 2 pi and establishes an upper bound on entropy and marginal deformations for specific classes of CFTs.

Findings

Entropy at inverse temperature 2 pi has a universal lower bound.

Upper bounds on entropy are proven for CFTs with c_left + c_right < 48 and no relevant operators.

Maximum number of marginal deformations is bounded by a function of central charge.

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 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. Under the same conditions we also prove that a CFT can have a number of marginal deformations no greater than ((c_left + c_right) / (48 - c_left - c_right)) e^(4 Pi) - 2.