Abstract
Modular invariance strongly constrains the spectrum of states of two dimensional conformal field theories. By summing over the images of the modular group, we construct candidate CFT partition functions that are modular invariant and have positive spectrum. This allows us to efficiently extract the constraints on the CFT spectrum imposed by modular invariance, giving information on the spectrum that goes beyond the Cardy growth of the asymptotic density of states. Some of the candidate modular invariant partition functions we construct have gaps of size (c-1)/12, proving that gaps of this size and smaller are consistent with modular invariance. We also revisit the partition function of pure Einstein gravity in AdS3 obtained by summing over geometries, which has a spectrum with two unphysical features: it is continuous, and the density of states is not positive definite. We show that both of these can be resolved by adding corrections to the spectrum which are subleading in the semi-classical (large central charge) limit.
Highlights
Introduction and summary1.1 Modular invariance in CFTFor conformal field theories in two dimensions, modular invariance — the invariance under large conformal transformations in Euclidean signature — strongly constrains the spectrum of the theory
By summing over the images of the modular group, we construct candidate CFT partition functions that are modular invariant and have positive spectrum. This allows us to efficiently extract the constraints on the CFT spectrum imposed by modular invariance, giving information on the spectrum that goes beyond the Cardy growth of the asymptotic density of states
Some of the candidate modular invariant partition functions we construct have gaps of size (c − 1)/12, proving that gaps of this size and smaller are consistent with modular invariance
Summary
For conformal field theories in two dimensions, modular invariance — the invariance under large conformal transformations in Euclidean signature — strongly constrains the spectrum of the theory. These are the states which, in the gravitational language, cannot be interpreted as BTZ black holes These states will play a special role for modular invariant partition functions, somewhat similar to that played by polar states in the theory of modular forms or weak. If we want to interpret our modular functions as partition functions of physical theories, we must demand that the density of states is positive Ensuring this is more subtle, and there is no reason to believe that an arbitrary censored configuration will give a positive spectrum. C/24 k=1 a−k q−k for some constants a−k To turn this polar part into a modular invariant function, we perform the holomorphic version of the Poincare series, known as a Rademacher sum. The Poincare series determines the free energy up to a finite piece
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