Abstract
We analyze modular invariance drawing inspiration from tauberian theorems. Given a modular invariant partition function with a positive spectral density, we derive lower and upper bounds on the number of operators within a given energy interval. They are most revealing at high energies. In this limit we rigorously derive the Cardy formula for the microcanonical entropy together with optimal error estimates for various widths of the averaging energy shell. We identify a new universal contribution to the microcanonical entropy controlled by the central charge and the width of the shell. We derive an upper bound on the spacings between Virasoro primaries. Analogous results are obtained in holographic 2d CFTs. We also study partition functions with a UV cutoff. Control over error estimates allows us to probe operators beyond the unity in the modularity condition. We check our results in the 2d Ising model and the Monster CFT and find perfect agreement.
Highlights
High energy estimates on various physical quantities are commonly stated locally even though they are only true on average
A few famous examples are: the Froissart bound on the growth of the cross-section [1, 2], high-frequency expansion of conductivity at finite temperature [3, 4], high energy asymptotics of the electromagnetic current spectral density in the context of the so-called quark-hadron duality [5], and the Cardy formula for two-dimensional CFTs [6]
The leading contribution to the on-shell action is insensitive to the ensemble choice, the state of the quantum fields in the black hole background changes which should be taken into account when computing the corrections to the leading Cardy formula, see e.g. [30, 31]
Summary
High energy estimates on various physical quantities are commonly stated locally even though they are only true on average. As further noticed in [15] the optimal error estimates can be obtained using the socalled complex tauberian theorems, which exploit the fact that physical quantities of interest are very often analytic functions in a complex domain This is the case for the modularity condition of 2d CFTs. In this note we apply methods of tauberian theory to modular invariance in 2d CFTs and rigorously derive the Cardy formula and corrections to it, where we explicitly keep track of the dependence on δ. In this note we apply methods of tauberian theory to modular invariance in 2d CFTs and rigorously derive the Cardy formula and corrections to it, where we explicitly keep track of the dependence on δ Combining these ideas with bounds on the partition function of Hartman, Keller and Stoica (HKS) [16] we find lower and upper bounds on the number of operators within a given window of finite conformal dimensions (∆ − δ, ∆ + δ). Though true at finite ∆, they are most revealing in the limit ∆ → ∞
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