Abstract
A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and c > 1. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin j ≠ 0. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity.
Highlights
Exact expression for the Fourier coefficients p(n) in terms of a series known as Rademacher expansion [2]
Rademacher expansions were first studied in the context of 2d CFT in [5], where they were used to give an exact expression for the Fourier coefficients of elliptic genera on CY manifolds
Z(q, q) = qh−c/24qh−c/24, h,h where c is the central charge of the theory, what are the constraints imposed by the full modular invariance
Summary
Note that it is important that Z(e−z) is defined for complex z, and the above conditions hold in the specified arcs To make this point clear, suppose the asymptotic expansion contains the term an = (−1)nbn, where bn grows at least as fast as the contribution in the theorem. For real and positive x these terms would not contribute to the exponential behaviour of Z(e−x) as x → 0 They would invalidate the second condition of the theorem. Going back to the problem of the number of partitions, covariance under different elements of the modular group give asymptotic expressions for the series n=0 p(n)ωne−zn, where ω is a root of unity This allows to write an asymptotic series for p(n), where each term is written in terms of modified Bessel functions, as above. In the case of a modular invariant function (or covariant with appropriate weight) we can replace this asymptotic series by an exact expression
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