Beyond logarithmic corrections to Cardy formula

Abstract

As shown by Cardy [1], modular invariance of the partition function of a given unitary non-singular 2d CFT with left and right central charges c _L and c _R , implies that the density of states in a microcanonical ensemble, at excitations Δ and $\bar{\Delta }$ and in the saddle point approximation, is ${\rho_0}\left( {\Delta, \bar{\Delta };{c_L},{c_R}} \right) = {c_L}\exp \left( {2\pi \sqrt {{{{{{c_L}\Delta }} \left/ {6} \right.}}} } \right) \cdot {c_R}\exp \left( {2\pi \sqrt {{{{{{c_R}\bar{\Delta }}} \left/ {6} \right.}}} } \right)$. In this paper, we extend Cardy’s analysis and show that in the saddle point approximation and up to contributions which are exponentially suppressed compared to the leading Cardy’s result, the density of states takes the form $\rho \left( {\Delta, \bar{\Delta };{c_L},{c_R}} \right) = f\left( {{c_L}\Delta } \right)f\left( {{c_R}\bar{\Delta }} \right){\rho_0}\left( {\Delta, \bar{\Delta };{c_L},{c_R}} \right)$, for a function f(x) which we specify. In particular, we show that (i) $\rho \left( {\Delta, \bar{\Delta };{c_L},{c_R}} \right)$ is the product of contributions of left and right movers and hence, to this approximation, the partition function of any modular invariant, non-singular unitary 2d CFT is holomorphically factorizable and (ii) ${{{\rho \left( {\Delta, \bar{\Delta };{c_L},{c_R}} \right)}} \left/ {{\left( {{c_L}{c_R}} \right)}} \right.}$ is only a function of c _L Δ and ${c_R}\bar{\Delta }$. In addition, treating $\rho \left( {\Delta, \bar{\Delta };{c_L},{c_R}} \right)$ as the density of states of a microcanonical ensemble, we compute the entropy of the system in the canonical counterpart and show that the function f(x) is such that the canonical entropy, up to exponentially suppressed contributions, is simply given by the Cardy’s result $\ln {\rho_0}\left( {\Delta, \bar{\Delta };{c_L},{c_R}} \right)$.