Monstrous Product CFTs in the Grand Canonical Ensemble

TL;DR

This paper explores the grand canonical ensemble of symmetric products of the Monster CFT, revealing enhanced modular symmetries, extending Cardy's formula, and uncovering phase transitions including Bose-Einstein condensation and wall-crossing phenomena.

Contribution

It introduces an extended modular symmetry in the grand canonical ensemble of the Monster CFT, allowing a deeper understanding of the spectrum, phase structure, and black hole entropy matching.

Findings

Extended O(2,2;Z) symmetry mixing  and  parameters.

Extension of Cardy's formula validity to AdS-scale.

Identification of a Bose-Einstein condensate transition and wall-crossing phenomena.

Abstract

We study symmetric products of the chiral 'Monster' conformal field theory with c=24 in the grand canonical ensemble by introducing a complex parameter \rho, whose imaginary part represents the chemical potential \mu conjugate to the number of copies of the CFT. The grand canonical partition function is given by the DMVV product formula in terms of the multiplicities of the seed CFT. It possesses an O(2,2;\ZZ) symmetry that enhances the familiar SL(2,\ZZ) modular invariance of the canonical ensemble and mixes the modular parameter \tau with the parameter \rho. By exploiting this enhanced modular symmetry and the pole structure of the DMVV product formula we are able to extend the region of validity of Cardy's formula, and explain why it matches the semi-classical Bekenstein-Hawking formula for black holes all the way down to the AdS-scale. We prove that for large c the spectrum contains a universal perturbative sector whose degeneracies obey Hagedorn growth. The transition from Cardy to Hagedorn growth is found to be due to the formation of a Bose-Einstein condensate of ground state CFTs at low temperatures. The grand canonical partition function has an interesting phase structure, which in addition to the standard Hawking-Page transition between low and high temperature, exhibits a wall-crossing transition that exchanges the roles of \tau and \rho.