Deconfinement of Gauge Theories at High Temperature

The deconfinement phase transition of large N gauge theories has been recognized as the Hawking-Page transition through AdS/CFT correspondence. To derive the phase transition, the calculation of the partition function is restricted to the singlet sector. This poster examines whether the singlet constraint of gauge theory at the deconfinement phase was a physical consequence. A model of multiple matrices quantum harmonic oscillators is considered in this poster as a toy model of large N, four dimensional SU(N) gauge theory. Our work aims to compare the partition function restricted to singlet sector to the known partition function of a group of non-interacting harmonic oscillators, which represent the model without imposing the singlet constraint. We compute the singlet partition function analytically for the model with a large number of matrices, and we also develop numerical methods using a quadratic programming algorithm to solve the partition function for the model with a small number of matrices. We assure that the eigenvalues are frozen at the saddle point at high temperature. However the singlet partition function has the free energy converging to $(m-1)N^2ln (1-q)$ in the large m limit while the free energy of harmonic oscillators is $mN^2ln (1-q)$. Changing the integration variables from group elements to eigenvalues explains how to derive such discrepancy of the free energy between imposing and relaxing the single constraint. Gauge theory cannot simply relax the singlet constraint at high temperature without adding a normalization factor.