Singlet sector of the vector model with angular potential

We study the singlet sector of large- N, O(N) vector model on S 1 β times S 2 r in presence of an angular potential Ω. We compute the free energy of the model both at the free fixed point as well as the non-trivial fixed point. The model undergoes a phase transition when the temperature T and angular potential attain the relation T 2r 2 about N (1-ω 2), where ω=rΩ and r is the radius of S squared, with Tr 1, at both the fixed points, with different numerical coefficients. The transition was studied earlier at zero angular potential at Tr about sqrt N. The holonomy eigenvalue distribution, having support on the full circle below the transition, develops a gap above it, called the Gross-Witten-Wadia transition. The angular potential introduces an additional parameter to such a transition, tuning it, the transition can occur even at a fixed temperature. We show that for the free model transition can occur as 1-ω 2 about 1/N while Tr about O(1). For the interacting model the computation is performed at a limit Tr 1, thus we show the same at a large but fixed temperature which does not scale as sqrt N. In the ungapped phase, the free energy exhibits a double pole near ω 2=1 for both the fixed points. This is possibly due to the fact that the theory projected onto the singlets is non-local and therefore need not obey the leading simple pole scaling of a local thermal effective action.