Finite N effect in the 1/D expansion of the one-dimensional large-N gauge theory

We consider finite N BFSS matrix model with D scalar fields and a gauge field given by N by N Hermitian matrix on a one-dimensional circled space. We introduce an auxiliary matrix field in our analysis. As a result, our action changes to a quadratic form for scalar fields with interaction terms. Integrating the scalar fields to the one-loop order, and also performing the gauge-fixing, we obtain the one-loop effective action with the auxiliary field and gauge field as the dynamical variables. We then evaluate the auxiliary field at a saddle point. As a result, a mass term appear, and the one-loop effective action changes to the Ginzburg-Landau type effective action (GL action) with Wilson line as its dynamical variable. We can see that there exists confinement/deconfinemnt transition and the third-order transition in the GL action. We investigate these transitions with finite N effect at large D. Here, we ignore all the 1/D corrections except for the part indispensable for determining the occurrence of the confinement/deconfinemnt transition to keep the feasibility for the analysis of the transition order. Further, our finite N is assumed to be no larger than breaking the validity of the saddle point method. Hence, our finite N is some correction from the large-N.