One dimensional Large-N gauge theory in a 1/D expansion

We study a dimensional reduction of d+D dimensional large-N YM theory to d dimension, especially d=0 and d=1 case.  Then the theory is described by d dimensional gauge field and D adjoint scalars.  We take a large D limit and evaluate the path-integral by expanding 1/D.  Then we find a non-trivial vacuum characterized by a mass gap for the adjoint scalars.

In d=0 case, we obtain an exact expression for the partition function up to 1/D in finite N.  In d=1 case, we derived an effective action for the gauge field and showed the existence of the confinement/deconfinment phase transition.  Remarkably we can sum over all loop correction in 1/D order and our results reproduce numerical results even in the strong 'tHooft coupling region.  We will also discuss the gravity interpretation of our results.