Perturbative analysis of twisted volume reduced theories

We discuss the perturbative expansion of SU(N) Yang-Mills theories defined on a d-dimensional torus of linear size l with twisted boundary conditions, generalizing previous results in the literature. For a specific class of twist tensors depending on a single integer flux value k, we show that perturbative results to all orders depend on the combination lN^(2/d) and a flux-dependent angle theta. This implies a new kind of volume independence that holds at finite N and for fixed values of theta. Our results also provide interesting information about the possible occurrence of tachyonic instabilities at one-loop order. We support the prescription that instabilities are avoided, if the large N limit is taken keeping theta > theta_c, and appropriately scaling the magnetic flux k with N. Numerical results in 2+1 dimensions provide a test of how these ideas extend into the non-perturbative regime.