Duality of large N Yang-Mills Theory on $T^2 \times R^n$

Abstract

We find aspects of electrically confining large $N$ Yang-Mills theories on $T^2 \times R^{d-2}$ which are consistent with a $GL(2,Z)$ duality. The modular parameter associated with this $GL(2,Z)$ is given by ${m\over N} + i\Lambda^2 A$, where $A$ is the area of the torus, $m$ is the t'Hooft twist on the torus, and $\Lambda^2$ is the string tension. $N$ is taken to infinity keeping $m\over N$ and $g^2N$ fixed. This duality may be interpreted as T-duality of the QCD string if one identifies the magnetic flux with a two-form background in the string theory. Our arguments make no use of supersymmetry. While we are not able to show that this is an exact self duality of conventional QCD, we conjecture that it may be applicable within the universality class of QCD. We discuss the status of the conjecture for the soluble case of pure two dimensional Euclidean QCD on $T^2$, which is almost but not exactly self dual. For higher dimensional theories, we discuss qualitative features consistent with duality. For $m=0$, such a duality would lead to an equivalence between pure QCD on $R^4$ and QCD on $R^2$ with two adjoint scalars. When $\Lambda^2 A << m^2/N^2$, the proposed duality includes exchanges of rank with twist. This exchange bears some resemblance, but is not equivalent, to Nahm duality. A proposal for an explicit perturbative map which implements duality in this limit is discussed.