Strings and Discrete Fluxes of QCD

We study discrete fluxes in four dimensional SU(N) gauge theories with a mass gap by using brane compactifications which give $\mathcal{N}=1$ or ${\cal{N}} = 0$ supersymmetry. We show that when such theories are compactified further on a torus, the t’Hooft magnetic flux $m$ is related to the NS two-form modulus $B$ by $B=2\pi \frac{m}{N}$. These values of $B$ label degenerate brane vacua, giving a simple demonstration of magnetic screening. Furthermore, for these values of $B$ one has a conventional gauge theory on a commutative torus, without having to perform any T-dualities. Because of the mass gap, a generic $B$ does not give a four dimensional gauge theory on a non-commutative torus. The Kaluza-Klein modes which must be integrated out to give a four dimensional theory decouple only when $B=2\pi \frac{m}{N}$. Finally we show that $2\pi {m\over N}$ behaves like a two form modulus of the QCD string. This confirms a previous conjecture based on properties of large $N$ QCD suggesting a T-duality invariance.