The N = 1 ∗ Theories on R 1+2 × S 1 with Twisted Boundary Conditions

We explore the N = 1 ∗ theories compactified on a circle with twisted boundary conditions. The gauge algebra of these theories are the so-called twisted affine Lie algebra. We propose the exact superpotentials by guessing the sum of all monopole-instanton contributions and also by requiring SL(2, Z) modular properties. The latter is inherited from the N = 4 theory, which will be justified in the M theory setting. Interestingly all twisted theories possess full SL(2, Z) invariance, even though none of them are simply-laced. We further notice that these superpotentials are associated with certain integrable models widely known as elliptic Calogero-Moser models. Finally, we argue that the glueball superpotential must be independent of the compactification radius, and thus of the twisting, and confirm this by expanding it in terms of glueball superfield in weak coupling expansion.