  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.

High Energy Physics - Theory