The arithmetic of supersymmetric vacua

We provide explicit formulas for the number of vacua of four-dimensional pure N=1 super Yang-Mills theories on a circle, with any simple gauge algebra and any choice of center and spectrum of line operators. The formula for the (SU(N)/ℤ_m)_n theory is a key ingredient in the semi-classical calculation of the number of massive vacua of N=1^∗ gauge theories with gauge algebra su(n), compactified on a circle. Using arithmetic, we express that number in an SL(2,ℤ) duality invariant manner. We confirm our tally of massive vacua of the N=1^∗ theories by a count of inequivalent extrema of the exact superpotential.