The Arithmetic of Supersymmetric Vacua
Résumé
We provide explicit formulas for the number of vacua of four-dimensional pure $ \mathcal{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 $ {\left(\mathrm{S}\mathrm{U}(N)/{\mathbb{Z}}_m\right)}_n $ theory is a key ingredient in the semi-classical calculation of the number of massive vacua of $ \mathcal{N} $ = 1$^{∗}$ gauge theories with gauge algebra $ \mathfrak{s}\mathfrak{u}(n) $ , compactified on a circle. Using arithmetic, we express that number in an $ \mathrm{S}\mathrm{L}\left(2,\mathbb{Z}\right) $ duality invariant manner. We confirm our tally of massive vacua of the $ \mathcal{N} $ = 1$^{∗}$ theories by a count of inequivalent extrema of the exact superpotential.