Abstract
We compute the number of massive vacua of N=4 supersymmetric Yang-Mills theory mass-deformed to preserve N=1 supersymmetry, for any gauge group G. We use semi-classical techniques and efficiently reproduce the known counting for A,B and C-type gauge groups, present the generating function for both O(2n) and SO(2n), and compute the supersymmetric index for gauge groups of exceptional type. A crucial role is played by the classification of nilpotent orbits, as well as global properties of their centralizers. We give illustrative examples of new features of our analysis for the D-type algebras.
Highlights
An analysis of commuting triples in the gauge group
We computed the number of massive vacua for N = 1∗ gauge theory on R4 for general gauge group
The main technique we used was to find a bijection between the classical vacuum expectation values for the three massive adjoint chiral multiplets and the nilpotent orbits in the gauge algebra
Summary
The relation between nilpotent elements and sl(2) triples is a bijection in the following sense: there is a one-toone correspondence between G-conjugacy classes of sl(2) triples in g and non-zero nilpotent G-orbits in g This follows for instance from Theorem 3.2.10 in [13] when G = Gad, and it remains true for connected gauge groups of any isogeny type (i.e. with non-zero center) because the adjoint action of the center is trivial. The bottom line is that it will be sufficient for us to study the nilpotent orbits of g in order to enumerate gauge inequivalent vacuum configurations for the triplet of adjoint scalars in the chiral multiplets.
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