Supersymmetric index of three-dimensional gauge theory (Q2755517)

Within the framework of \(N=1\) super Yang-Mills theory in three spacetime dimensions, with a simple gauge group \(G\) and a Chern-Simons interaction of level \(k\), the author shows that the supersymmetric index \(Tr(-1)^F\) can be computed by making a relation to a pure Chern-Simons theory or microscopically by an explicit Born-Oppenheimer calculation on a two-torus. In this process it is reviewed the anomaly that sometimes shifts \(k\) to half-integer values occur and an explanation of the failure of a plausible attempt to disprove the hypothesis of symmetry breaking in the gap via \(SU(n)/\mathbb Z_n\) gauge theory is provided. This result shows that supersymmetrtry is unbroken if \(|k|\geq h/2\) (where \(h\) is the dual Coxeter number of \(G\)) and suggests that dynamical supersymmetry breaking occurs for \(k<h/2\). Considering the three-dimensional Chern-Simons theories in the light of the familiar classification of massive phases of gauge theories, it is pointed out that the theories with large \(|k|\) are massive gauge theories whose universality class is not fully described by the usual criteria, and that the full classification of massive phases is particularly rich in three dimensions.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00052].