Supersymmetric M5 brane theories on R × CP²

Abstract

We propose 4 and 12 supersymmetric conformal Yang-Mills-Chern-Simons theories on R × CP² as multiple representations of the theory on M5 branes. These theories are obtained by twisted Z_k modding and dimensional reduction of the 6d (2,0) superconformal field theory on R × S⁵ and have a discrete coupling constant $\frac{1}{{g_{{Y\;M}}^2}}=\frac{k}{{4{\pi^2}}}$ with natural number k. Instantons in these theories are expected to represent the Kaluza-Klein modes. For the k = 1, 2 cases, we argue that the number of supersymmetries in our theories should be enhanced to 32 and 16, respectively. For the k = 3 case, only the 4 supersymmetric theory gets the supersymmetric enhancement to 8. For the 4 supersymmetric case, the vacuum structure becomes more complicated as there are degenerate supersymmetric vacua characterized by fuzzy spheres. We calculate the perturbative part of the SU(N ) gauge group Euclidean path integral for the index function at the symmetric phase of the 4 supersymmetric case and confirm it with the known half-BPS index. From the similar twisted Z _k modding of the AdS₇ × S⁴ geometry, we speculate that the M region is for k ≲ N ^1/3 and the type IIA region is N ^1/3 ≲ k ≲ N. When nonperturbative corrections are included, our theories are expected to produce the full index of the 6d (2,0) theory.