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Supersymmetric M5 Brane Theories on R x CP2

We propose 4 and 12 supersymmetric Yang-Mills-Chern-Simons theories on $\mathrm{R\times CP^2}$ obtained by twisted $\mathrm{Z}_k$ moddings and dimensional reduction of the 6d (2,0) superconformal field theories on $\mathrm{R\times S^5}$. These theories have a discrete coupling constant $\frac{1}{g^2_{YM}} =\frac{k}{4π^2}$ so that instantons represent the Kaluza-Klein modes correctly. We calculate the perturbative part of the SU(N) gauge group Euclidean path integral for the index function and confirm it with the known half-BPS index. The scalar and fermionic fields have the conformal dimension prescribed by the 6d theory. From the similar twisted $Z_k$ modding of the $\mathrm{AdS_7\times S^4}$ geometry, we speculate that the $M$ region is for $k\lesssim N^{1/3}$ and the type IIA region is $N^{1/3}\lesssim k \lesssim N$. When nonperturbative corrections are included, our theory is expected to produce the full index of the 6d (2,0) theory.