Supersymmetric M5 Brane Theories on R x CP2

TL;DR

This work constructs two 5d Yang-Mills-Chern-Simons theories on $\mathbb{R}\times \mathbb{CP}^2$ by twisted $\mathbb{Z}_k$ modding of the 6d $(2,0)$ theory on $\mathbb{R}\times S^5$ and subsequent dimensional reduction, yielding discrete couplings $\frac{1}{g_{YM}^2}=\frac{k}{4\pi^2 r}$ and two models with 4 or 12 preserved supercharges. Instantons in these theories encode KK modes of the original 6d theory, and the perturbative SU($N$) index reproduces the known half-BPS index, with nonperturbative corrections expected to generate the full 6d index; a mapping to M-theory vs Type IIA regimes is proposed in terms of $k$ and $N$. The analysis connects to AdS$_7\times$S$^4$ holography, exploring parameter regions where M-theory or Type IIA descriptions dominate, and outlines a program to enrich the index with instanton contributions to recover the full 6d spectrum. This framework provides a UV-structured 5d proxy for probing the elusive 6d $(2,0)$ theory and its nonperturbative BPS structure.

Abstract

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.