The $M5$-brane conjecture for $(1,0)$ BCOV theory

Consider a single $M5$ brane on $\mathbb{R}^7\times(\mathbb{R}^4/\mathbb{Z}_N)$, located at the singular point of the $A_N$ singularity and spanning six of the seven directions in $\mathbb{R}^7$. Its world-volume supports a theory with $(1,0)$ supersymmetry. M5-brane conjecture. The holomorphic twist of this theory is $(1,0)$ BCOV theory coupled to $\mathfrak{sl}(N\mid N)$ holomorphic Chern–Simons theory. This conjecture relates the theory on the singularity-supported $M5$ brane to the coupled holomorphic theory constructed in the paper; it was suggested by Davide Gaiotto and is not proved here.