The general M5-brane superconformal index

TL;DR

The paper develops a manifestly index-form 6d $$(2,0)$$ superconformal index by constructing a class of 5d $\mathcal{N}=2$ SYM theories on $\mathbb{CP}^2\times\mathbb{R}$ via reductions of $S^5/\mathbb{Z}_K\times\mathbb{R}$ and localizing the path integral on $\mathbb{CP}^2\times S^1$.The index is organized as a triple Nekrasov-like factorization associated with the three fixed points of $\mathbb{CP}^2$ and a contour integral over holonomies, with instanton sums weighted by $e^{-Kk\beta(1+a_i)}$, yielding a finite $k$-graded expansion that can be matched to gravity.For Abelian theories and finite $N$, the results reproduce known 6d indices and spaces of BPS states, including an $N^3$-scaling Casimir energy, and they agree with the gravity dual $AdS_7\times S^4$ for $k\le N$; the framework also extends to general $K$ with new predictions at finite $N$.The work also discusses S-duality, regulator choices, and potential applications to 6d $$(1,0)$$ theories, outlining a path toward systematic studies of higher-dimensional SCFT indices.

Abstract

We calculate and study the general superconformal index for the 6d U(N) (2,0) theory with four chemical potentials, from the indices of gauge theories on CP^2 x R. Our index agrees with the large N supergravity index on AdS_7 x S^4 at low energies, and also yields the negative 'Casimir energy' with an N^3 scaling which was recently calculated from a QFT on S^5. Our approach also suggests a systematic study of the (1,0) superconformal indices.