Indices for 6 dimensional superconformal field theories

TL;DR

This article surveys how to access BPS spectra of 6d $(2,0)$ SCFTs via Witten indices, outlining both the Coulomb-branch indices and the 6d superconformal index. It details two complementary computational frameworks for Coulomb-branch indices—1d quantum-mechanical instanton counting from D0-D4 systems and 2d elliptic genera of self-dual string worldsheets—and presents two realizations of the 6d index on $S^5\times S^1$ and on $\mathbb{CP}^2\times S^1$, including Abelian and non-Abelian refinements and the emergence of $W_G$ vacuum characters and a supersymmetric Casimir energy in particular limits. The work connects 6d SCFT data to 5d SYM instantons, topological-string perspectives, and holographic expectations, while highlighting exact results in tractable limits and the modular structure of these indices. It also identifies open questions about intrinsic formulations that do not rely on 5d descriptions and the potential unification of the Coulomb-branch and conformal-phase computations within a broader string/M-theory framework.

Abstract

We review some recent developments in the 6 dimensional (2, 0) superconformal field theories, focusing on their BPS spectra in the Coulomb and symmetric phases computed by various Witten indices. We shall discuss the instanton partition function of 5d maximal super-Yang-Mills theory, and the 6d superconformal index.