Indices for Superconformal Field Theories in 3,5 and 6 Dimensions

TL;DR

<3-5 sentence high-level summary> This work provides a unified framework for protected indices in superconformal field theories across dimensions 3, 5, and 6 by constructing a Witten-type trace formula that counts states annihilated by a chosen supercharge. It systematically analyzes unitary representations, distinguishing regular and isolated short multiplets, and derives exact character decompositions that capture all protected indices via subalgebra representations. The authors compute explicit indices for M2- and M5-brane worldvolume theories and for large-N Chern-Simons matter theories, revealing cancellations relative to naive supersymmetric state counting and, in some cases, large-N phase transitions as a function of chemical potentials. This framework offers a powerful, algebraic handle on the space of 3d/5d/6d SCFTs and their holographic duals, with potential applications to entropy-like growth and localization techniques in higher dimensions.

Abstract

We present a trace formula for a Witten type Index for superconformal field theories in d=3,5 and 6 dimensions, generalizing a similar recent construction in d=4. We perform a detailed study of the decomposition of long representations into sums of short representations at the unitarity bound to demonstrate that our trace formula yields the most general index (i.e. quantity that is guaranteed to be protected by superconformal symmetry alone) for the corresponding superalgebras. Using the dual gravitational description, we compute our index for the theory on the world volume of N M2 and M5 branes in the large N limit. We also compute our index for recently constructed Chern Simons theories in three dimensions in the large N limit, and find that, in certain cases, this index undergoes a large N phase transition as a function of chemical potentials.