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

TL;DR

This paper develops a trace formula for superconformal indices in 3, 5, and 6 dimensions, demonstrating its effectiveness through calculations in M-brane theories and Chern-Simons theories, revealing phase transitions at large N.

Contribution

It generalizes a superconformal index construction to higher dimensions and applies it to M-brane and Chern-Simons theories, uncovering phase transitions.

Findings

Derived a general index formula for superconformal theories in 3, 5, and 6 dimensions.

Computed indices for M2 and M5 brane theories in the large N limit.

Discovered large N phase transitions in certain 3D Chern-Simons theories.

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.