On Superconformal Characters and Partition Functions in Three Dimensions

TL;DR

This paper derives character formulas for superconformal representations in three dimensions, simplifies them via limits, and applies these to compute partition functions and BPS operator counts in superconformal theories and their duals.

Contribution

It provides new character formulas for Osp(2N|4) superconformal representations and applies them to obtain closed-form partition functions for BPS operators in superconformal Chern-Simons theories.

Findings

Derived character formulas consistent with conformal group decompositions.

Obtained closed-form generating functions for BPS operators in large N limit.

Demonstrated consistency between free field and supergravity operator counts.

Abstract

Possible short and semi-short positive energy, unitary representations of the Osp(2N|4) superconformal group in three dimensions are discussed. Corresponding character formulae are obtained, consistent with character formulae for the SO(3,2) conformal group, revealing long multiplet decomposition at unitarity bounds in a simple way. Limits, corresponding to reduction to various Osp(2N|4) subalgebras, are taken in the characters that isolate contributions from fewer states, at a given unitarity threshold, leading to considerably simpler formulae. Via these limits, applied to partition functions, closed formulae for the generating functions for numbers of BPS operators in the free field limit of superconformal U(n)\times U(n) \N=6 Chern Simons theory and its supergravity dual are obtained in the large n limit. Partial counting of semi-short operators is performed and consistency between operator counting for the free field and supergravity limits with long multiplet decomposition rules is explicitly demonstrated. Partition functions counting certain protected scalar primary semi-short operators, and their superconformal descendants, are proposed and computed for large n. Certain chiral ring partition functions are discussed from a combinatorial perspective.