On superconformal characters and partition functions in three dimensions

Abstract

Possible short and semishort positive energy, unitary representations of the Osp(2N|4) superconformal group in three dimensions are discussed. Corresponding character formulas are obtained, consistent with character formulas 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 formula. Via these limits, applied to partition functions, closed formula for the generating functions for numbers of BPS operators in the free field limit of superconformal U(n)×U(n) N=6 Chern–Simons theory and its supergravity dual are obtained in the large n limit. Partial counting of semishort 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 semishort operators, and their superconformal descendants, are proposed and computed for large n. Certain chiral ring partition functions are discussed from a combinatorial perspective.