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On Superconformal Characters and Partition Functions in Three Dimensions

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.