N=4 Superconformal Characters and Partition Functions

TL;DR

This paper derives character formulae for N=4 superconformal representations, explores their expansions and limits, and applies these to compute partition functions and BPS operator counts in super Yang Mills theory.

Contribution

It introduces new character formulae for N=4 superconformal representations and applies them to analyze super Yang Mills partition functions and BPS operator counting.

Findings

Derived explicit character formulae for N=4 superconformal representations.

Obtained exact counts of BPS operators in free super Yang Mills.

Showed how certain short operators are protected dynamically.

Abstract

Character formulae for positive energy unitary representations of the N=4 superconformal group are obtained through use of reduced Verma modules and Weyl group symmetry. Expansions of these are given which determine the particular representations present and results such as dimensions of superconformal multiplets. By restriction of variables various `blind' characters are also obtained. Limits, corresponding to reduction to particular subgroups, in the characters isolate contributions from particular subsets of multiplets and in many cases simplify the results considerably. As a special case, the index counting short and semi-short multiplets which do not form long multiplets found recently is shown to be related to particular cases of reduced characters. Partition functions of N=4 super Yang Mills are investigated. Through analysis of these, exact formulae are obtained for counting half and some quarter BPS operators in the free case. Similarly, partial results for the counting of semi-short operators are given. It is also shown in particular examples how certain short operators which one might combine to form long multiplets due to group theoretic considerations may be protected dynamically.