Counting BPS operators in N=4 SYM

Understand

The free field partition function for a generic U(N) gauge theory, where the fundamental fields transform in the adjoint representation, is analysed in terms of symmetric polynomial techniques.

• It is shown by these means how this is related to the cycle polynomial for the symmetric group and how the large N result may be easily recovered.
• Higher order corrections for finite N are also discussed in terms of symmetric group characters.
• For finite N, the partition function involving a single bosonic fundamental field is recovered and explicit counting of multi-trace quarter BPS operators in free \N=4 super Yang Mills discussed, including a general result for large N.