# Counting BPS operators in N=4 SYM

## TL;DR

This paper develops a symmetric polynomial framework to count BPS operators in free  super Yang-Mills theory, providing explicit formulas for finite and large N, and analyzing their asymptotic behavior.

## Contribution

It introduces a novel symmetric polynomial approach to enumerate BPS operators in  SYM, including finite N corrections and asymptotic counting methods.

## Key findings

- Explicit counting formulas for BPS operators at finite N.
- Large N limit results for BPS operator enumeration.
- Asymptotic analysis of BPS operators with different R-charges.

## Abstract

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. The partition function for BPS operators in the chiral ring of \N=4 super Yang Mills is analysed in terms of plane partitions. Asymptotic counting of BPS primary operators with differing R-symmetry charges is discussed in both free \N=4 super Yang Mills and in the chiral ring. Also, general and explicit expressions are derived for SU(2) gauge theory partition functions, when the fundamental fields transform in the adjoint, for free field theory.