On the degeneracy of $SU(3)_k$ topological phases

TL;DR

This paper investigates the ground state degeneracy of $SU(3)_k$ topological phases, providing combinatorial analysis, generating functions, and solutions for special cases, advancing understanding in quantum physics and mathematics.

Contribution

It offers a combinatorial framework and partial solutions for calculating degeneracies in $SU(3)_k$ topological phases, addressing an open problem for N>2.

Findings

Derived generating functions for degeneracy calculations

Solved specific cases of the degeneracy problem

Provided a combinatorial perspective on topological phase degeneracy

Abstract

The ground state degeneracy of an $SU(N)_k$ topological phase with $n$ quasiparticle excitations is relevant quantity for quantum computation, condensed matter physics, and knot theory. It is an open question to find a closed formula for this degeneracy for any $N > 2$. Here we present the problem in an explicit combinatorial way and analyze the case N=3. While not finding a complete closed-form solution, we obtain generating functions and solve some special cases.