Fermion-Number Variation for Several Coupling Schemes in an $\mathrm{SU}(N)$ Chiral Soliton Background

Abstract

When topological solitons are coupled to fermions, they often induce a nontrivial fermion number. In the chiral quark soliton model (CQSM), this phenomenon underlies the identification of soliton topology with baryon number. However, whether this topology--fermion-number correspondence is determined solely by soliton topology or also depends on the fermion--soliton coupling scheme remains unclear. In this paper, we investigate this question by comparing several coupling schemes to the same $\mathrm{SU}(N)$ chiral soliton background. To treat the corresponding Dirac Hamiltonians within a common framework, we evaluate the variation of the regularized fermion-number along a model-parameter space. For the standard chiral Yukawa coupling of the CQSM, this variation reproduces the topological charge $Q$ associated with $π_3(\mathrm{SU}(N))$. The same result is obtained for an axial-vector coupling, whereas the fermion number variation vanishes for a vector coupling, despite the nontrivial topology of the chiral soliton background. These results provide an explicit example in which soliton topology alone does not determine the fermionic response; the topology--fermion-number correspondence also depends on how the background field is coupled to fermions.