Quasi-particles and the Kanade-Russell and Kurşungöz formula for Capparelli's identity

We construct a quasi-particle basis of the integrable highest weight module of highest weight $3{\mathrm{\Lambda }}_0$ for the twisted affine Lie algebra of type $A_2^{(2)}$ in the principal realization. More specifically, by introducing the concept of polychromatic quasi-particle and finding relations among quasi-particles, we construct the spanning set of the standard module. Finally, its linear independence is proved by using Kanade-Russell and Kurşungöz’s Andrews-Gordon type series of Capparelli’s identities.