3D fermions and affine Yangian

Abstract

2D (dimensional) Boson-Fermion correspondence is a well-known object. In this paper, we define 3D Fermions Γm→ and Γm→⁎ for any vertex m→ and the representation space F, we find that F is isomorphic to the vector space of 3D Young diagrams. Operators Γm→ and Γm→⁎ are defined with amplitudes, which are described by complex numbers e1,e2,e3 satisfying e1+e2+e3=0, then we find that Γm→ and Γm→⁎ have relations with the generators in affine Yangian, and states in F are corresponding to 3-Schur functions. We also discuss the slice of state in F, which corresponds to the slice of 3D Young diagram. Finally, we show that in special case, 3D Fermions and the representation space become ordinary 2D Fermions and charged zero Fermionic Fock space respectively.