Fermionic Construction in the Supersymmetric Coset Model

TL;DR

This paper computes and fully determines the operator product expansions between specific multiplets in an ${ m N}=3$ supersymmetric coset model, revealing detailed multiplet structures and superspace descriptions.

Contribution

It provides a complete calculation of the ${ m N}=3$ supersymmetric OPEs between the first and second multiplets using fermionic realizations and superspace methods.

Findings

Explicit ${ m N}=3$ supersymmetric OPEs derived

Identification of ${ m N}=3$ multiplet structures in the OPEs

Description of ${ m N}=2$ superspace formulation and decoupling

Abstract

It is known previously that the operator product expansion (OPE) between the first ${\cal N}=3 $ multiplet and itself contains the second ${\cal N}=3$ multiplet in the supersymmetric coset model. In this paper, by using their realizations in terms of various fermions, we compute the four kinds of OPEs between the first and the second ${\cal N}=3$ multiplets for fixed $N$ and $M$ where the group of the coset contains $SU(N+M)$. By supersymmetrizing the above OPEs in ${\cal N}=3$ superspace and using the various Jacobi identities between the currents, we determine the ${\cal N}=3$ supersymmetric OPE between the first and the second ${\cal N}=3$ multiplets completely. The right hand side of this OPE contains the various ${\cal N}=3$ multiplets: the $SO(3)$ singlet ${\cal N}=3$ multiplets of superspin-$\frac{3}{2},2,3,4$ and the $SO(3)$ triplet ${\cal N}=3$ multiplets of superspin-$\frac{5}{2},3,\frac{7}{2}$. The ${\cal N}=2$ superspace description and the decoupling of the spin-$\frac{1}{2}$ current of the ${\cal N}=3$ superconformal algebra are also described.