Explicit Construction of Spin 4 Casimir Operator in the Coset Model $ \hat{SO} (5)_{1} \times \hat{SO} (5)_{m} / \hat{SO} (5)_{1+m} $

TL;DR

This paper extends the GKO coset construction to include a spin 4 Casimir operator in a specific $ ext{SO}(5)$ coset model, providing explicit calculations for the fourth order invariant in a family of minimal models.

Contribution

It introduces a generalized coset construction for a dimension 5/2 operator in $ ext{SO}(5)$ and computes the fourth order Casimir invariant for a series of minimal models with $c<5/2$.

Findings

Explicit form of the spin 4 Casimir operator in the coset model.

Calculation of the fourth order Casimir invariant for minimal models.

Connection to the $c=5/2$ free fermion model.

Abstract

We generalize the Goddard-Kent-Olive (GKO) coset construction to the dimension 5/2 operator for $ \hat{so} (5) $ and compute the fourth order Casimir invariant in the coset model $\hat{SO} (5)_{1} \times \hat{SO} (5)_{m} / \hat{SO} (5)_{1+m} $ with the generic unitary minimal $ c < 5/2 $ series that can be viewed as perturbations of the $ m \rightarrow \infty $ limit, which has been investigated previously in the realization of $ c= 5/2 $ free fermion model.