Higher spin currents in the $\mathcal{N}$ = 1 stringy coset minimal model

Abstract

This study reconsidered the $\mathcal{N}$ = 1 supersymmetric extension of the W ₃ algebra which was studied previously. This extension consists of seven higher spin supercurrents (fourteen higher spin currents in the components) as well as the $\mathcal{N}$ = 1 stress energy tensor of spins $\left( {\frac{3}{2},2} \right)$. Thus far, the complete expressions for the higher spin currents have not been derived.

This paper constructs them explicitly in both the c = 4 eight free fermion model and the supersymmetric coset model based on $\left( {A_2^{(1)}\oplus A_2^{(1) },A_2^{(1) }} \right)$ at level (3, k). By acting with the above $\mathrm{spin}\hbox{-}\frac{3}{2}$ current on the higher spin-3 Casimir current, its fermionic partner, the higher $\mathrm{spin}\hbox{-}\frac{5}{2}$ current, can be generated and combined as a first higher spin supercurrent with spins $\left( {\frac{5}{2},3} \right)$. By calculating the operator product expansions (OPE) between the higher spin supercurrent and itself, the next two higher spin supercurrents can be generated with spins $\left( {\frac{7}{2},4} \right)$ and $\left( {4,\frac{9}{2}} \right)$. Moreover, the other two higher spin supercurrents with spins $\left( {4,\frac{9}{2}} \right)$ and $\left( {\frac{9}{2},5} \right)$ can be generated by calculating the OPE between the first higher spin supercurrent with spins $\left( {\frac{5}{2},3} \right)$ and the second higher spin supercurrent with spins $\left( {\frac{7}{2},4} \right)$. Finally, the higher spin supercurrents, $\left( {\frac{11 }{2},6} \right)$ ^and $\left( {6,\frac{13 }{2}} \right)$, can be extracted from the right hand side of the OPE between the higher spin supercurrents, $\left( {\frac{5}{2},3} \right)$ and $\left( {4,\frac{9}{2}} \right)$.