Higher spin currents in the (N) = 1 stringy coset minimal model

This study reconsidered the (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 (N) = 1 stress energy tensor of spins (3/2,2). 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 (A₂ ⁽¹⁾⊕ A₂⁽¹⁾,A₂⁽¹⁾) at level (3, k). By acting with the above spin-3/2 current on the higher spin-3 Casimir current, its fermionic partner, the higher spin-5/2 current, can be generated and combined as a first higher spin supercurrent with spins (5/2,3). 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 (7/2,4) and (4,9/2). Moreover, the other two higher spin supercurrents with spins (4,9/2) and (9/2,5) can be generated by calculating the OPE between the first higher spin supercurrent with spins (5/2,3) and the second higher spin supercurrent with spins (7/2,4). Finally, the higher spin supercurrents, (11/2,6) ^and (6,13/2), can be extracted from the right hand side of the OPE between the higher spin supercurrents, (5/2,3) and (4,9/2).