Adding complex fermions to the Grassmannian-like coset model

Abstract

In the $${{\mathcal {N}}}=2$$ N = 2 supersymmetric coset model, $$\frac{SU(N+M)_k \times SO(2 N M)_1}{ SU(N)_{k+M} \times U(1)_{ N M (N+M)(k+N+M)}}$$ S U ( N + M ) k × S O ( 2 N M ) 1 S U ( N ) k + M × U ( 1 ) N M ( N + M ) ( k + N + M ) , we construct the SU(M) nonsinglet $${{\mathcal {N}}}=2$$ N = 2 multiplet of spins $$(1, \frac{3}{2}, \frac{3}{2}, 2)$$ ( 1 , 3 2 , 3 2 , 2 ) in terms of coset fields. The next SU(M) singlet and nonsinglet $${{\mathcal {N}}}=2$$ N = 2 multiplets of spins $$(2, \frac{5}{2}, \frac{5}{2}, 3)$$ ( 2 , 5 2 , 5 2 , 3 ) are determined by applying the $${{\mathcal {N}}}=2$$ N = 2 supersymmetry currents of spin $$\frac{3}{2}$$ 3 2 to the bosonic singlet and nonsinglet currents of spin 3 in the bosonic coset model. We also obtain the operator product expansions (OPEs) between the currents of the $${{\mathcal {N}}}=2$$ N = 2 superconformal algebra and above three kinds of $${{\mathcal {N}}}=2$$ N = 2 multiplets. These currents in two dimensions play the role of the asymptotic symmetry, as the generators of $${{\mathcal {N}}}=2$$ N = 2 “rectangular W-algebra”, of the $$M \times M$$ M × M matrix generalization of $$\mathcal{N}=2$$ N = 2 $$AdS_3$$ A d S 3 higher spin theory in the bulk. The structure constants in the right hand sides of these OPEs are dependent on the three parameters k, N and M explicitly. Moreover, the OPEs between SU(M) nonsinglet $${{\mathcal {N}}}=2$$ N = 2 multiplet of spins $$(1, \frac{3}{2}, \frac{3}{2}, 2)$$ ( 1 , 3 2 , 3 2 , 2 ) and itself are analyzed in detail. The complete OPE between the lowest component of the SU(M) singlet $${{\mathcal {N}}}=2$$ N = 2 multiplet of spins $$(2, \frac{5}{2}, \frac{5}{2}, 3)$$ ( 2 , 5 2 , 5 2 , 3 ) and itself is described. In particular, when $$M=2$$ M = 2 , it is known that the above $${{\mathcal {N}}}=2$$ N = 2 supersymmetric coset model provides the realization of the extension of the large $${{\mathcal {N}}}=4$$ N = 4 nonlinear superconformal algebra. We determine the currents of the large $${{\mathcal {N}}}=4$$ N = 4 nonlinear superconformal algebra and the higher spin-$$\frac{3}{2}, 2$$ 3 2 , 2 currents of the lowest $${{\mathcal {N}}}=4$$ N = 4 multiplet for generic k and N in terms of the coset fields. For the remaining higher spin-$$\frac{5}{2},3$$ 5 2 , 3 currents of the lowest $$\mathcal{N}=4$$ N = 4 multiplet, some of the results are given.

The ${\cal N}=4$ Supersymmetric Linear $W_{\infty}[λ]$ Algebra

“…Compared to the N = 4 primary condition [11] of the first N = 4 multiplet, there are additional terms in (B.1): either central terms or the field contents of the N = 4 stress energy tensor 17 . In other words, the first N = 4 multiplet is not N = 4 primary.…”