Narrow Results

In this paper we analyze the structure of the BRST charge of nonlinear superalgebras. We consider quadratic nonlinear superalgebras where a commutator (in terms of (super-)Poisson brackets) of the generators is a quadratic polynomial of the generators. We find the explicit form of the BRST charge up to cubic order in Faddeev–Popov ghost fields for arbitrary quadratic nonlinear superalgebras. We point out the existence of constraints on structure constants of the superalgebra when the nilpotent BRST charge is quadratic in Faddeev–Popov ghost fields. The general results are illustrated by simple examples of superalgebras.

We construct the lowest higher spin-2 current in terms of the spin-1 and the spin-$\frac{1}{2}$ currents living in the orthogonal $\frac{SO(N+4)}{SO(N)\times SO(4)}$ Wolf space coset theory for general $N$. The remaining 15 higher spin currents are determined. We obtain the three-point functions of bosonic (higher) spin currents with two scalars for finite $N$ and $k$ (the level of the spin-1 current). By multiplying $SU(2)\times U(1)$ into the above Wolf space coset theory, the other 15 higher spin currents together with the above lowest higher spin-2 current are realized in the extension of the large ${𝒩}=4$ linear superconformal algebra. Similarly, the three-point functions of bosonic (higher) spin currents with two scalars for finite $N$ and $k$ are obtained. Under the large $N$ ’t Hooft limit, the two types of three-point functions in the nonlinear and linear versions coincide as in the unitary coset theory found previously.

The large ${𝒩}=4$ nonlinear superconformal algebra is generated by six spin-$1$ currents, four spin-$\frac{3}{2}$ currents and one spin-$2$ current. The simplest extension of these $11$ currents is described by the $16$ higher spin currents of spins $(1,\frac{3}{2},\frac{3}{2},\frac{3}{2},\frac{3}{2},2,2,2,2,2,2,\frac{5}{2},\frac{5}{2},\frac{5}{2},\frac{5}{2},3)$. In this paper, by using the defining operator product expansions (OPEs) between the $11$ currents and $16$ higher spin currents, we determine the $16$ higher spin currents (the higher spin-$1,\frac{3}{2}$ currents were found previously) in terms of affine Kac–Moody spin-$\frac{1}{2}$, one currents in the Wolf space coset model completely. An antisymmetric second rank tensor, three antisymmetric almost complex structures or the structure constant are contracted with the multiple product of spin-$\frac{1}{2},1$ currents. The eigenvalues are computed for coset representations containing at most four boxes, at finite $N$ and $k$. After calculating the eigenvalues of the zeromode of the higher spin-$3$ current acting on the higher representations up to three (or four) boxes of Young tableaux in $SU(N+2)$ in the Wolf space coset, we obtain the corresponding three-point functions with two scalar operators at finite $(N,k)$. Furthermore, under the large $(N,k)$ ’t Hooft-like limit, the eigenvalues associated with any boxes of Young tableaux are obtained and the corresponding three-point functions are written in terms of the ’t Hooft coupling constant in simple form in addition to the two-point functions of scalars and the number of boxes.

By computing the operator product expansions between the first two ${𝒩}=4$ higher spin multiplets in the unitary coset model, the (anti-)commutators of higher spin currents are obtained under the large $(N,k)$ ’t Hooft-like limit. The free field realization with complex bosons and fermions is presented. The (anti-)commutators for generic spins $s_1$ and $s_2$ with manifest $SO(4)$ symmetry at vanishing ’t Hooft-like coupling constant are completely determined. The structure constants can be written in terms of the ones in the ${𝒩}=2$${{𝒲}}_{\infty }$ algebra found by Bergshoeff, Pope, Romans, Sezgin and Shen previously, in addition to the spin-dependent fractional coefficients and two $SO(4)$ invariant tensors. We also describe the ${𝒩}=4$ higher spin generators, by using the above coset construction results, for general superspin $s$ in terms of oscillators in the matrix generalization of ${\mathrm{\text{AdS}}}_3$ Vasiliev higher spin theory at nonzero ’t Hooft-like coupling constant. We obtain the ${𝒩}=4$ higher spin algebra for low spins and present how to determine the structure constants, which depend on the higher spin algebra parameter, in general, for fixed spins $s_1$ and $s_2$.

The complete structure of the Casimir ${{𝒲}{𝒜}}_N$ algebras is shown to exist in such a way that the Casimir ${{𝒲}{𝒜}}_N$ algebra is a kind of truncated type of ${{𝒲}}_{\infty }$ algebra both in the primary and in the quadratic basis, first using the associativity conditions in the basis of primary fields and second using the Miura basis coming from the free field realization as a different basis. We can conclude that the Casimir ${{𝒲}{𝒜}}_N$ algebra is a kind of truncated type of ${{𝒲}}_{\infty }$ algebra, so it is clear from any construction of ${{𝒲}}_{\infty }$ algebra that by putting infinite number of fields $W_s$ with $s>N$ to zero, we arrive at the Casimir ${{𝒲}{𝒜}}_N$ algebra. We concentrated in this work only for the particular case of ${{𝒲}{𝒜}}_5$ algebra since this example gives us explicitly a method on how to deal with the general case $N$.

Gravity, itself a gauge theory of a spin 2 field, can be extended to a higher spin gauge theory on AdS spaces. Recently, Gaberdiel and Gopakumar conjectured that a large N limit of a 2d minimal model is dual to a bosonic subsector of a higher spin supergravity theory on 3d AdS space. We propose and test the untruncated supersymmetric version of this conjecture where the dual CFT is a large N limit of the CP^N Kazama-Suzuki model.