Abstract
In the Grassmannian-like coset model, frac{mathrm{SU}{left(N+Mright)}_k}{mathrm{SU}{(N)}_ktimes mathrm{U}{(1)}_{kNMleft(N+Mright)}} , Creutzig and Hikida have found the charged spin-2, 3 currents and the neutral spin-2, 3 currents previously. In this paper, as an extension of Gaberdiel-Gopakumar conjecture found ten years ago, we calculate the operator product expansion (OPE) between the charged spin-2 current and itself, the OPE between the charged spin-2 current and the charged spin-3 current and the OPE between the neutral spin-3 current and itself for generic N, M and k. From the second OPE, we obtain the new charged quasi primary spin-4 current while from the last one, the new neutral primary spin-4 current is found implicitly. The infinity limit of k in the structure constants of the OPEs is described in the context of asymptotic symmetry of M×M matrix generalization of AdS3 higher spin theory. Moreover, the OPE between the charged spin-3 current and itself is determined for fixed (N, M) = (5, 4) with arbitrary k up to the third order pole. We also obtain the OPEs between charged spin-1, 2, 3 currents and neutral spin-3 current. From the last OPE, we realize that there exists the presence of the above charged quasi primary spin-4 current in the second order pole for fixed (N, M) = (5, 4). We comment on the complex free fermion realization.
Highlights
The Grassmannian-like coset model is described by [1] SU(N + M )kSU(N )k × U(1)kNM(N+M) (1.1)By introducing the ’tHooft-like coupling constant λ ≡ k (k+N )and taking the infinity limit of N with fixed λ and M, it has been proposed in [2] that the above coset model is dual to M × M matrix generalization of AdS3 Vasiliev higher spin theory [3, 4]
Hooft-like coupling constant λ and taking the infinity limit of N with fixed λ and M, it has been proposed in [2] that the above coset model is dual to M × M matrix generalization of AdS3 Vasiliev higher spin theory [3, 4]
How do we construct the higher spin-3 current which is neutral under the spin-1 current? We should write down the possible composite spin-3 operators and determine the relative coefficients by imposing the basic conditions coming from the coset (1.1)
Summary
Taking the infinity limit of N with fixed λ and M , it has been proposed in [2] that the above coset model is dual to M × M matrix generalization of AdS3 Vasiliev higher spin theory [3, 4]. By analyzing the first order pole of this OPE, we will determine the new quasi primary charged spin-4 current in terms of coset realization. From the explicit result for the OPE between the neutral spin-3 current and itself for fixed (N, M ) values, we will extract this OPE for generic (N, M ) case and at the second order pole of this OPE we will observe that there should be new primary neutral spin-4 current in terms of coset realization In obtaining this result, we realize that the k-dependent structure constant can be rewritten as the modified central charge which is equal to the coset central charge subtracted by the central term due to the stress energy tensor for the quadratic Sugawara term in the spin-1 current of SU(M ). The remaining ones were found in [2] previously
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