The operator product expansion between the 16 lowest higher spin currents in the $$\mathcal{N}=4$$ N = 4 superspace

Changhyun Ahn,Man Hea Kim

doi:10.1140/epjc/s10052-016-4234-2

Changhyun Ahn, Man Hea Kim

Open Access

https://doi.org/10.1140/epjc/s10052-016-4234-2

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Abstract

Some of the operator product expansions (OPEs) between the lowest 16 higher spin currents of spins (1, 3/2, 3/2, 3/2, 3/2, 2, 2, 2, 2, 2, 2, 5/2, 5/2, 5/2, 5/2, 3) in an extension of the large N=4 linear superconformal algebra were constructed in the N=4 superconformal coset SU(5)/SU(3) theory previously. In this paper, by rewriting the above OPEs in the N=4 superspace developed by Schoutens (and other groups), the remaining undetermined OPEs where the corresponding singular terms possess the composite fields with spins s =7/2, 4, 9/2, 5 are completely determined. Furthermore, by introducing the arbitrary coefficients in front of the composite fields in the right hand sides of the above complete 136 OPEs, reexpressing them in the N=2 superspace and using the N=2 OPEs mathematica package by Krivonos and Thielemans, the complete structures of the above OPEs with fixed coefficient functions are obtained with the help of various Jacobi identities. Then one obtains ten N=2 super OPEs between the four N=2 higher spin currents denoted by (1, 3/2, 3/2, 2), (3/2, 2, 2, 5/2), (3/2, 2, 2, 5/2) and (2, 5/2, 5/2, 3) (corresponding 136 OPEs in the component approach) in the N=4 superconformal coset SU(N+2)/SU(N) theory. Finally, one describes them as one single N=4 super OPE between the above sixteen higher spin currents in the N=4 superspace. The fusion rule for this OPE contains the next 16 higher spin currents of spins of (2, 5/2, 5/2, 5/2, 5/2, 3, 3, 3, 3, 3, 3, 7/2, 7/2, 7/2, 7/2, 4) in addition to the quadratic N=4 lowest higher spin multiplet and the large N=4 linear superconformal family of the identity operator. The various structure constants (fixed coefficient functions) appearing in the right hand side of this OPE depend on N and the level k of the bosonic spin-1 affine Kac-Moody current.

Highlights

In the large N = 4 holography observed in [1], the duality between matrix extended higher spin theories on AdS3 space with large N = 4 supersymmetry and large N = 4 coset theory in two-dimensional conformal field theory (CFT) was proposed.1 One of the consistency checks for this duality is based on the matching of correlation functions
The zero-mode eigenvalue equations of the higher spin Casimir current in the two-dimensional N = 4 coset model should coincide with the zero-mode eigenvalue equations of the higher spin field in an asymptotic symmetry algebra of higher spin theory on the AdS3 space
2 Review of the operator product expansions (OPEs) between the 16 currents in N = 4 superspace we describe the 16 currents of the large N = 4 linear superconformal algebra in the N = 4 superspace, where SO(4) symmetry is manifest

Summary

Introduction

The corresponding large N = 4 linear superconformal algebra, which consists of 13 nontrivial OPEs in the component approach (in Appendix A), can be expressed in terms of a single N = 4 (super) OPE. The positive integer n can be n = 1, 2, 3, or 4 for the 16 currents of the large N = 4 linear superconformal algebra because the highest spin among them is given by 2 and the highest singular term is the fourth-order pole in the OPE.. In this way, we can check that all the component results in Appendix A can be rewritten in terms of a single OPE (2.3) in the N = 4 superspace. Appendix B is given by the expression G1(z) (−i) Q4(w), which contains the first-order pole (−i )

We used the fact that there are no singular terms in the OPE of

Explicit relations between the higher spin currents in different bases

Ansatz from the 136 OPEs in the component approach

Jacobi identities

Determination of the structure constant

Conclusions and outlook