The Coset Spin-4 Casimir Operator and Its Three-Point Functions with Scalars

TL;DR

This paper constructs the GKO coset realization of a dimension-$4$ Casimir built from quartic SU($N$) WZW currents, using completely symmetric $d$-tensors to form spin-$4$ operators. By enforcing regularity of the diagonal current OPE and primarity under the coset Virasoro generator, the spin-$4$ current coefficients are fixed up to two constants, which are then determined by the self-OPE of the spin-$3$ current; the resulting coset spin-$4$ current is analyzed in the large-$N$ 't Hooft limit, where explicit forms simplify and eigenvalues on primary states are obtained. The work then constructs a generalized coset spin-$4$ current by including an extra term $J^a J^b K^a K^b$, derives the constraints that fix the remaining constants, and provides their large-$N$ scalings, showing that the zero-mode eigenvalues reproduce the expected $(1oldsymbol{\pmoldsymbol{\lambda}})(2oldsymbol{\pmoldsymbol{\lambda}})(3oldsymbol{\pmoldsymbol{\lambda}})$ structure. The computed three-point functions with scalars at all values of the 't Hooft coupling $oldsymbol{\lambda}=N/(N+k)$ agree, in the large-$N$ limit, with the results of Chang and Yin for the undeformed AdS$_3$ bulk theory and extend to finite $oldsymbol{\lambda}$. These results reinforce the higher-spin AdS$_3$/CFT$_2$ correspondence in the spin-$4$ sector and motivate further exploration of spin-$>4$ coset generators and their bulk duals.

Abstract

We find the GKO coset construction of the dimension 4 Casimir operator that contains the quartic WZW currents contracted with completely symmetric SU(N) invariant tensors of ranks 4, 3, and 2. The requirements, that the operator product expansion with the diagonal current is regular and it should be primary under the coset Virasoro generator of dimension 2, fix all the coefficients in spin-4 current, up to two unknown coefficients. The operator product expansion of coset primary spin-3 field with itself fixes them completely. We compute the three-point functions with scalars for all values of the 't Hooft coupling in the large N limit. At fixed 't Hooft coupling, these three-point functions are dual to that found by Chang and Yin recently in the undeformed AdS_3 bulk theory (higher spin gravity with matter).