The Primary Spin-4 Casimir Operators in the Holographic SO(N) Coset Minimal Models

TL;DR

This paper constructs primary higher spin-4 Casimir operators in SO(N) minimal models, relating them to known minimal models and exploring their three-point functions in the large N limit, relevant for higher spin AdS_3 gravity duals.

Contribution

It introduces explicit constructions of higher spin-4 Casimir operators in SO(N) minimal models and connects them to three-point functions in the holographic duality framework.

Findings

Constructed two lowest primary higher spin-4 Casimir operators.

Linked these operators to currents in specific minimal models depending on N.

Derived three-point functions in the large N limit for holographic correspondence.

Abstract

Starting from SO(N) current algebra, we construct two lowest primary higher spin-4 Casimir operators which are quartic in spin-1 fields. For N is odd, one of them corresponds to the current in the WB_{\frac{N-1}{2}} minimal model. For N is even, the other corresponds to the current in the WD_{\frac{N}{2}} minimal model. These primary higher spin currents, the generators of wedge subalgebra, are obtained from the operator product expansion of fermionic (or bosonic) primary spin-N/2 field with itself in each minimal model respectively. We obtain, indirectly, the three-point functions with two real scalars, in the large N 't Hooft limit, for all values of the 't Hooft coupling which should be dual to the three-point functions in the higher spin AdS_3 gravity with matter.