Abstract
Correlation functions of the higher-spin current operators in large N Chern-Simons theories are important to understand approximate higher-spin symmetries in these theories. Moreover, they also provide stronger checks for conjectured dualities in these theories. In this paper, we compute the two, three and four-point functions of the operators in the spin zero multiplet of mathcal{N} = 2 Supersymmetric vector matter Chern-Simons theory at large N to all orders of ’t Hooft coupling. While the two- and three-point functions are computed by solving the Schwinger-Dyson equation, this method becomes intractable for the computation of the four-point functions. Thereby, we use bootstrap method to evaluate four-point function of scalar operator {J}_0^f=overline{psi}psi and {J}_0^b=overline{phi}phi . Interestingly, because leftlangle {J}_0^f{J}_0^f{J}_0^brightrangle is a contact term, the four point function of {J}_0^f operator resembles the corresponding correlation function in the free theory, up to overall coupling constant dependent factors and up to some ‘bulk AdS’ contact terms. On the other hand the {J}_0^b four-point function receives an additional contribution compared to the free theory expression due to the {J}_0^f exchange. We find that the double discontinuity of this single trace operator {J}_0^f vanishes and hence it only contributes to AdS-contact term.
Highlights
Correlation functions of the higher-spin current operators in large N ChernSimons theories are important to understand approximate higher-spin symmetries in these theories
The self duality of this supersymmetric theory serves as a parent duality for the non-supersymmetric bosonization dualities mentioned above since they can obtained from the supersymmetric theory via RG flows seeded by mass deformations [41, 42]
Many interesting non supersymmetric physical observables, as mentioned above, are amenable to direct exact computations by solving corresponding DysonSchwinger equations, the computation of 4-point correlation function of even the simplest of single trace operators, namely the scalar operators φφ and ψψ, appears prohibitively difficult2 to compute via this direct approach
Summary
We compute the two and three point correlation function of the J0(θ, p) operator in momemtum space. Two of the main ingredients for these computations are the exact propagator (3.2) and the renormalized four point vertex for the fundamental superfield Φ(θ, p) (ν4 in (3.3)). These were computed in [27] for a more general class of theories with N = 1 supersymmetry which can be thought of as one parameter deformation of the N = 2 theory of interest in this paper. While the momenta p and k are arbitrary.4 For this reason, our computation of correlation functions will be restricted configuration in which the momentum of J0 operators are restricted to lie only in the 3-direction. The exact four point vertex will be represented as in figure 1
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