Three‐Point Functions of Higher‐Spin Supercurrents in 4D N=1${{\cal N}}=1$ Superconformal Field Theory

Abstract

We develop a general formalism to study the three‐point correlation functions of conserved higher‐spin supercurrent multiplets Jαfalse(rfalse)α̇false(rfalse)$J_{\alpha (r) \dot{\alpha }(r)}$ in 4D scriptN=1${\cal N}=1$ superconformal theory. All the constraints imposed by scriptN=1${\cal N}=1$ superconformal symmetry on the three‐point function false⟨Jα(r1)trueα̇(r1)Jβ(r2)trueβ̇(r2)Jγ(r3)trueγ̇(r3)false⟩$\langle J_{\alpha (r_1) \dot{\alpha }(r_1)} J_{\beta (r_2) \dot{\beta }(r_2) }J_{\gamma (r_3) \dot{\gamma }(r_3)}\rangle$ are systematically derived for arbitrary r1,r2,r3$r_1, r_2, r_3$, thus reducing the problem mostly to computational and combinatorial. As an illustrative example, we explicitly work out the allowed tensor structures contained in false⟨Jα(r)trueα̇(r)Jβtrueβ̇Jγtrueγ̇false⟩$\langle J_{\alpha (r) \dot{\alpha }(r)} J_{\beta \dot{\beta } } J_{\gamma \dot{\gamma }}\rangle$, where Jαα̇$J_{\alpha \dot{\alpha }}$ is the supercurrent. We find that this three‐point function depends on two independent tensor structures, though the precise form of the correlator depends on whether r is even or odd. The case r=1$r=1$ reproduces the three‐point function of the ordinary supercurrent derived by Osborn. Additionally, we present the most general structure of mixed correlators of the form false⟨LLJα(r)trueα̇(r)false⟩$\langle L L J_{\alpha (r) \dot{\alpha }(r)}\rangle$ and false⟨Jα(r1)trueα̇(r1)Jβ(r2)trueβ̇(r2)Lfalse⟩$\langle J_{\alpha (r_1) \dot{\alpha }(r_1)} J_{\beta (r_2) \dot{\beta }(r_2)} L \rangle$, where L is the flavour current multiplet.

Three-point functions of conserved currents in 3D CFT: General formalism for arbitrary spins

“…An open question is whether it is possible to find generating functions for arbitrary superspins that encapsulate the results in this paper, similar to the ones found in non-supersymmetric theories [16,17]. It would also be interesting to apply our methods to superconformal theories in higher dimensions (see [44,45,49] for recent progress) and to N -extended superconformal theories. Correlation functions of higher-spin currents in conformal theories with extended supersymmetry have practically not been studied, however recent progress has been reported in [47].…”

“…An important difference compared to the N = 1 case is that conserved currents can carry indices of the R-symmetry group. Concerning the study of three-point functions in four dimensions, in [49] a method was introduced to study three-point functions of conserved supercurrents J α(r) α(r) for arbitrary superspins in 4D N = 1 superconformal field theories. Explicit solutions were constructed for three-point functions involving higher-spin supercurrents and flavour current multiplets.…”

“…Explicit solutions were constructed for three-point functions involving higher-spin supercurrents and flavour current multiplets. The method of [49] was an extension of the one used in [45] where the classification problem was solved for generic three-point functions of conserved fermionic currents S α(k) of arbitrary rank in 4D N = 1 SCFT. We believe that the formalism developed in the present paper will generalise directly to 4D N = 1 theories and will allow us to extend the results of [49].…”

“…The method of [49] was an extension of the one used in [45] where the classification problem was solved for generic three-point functions of conserved fermionic currents S α(k) of arbitrary rank in 4D N = 1 SCFT. We believe that the formalism developed in the present paper will generalise directly to 4D N = 1 theories and will allow us to extend the results of [49]. We leave these considerations for a future study.…”

“…A natural follow-up question arises: in conformal field theories, what are the implications of supersymmetry on the general structure of three-point correlation functions? The study of correlation functions in superconformal theories has been carried out in diverse dimensions using the group-theoretic approach developed in the following publications [33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48][49]. It has been shown that superconformal symmetry imposes additional restrictions on the three-point functions of conserved currents compared to non-supersymmetric theories.…”