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

• Published in: Fortschritte der Physik 2022-12, Vol.70 (12), p.n/a
• Main Authors: Buchbinder, Evgeny I., Hutomo, Jessica, Tartaglino‐Mazzucchelli, Gabriele
• Format: Article
• Language: English
• Subjects: Combinatorial analysis; Correlators; Field theory; Flavors; higher spin supercurrents; Mathematical analysis; superconformal field theory; Tensors
• ISSN: 0015-8208; 1521-3978
• DOI: 10.1002/prop.202200133

We develop a general formalism to study the three‐point correlation functions of conserved higher‐spin supercurrent multiplets Jα(r)α̇(r)$J_{\alpha (r) \dot{\alpha }(r)}$ in 4D N=1${\cal N}=1$ superconformal theory. All the constraints imposed by N=1${\cal N}=1$ superconformal symmetry on the three‐point function ⟨Jα(r1)α̇(r1)Jβ(r2)β̇(r2)Jγ(r3)γ̇(r3)⟩$\langle J_{\alpha (r_1) \dot{\alpha }(r_1)} J_{\beta (r_2) \dot{\beta }(r_2) }J_{\gamma (r_3) \dot{\gamma }(r_3)}\rangle$ 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 ⟨Jα(r)α̇(r)Jββ̇Jγγ̇⟩$\langle J_{\alpha (r) \dot{\alpha }(r)} J_{\beta \dot{\beta } } J_{\gamma \dot{\gamma }}\rangle$, 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 ⟨LLJα(r)α̇(r)⟩$\langle L L J_{\alpha (r) \dot{\alpha }(r)}\rangle$ and ⟨Jα(r1)α̇(r1)Jβ(r2)β̇(r2)L⟩$\langle J_{\alpha (r_1) \dot{\alpha }(r_1)} J_{\beta (r_2) \dot{\beta }(r_2)} L \rangle$, where L is the flavour current multiplet. The authors develop a general formalism to study the three‐point correlation functions of conserved higher‐spin supercurrent multiplets in 4D N=1${\cal N}=1$ superconformal theory. All the constraints imposed by N=1${\cal N}=1$ superconformal symmetry on the three‐point function are systematically derived, thus reducing the problem mostly to a computational and combinatorial one. As an illustrative example, the authors explicitly work out the allowed tensor structures. They find that the three‐point function considered depends on two independent tensor structures, though the precise form of the correlator depends on whether the argument r is even or odd. As special case (r = 1) one reproduces the three‐point function of the ordinary supercurrent derived by Osborn.