Universality of Toda equation in N = 2 $$ \mathcal{N}=2 $$ superconformal field theories

Abstract

Abstract We show that extremal correlators in all Lagrangian N = 2 $$ \mathcal{N}=2 $$ superconformal field theories with a simple gauge group, when suitably defined the S 4 $$ {\mathbb{S}}^4 $$ , are governed by the same universal Toda system of equations, which dictates the structure of extremal correlators to all orders in the perturbation series. A key point is the construction of a convenient orthogonal basis for the chiral ring, by arranging towers of operators in order of increasing dimension, which has the property that the associated two-point functions satisfy decoupled Toda chain equations. We explicitly verify this in all known SCFTs based on SU(N) gauge groups as well as in superconformal QCD based on orthogonal and symplectic groups. As a by-product, we find a surprising non-renormalization property for the N = 2 $$ \mathcal{N}=2 $$ SU(N) SCFT with one hypermultiplet in the rank-2 symmetric representation and one hypermultiplet in the rank-2 antisymmetric representation, where the two-loop terms of a large class of supersymmetric observables identically vanish.

Keywords

• Conformal Field Theory
• Supersymmetric Gauge Theory