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

TL;DR

This paper demonstrates that extremal correlators in all Lagrangian ${ m N}=2$ superconformal field theories with simple gauge groups are universally governed by Toda equations, revealing a deep mathematical structure and non-renormalization properties.

Contribution

It establishes a universal Toda system governing extremal correlators in ${ m N}=2$ SCFTs and constructs an orthogonal basis for the chiral ring to facilitate this.

Findings

Extremal correlators follow Toda equations across all studied ${ m N}=2$ SCFTs.

Constructed an orthogonal basis for the chiral ring with decoupled Toda chain equations.

Discovered a non-renormalization property in specific ${ m N}=2$ SCFTs with certain hypermultiplet representations.

Abstract

We show that extremal correlators in all Lagrangian ${\cal N}=2$ superconformal field theories with a simple gauge group 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 $\mathrm{SU}(N)$ 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 ${\cal N}=2$ $\mathrm{SU}(N)$ 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.