Superconformal blocks in diverse dimensions and $BC$ symmetric functions

TL;DR

This paper establishes a deep connection between superconformal blocks across various dimensions and symmetric functions related to the $BC$ root system, revealing new mathematical structures and properties in superconformal field theories.

Contribution

It introduces a novel relation between superblocks and $BC$ symmetric functions, including a supersymmetric uplift of $BC_n$ Jacobi polynomials and new identities linking superconformal blocks to hypergeometric functions.

Findings

Superblocks are eigenfunctions of the super Casimir matching $BC_{m|n}$ CMS Hamiltonian.

The paper introduces a supersymmetric uplift of $BC_n$ Jacobi polynomials with a stability property.

A new Cauchy identity pairs superconformal blocks with Sergeev-Veselov super Jacobi polynomials.

Abstract

We uncover a precise relation between superblocks for correlators of superconformal field theories (SCFTs) in various dimensions and symmetric functions related to the $BC$ root system. The theories we consider are defined by two integers $(m,n)$ together with a parameter $\theta$ and they include correlators of all half-BPS correlators in 4d theories with ${\cal N}=2n$ supersymmetry, 6d theories with $(n,0)$ supersymmetry and 3d theories with ${\cal N}=4n$ supersymmetry, as well as all scalar correlators in any non SUSY theory in any dimension, and conjecturally various 5d, 2d and 1d superconformal theories. The superblocks are eigenfunctions of the super Casimir of the superconformal group whose action we find to be precisely that of the $BC_{m|n}$ Calogero-Moser-Sutherland (CMS) Hamiltonian. When $m=0$ the blocks are polynomials, and we show how these relate to $BC_n$ Jacobi polynomials. However, differently from $BC_n$ Jacobi polynomials, the $m=0$ blocks possess a crucial stability property that has not been emphasised previously in the literature. This property allows for a novel supersymmetric uplift of the $BC_n$ Jacobi polynomials, which in turn yields the $(m,n;\theta)$ superblocks. Superblocks defined in this way are related to Heckman-Opdam hypergeometrics and are non polynomial functions. A fruitful interaction between the mathematics of symmetric functions and SCFT follows, and we give a number of new results on both sides. One such example is a new Cauchy identity which naturally pairs our superconformal blocks with Sergeev-Veselov super Jacobi polynomials and yields the CPW decomposition of any free theory diagram in any dimension.