Superconformal partial waves in Grassmannian field theories

TL;DR

<p>We unify conformal and superconformal partial waves for scalar four-point functions across a broad class of theories by formulating them on the Grassmannian $Gr(m|n,2m|2n)$ and expressing the results in terms of group-independent Schur-polynomial data. The approach lifts known ${ m GL}(4)$ partial waves to ${ m GL}(2m)$ and then to ${ m GL}(2m|2n)$, delivering a universal set of OPE coefficients $A^{p_1p_2p_3p_4}_{oldsymbol{ u}}$ through a Schur-polynomial expansion with coefficients $R^{oldsymbol{eta}oldsymbol{eta}oldsymbol{ u}}_{oldsymbol{\mu}}$. Specialising to ${ m N}=4$ SYM ($m=n=2$) and analyzing mixed-charge four-point functions, we perform a detailed free-theory SCPW analysis, derive explicit free OPE coefficients, and implement multiplet recombination to separate protected and unprotected sectors, revealing a nontrivial protected twist-four sector in $raket{3333}$ determined via $raket{2233}$. Our results provide robust, group-independent input for bootstrap and integrability tests and yield compact determinant forms useful for analytic and numeric applications across several superconformal regimes.</p>

Abstract

We derive superconformal partial waves for all scalar four-point functions on a super Grassmannian space Gr(m|n,2m|2n) for all m,n. This family of four-point functions includes those of all (arbitrary weight) half BPS operators in both N=4 SYM (m=n=2) and in N=2 superconformal field theories in four dimensions (m=2,n=1) on analytic superspace. It also includes four-point functions of all (arbitrary dimension) scalar fields in non-supersymmetric conformal field theories (m=2,n=0) on Minkowski space, as well as those of a certain class of representations of the compact SU(2n) coset spaces. As an application we then specialise to N=4 SYM and use these results to perform a detailed superconformal partial wave analysis of the four- point functions of arbitrary weight half BPS operators. We discuss the non-trivial separation of protected and unprotected sectors for the <2222>, <2233> and <3333> cases in an SU(N) gauge theory at finite N. The <2233> correlator predicts a non-trivial protected twist four sector for <3333> which we can completely determine using the knowledge that there is precisely one such protected twist four operator for each spin.