Superconformal partial waves in Grassmannian field theories

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.

Loop corrections for Kaluza-Klein AdS amplitudes

“…We follow the notation introduced in [24], where the superblocks are specified by three integers α, β, γ…”

“…We briefly review the basics of a superblock expansion in N = 4 following [24]. The OPE implies the following decomposition of a four point function,…”

Bootstrap equations for N $$ \mathcal{N} $$ = 4 SYM with defects

“…A unitary highest weight representation χ blk of psu(2, 2|4) is usually identified using the Dynkin labels {∆, ( ,¯ ), [q, p, r]}, where ,¯ and [q, p, r] are spin and su(4) labels respectively. For the discussion of correlation functions it is convenient to reorganize these labels into a representation R blk of ps(u(2|2)⊕u(2|2)), see [36]. Similarly, a unitary highest weight representation χ bdy of osp(4|4) can be identified using the Dynkin labels {δ, s, (k + , k − )}.…”

“…We give the complete list of representations however, only a few of them will appear in the boundary OPE of a 1 2 -BPS bulk operator. We will therefore simplify the notation a bit following [47]: , Q}, where Q = ∆ − −¯ is the conformal twist, see [36] and references therein for more details.…”

Bootstrapping N = 3 $$ \mathcal{N}=3 $$ superconformal theories

“…A basic requirement in any superconformal bootstrap analysis is the computation of the superconformal blocks relevant for the theory in question, although correlation functions of half-BPS operators in various dimensions have been studied [31][32][33], the case of N = 3 has not yet been considered, and calculating the necessary blocks will be one of the goals of this paper. For literature on superconformal blocks see [31][32][33][34][35][36][37][38][39][40].…”