Covariant Approaches to Superconformal Blocks

TL;DR

This work develops two covariant frameworks for 4d superconformal blocks: a super-Casimir differential equation method and a supershadow projector method implemented in superembedding space. For configurations with two chiral and two anti-chiral operators, it yields compact expressions for 4d blocks and demonstrates their consistency across approaches, with key results such as $G_{ m N}(u,v)=u^{-{ m N}/2}\,g_{ riangle+{ m N}, ll}^{ riangle_{12}= riangle_{34}={ m N}}(u,v)$ in the chiral sector. The supershadow formalism is formulated in a manifestly covariant embedding-space language, enabling dimensionless projectors and tractable integrals that reduce to known bosonic conformal blocks after appropriate Grassmann coordinate simplifications. Collectively, these methods provide practical, covariant tools for the 4d ${ m N}$-extended SCFT bootstrap and lay groundwork for blocks of more general multiplets via embedding-space and shadow techniques.

Abstract

We develop techniques for computing superconformal blocks in 4d superconformal field theories. First we study the super-Casimir differential equation, deriving simple new expressions for superconformal blocks for 4-point functions containing chiral operators in theories with N-extended supersymmetry. We also reproduce these results by extending the "shadow formalism" of Ferrara, Gatto, Grillo, and Parisi to supersymmetric theories, where superconformal blocks can be represented as superspace integrals of three-point functions multiplied by shadow three-point functions.