Towards general super Casimir equations for $4D$ ${\mathcal N}=1$ SCFTs

TL;DR

This paper develops super Casimir equations for 4D ${ m extbf{N}=1}$ superconformal field theories, enabling the calculation of superconformal blocks by extending the Casimir approach to supersymmetric contexts.

Contribution

It introduces the super Casimir equations for 4D ${ m extbf{N}=1}$ SCFTs and constructs superconformal blocks as sums of conformal blocks, including cases with shortening conditions.

Findings

Derived super Casimir equations for specific four-point functions.

Expressed superconformal blocks as sums of conformal blocks.

Included analysis of operators with shortening conditions.

Abstract

Applying the Casimir operator to four-point functions in CFTs allows us to find the conformal blocks for any external operators. In this work, we initiate the program to find the superconformal blocks, using the super Casimir operator, for $4D$ ${\mathcal N}=1$ SCFTs. We begin by finding the most general four-point function with zero $U(1)_R$-charge, including all the possible nilpotent structures allowed by the superconformal algebra. We then study particular cases where some of the operators satisfy shortening conditions. Finally, we obtain the super Casimir equations for four point-functions which contain a chiral and an anti-chiral field. We solve the super Casimir equations by writing the superconformal blocks as a sum of several conformal blocks.