Abstract
In this work we launch a systematic theory of superconformal blocks for fourpoint functions of arbitrary supermultiplets. Our results apply to a large class of superconformal field theories including 4-dimensional models with any number mathcal{N} of supersymmetries. The central new ingredient is a universal construction of the relevant Casimir differential equations. In order to find these equations, we model superconformal blocks as functions on the supergroup and pick a distinguished set of coordinates. The latter are chosen so that the superconformal Casimir operator can be written as a perturbation of the Casimir operator for spinning bosonic blocks by a fermionic (nilpotent) term. Solu tions to the associated eigenvalue problem can be obtained through a quantum mechanical perturbation theory that truncates at some finite order so that all results are exact. We illustrate the general theory at the example of d = 1 dimensional theories with mathcal{N} = 2 supersymmetry for which we recover known superblocks. The paper concludes with an outlook to 4-dimensional blocks with mathcal{N} = 1 supersymmetry.
Highlights
Osborn characterized the blocks through second order Casimir differential equations and thereby managed to uncover many of their properties, including explicit formulas for scalar four-point blocks in even dimensions
Solutions to the associated eigenvalue problem can be obtained through a quantum mechanical perturbation theory that truncates at some finite order so that all results are exact
The connection uncovered in [17] implies that conformal blocks are certain multivariable hypergeometric functions with properties that follow from integrability of the quantum mechanical system and are very similar to those of ordinary hypergeometrics in one variable
Summary
Following the strategy laid out in [19], our first task is to realize superconformal partial waves as functions on the superconformal group. The observation which allows us to generalize many of the results from [19] to supersymmetric setup is that one can pass from the notion of an induced representations of a Lie group to that of a coinduced representation. Once this is done, adding supersymmetry largely amounts to inserting appropriate minus signs into the formulas. We shall initially consider correlators of generic (long) multiplets and discuss the effect of shortening conditions in the final subsection
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