Harmonic Analysis in d-dimensional Superconformal Field Theory

TL;DR

This paper reviews harmonic analysis methods on the superconformal group to derive superconformal blocks and crossing equations, enabling finite sums of spinning bosonic blocks in superconformal field theories.

Contribution

It introduces a harmonic analysis approach to superconformal blocks, providing compact expressions and finite sums of bosonic blocks, advancing the computational framework in superconformal theories.

Findings

Derived compact crossing equations using harmonic analysis.

Expressed superconformal blocks as finite sums of bosonic blocks.

Provided a systematic method for analyzing superconformal four-point functions.

Abstract

Superconformal blocks and crossing symmetry equations are among central ingredients in any superconformal field theory. We review the approach to these objects rooted in harmonic analysis on the superconformal group that was put forward in [J. High Energy Phys. 2020 (2020), no. 1, 159, 40 pages, arXiv:1904.04852] and [J. High Energy Phys. 2020 (2020), no. 10, 147, 44 pages, arXiv:2005.13547]. After lifting conformal four-point functions to functions on the superconformal group, we explain how to obtain compact expressions for crossing constraints and Casimir equations. The later allow to write superconformal blocks as finite sums of spinning bosonic blocks.