Universal Spinning Casimir Equations and Their Solutions

TL;DR

This paper develops universal Casimir differential equations for spinning conformal blocks in any dimension, providing new tools and solutions that generalize existing methods and enable applications to defect conformal field theories.

Contribution

It introduces a universal approach to spinning conformal blocks using Harish-Chandra's radial component map and differential shifting operators, applicable across all dimensions.

Findings

Constructed universal Casimir equations for spinning blocks in any dimension.

Developed differential shifting operators to generate solutions from seed functions.

Applied methods to defect CFT, including two-point functions and bulk-bulk-defect three-point functions.

Abstract

Conformal blocks are a central analytic tool for higher dimensional conformal field theory. We employ Harish-Chandra's radial component map to construct universal Casimir differential equations for spinning conformal blocks in any dimension $d$ of Euclidean space. Furthermore, we also build a set of differential ``shifting'' operators that allow to construct solutions of the Casimir equations from certain seeds. In the context of spinning four-point blocks of bulk conformal field theory, our formulas provide an elegant and far reaching generalisation of existing expressions to arbitrary tensor fields and arbitrary dimension $d$. The power of our new universal approach to spinning blocks is further illustrated through applications to defect conformal field theory. In the case of defects of co-dimension $q=2$ we are able to construct conformal blocks for two-point functions of symmetric traceless bulk tensor fields in both the defect and the bulk channel. This opens an interesting avenue for applications to the defect bootstrap. Finally, we also derive the Casimir equations for bulk-bulk-defect three-point functions in the bulk channel.