Metric projective geometry, BGG detour complexes and partially massless gauge theories

TL;DR

This paper develops a geometric framework combining projective and metric structures to generate invariant gauge theories, providing new insights into higher spin propagation and partially massless models in Einstein spaces.

Contribution

It introduces a novel use of BGG detour complexes with Yang-Mills structures to produce invariant gauge theories and characterize higher spin obstructions in Einstein geometries.

Findings

Curved BGG detour complexes characterize higher spin propagation obstructions.

The framework generates gauge invariances and constraints for partially massless models.

Extension of log-radial reduction method to Einstein backgrounds.

Abstract

A projective geometry is an equivalence class of torsion free connections sharing the same unparametrised geodesics; this is a basic structure for understanding physical systems. Metric projective geometry is concerned with the interaction of projective and pseudo-Riemannian geometry. We show that the BGG machinery of projective geometry combines with structures known as Yang-Mills detour complexes to produce a general tool for generating invariant pseudo-Riemannian gauge theories. This produces (detour) complexes of differential operators corresponding to gauge invariances and dynamics. We show, as an application, that curved versions of these sequences give geometric characterizations of the obstructions to propagation of higher spins in Einstein spaces. Further, we show that projective BGG detour complexes generate both gauge invariances and gauge invariant constraint systems for partially massless models: the input for this machinery is a projectively invariant gauge operator corresponding to the first operator of a certain BGG sequence. We also connect this technology to the log-radial reduction method and extend the latter to Einstein backgrounds.