Yes-go cross-couplings in collections of tensor fields with mixed symmetries of the type (3,1) and (2,2)

TL;DR

This paper investigates the possible consistent interactions between tensor fields with mixed symmetries (3,1) and (2,2) using a cohomological deformation approach within the antifield-BRST formalism, revealing conditions for their cross-couplings.

Contribution

It provides a systematic analysis of cross-couplings between mixed symmetry tensor fields, identifying when such interactions are consistent and how they deform gauge structures.

Findings

Cross-couplings exist at first and second order in the coupling constant.

The existence of cross-couplings depends on the definiteness of the kinetic quadratic form.

Some gauge generators and reducibility functions are deformed by the interactions.

Abstract

Under the hypotheses of analyticity, locality, Lorentz covariance, and Poincare invariance of the deformations, combined with the requirement that the interaction vertices contain at most two space-time derivatives of the fields, we investigate the consistent cross-couplings between two collections of tensor fields with the mixed symmetries of the type (3,1) and (2,2). The computations are done with the help of the deformation theory based on a cohomological approach in the context of the antifield-BRST formalism. Our results can be synthesized in: 1. there appear consistent cross-couplings between the two types of field collections at order one and two in the coupling constant such that some of the gauge generators and of the reducibility functions are deformed, and 2. the existence or not of cross-couplings among different fields with the mixed symmetry of the Riemann tensor depends on the indefinite or respectively positive-definite behaviour of the quadratic form defined by the kinetic terms from the free Lagrangian.