Tensor gauge fields in arbitrary representations of GL(D,R) : duality & Poincare lemma
Xavier Bekaert, Nicolas Boulanger
TL;DR
The work develops a unified framework for free tensor gauge fields in arbitrary $GL(D,\mathbb{R})$ representations by extending the de Rham calculus to $N$-complexes built from Young diagrams and Schur modules. It proves a generalized Poincaré lemma, constructs generalized cohomology $H_{(m)}(d)$, and uses multiforms with multiple Hodge dualities to systematically derive gauge structures, Bianchi identities, and dualities for both symmetric and mixed-symmetry fields. The results yield explicit dualities between linearized Riemann-type curvatures and dual gauge fields, describe reducibilities, and provide field equations for arbitrary Young symmetry types, thereby offering a robust mathematical foundation for linearized gravity and higher-spin gauge theories in any dimension. This framework has potential to streamline the analysis of dualities and gauge structures in string theory and related higher-spin constructs across dimensions, by providing exact cohomological control over gauge redundancies and dual formulations.
Abstract
Using a mathematical framework which provides a generalization of the de Rham complex (well-designed for p-form gauge fields), we study the gauge structure and duality properties of theories for free gauge fields transforming in arbitrary irreducible representations of GL(D,R). We prove a generalization of the Poincare lemma which enables us to solve the above-mentioned problems in a systematic and unified way.