Generalized cohomology for irreducible tensor fields of mixed Young symmetry type
Michel Dubois-Violette, Marc Henneaux
TL;DR
The paper extends the exterior calculus from differential forms to irreducible tensors with mixed Young symmetry by constructing $N$-complexes $(\Omega_N(\mathbb{R}^D),d)$ with $d^N=0$ and analyzing the generalized cohomologies $H_{(k)}$. A generalized Poincaré lemma is established, showing vanishing of $H_{(N-1)n}^{(k)}$ for $n\ge1$ and identifying $H^{0}_{(k)}$ with polynomials of degree $\le k-1$, while cohomology at other degrees is generically nontrivial. The formalism naturally describes higher-spin gauge fields, with curvatures $R=d^S h$ and gauge invariance encoded by $d^{S+1}=0$, and yields a generalized duality (Hodge-like) linking divergenceless tensors to higher-rank potentials. The authors also discuss a manifold generalization via a connection $\nabla$, where $d_\nabla=\mathbf{Y}\circ\nabla$ provides a curved-space extension with curvature-torsion obstructions to nilpotency. Overall, this framework unifies cohomological methods, higher-spin dynamics, and dualities beyond the classical $N=2$ setting.
Abstract
We construct N-complexes of non completely antisymmetric irreducible tensor fields on $\mathbb R^D$ generalizing thereby the usual complex (N=2) of differential forms. These complexes arise naturally in the description of higher spin gauge fields. Although, for $N\geq 3$, the generalized cohomology of these N-complexes is non trivial, we prove a generalization of the Poincaré lemma. Several results which appeared in various contexts are shown to be particular cases of this generalized Poincaré lemma.