Tensor fields of mixed Young symmetry type and N-complexes
Michel Dubois-Violette, Marc Henneaux
TL;DR
The authors extend differential calculus to irreducible tensor fields with mixed Young symmetry by constructing $N$-complexes $\Omega_N(\mathbb{R}^D)$ with differential $d=(-1)^p {\mathbf Y}_{p+1}\circ \stackrel{(0)}{\nabla}$ satisfying $d^N=0$, thereby generalizing differential forms ($N=2$). They prove a generalized Poincaré lemma: $H^{(N-1)n}_{(k)}(\Omega_N(\mathbb{R}^D))=0$ for $n\ge1$ and identify $H^0_{(k)}(\Omega_N(\mathbb{R}^D))$ as polynomials of degree $<k$, with nontrivial cohomology appearing in other degrees, including infinite-dimensional cases for $m$ not in $(N-1)\mathbb{N}$. The work develops a related multicomplex, dualities, and associative/nonassociative algebraic structures ($\wedge_{(Y)}E$, $\wedge_{[(Y)]}E$, $\wedge_{[N]}E$) to unify the framework and provides non-simplicial examples of $N$-complexes; it also relates to gauge-field theories (e.g., linearized spin-2) and hints at Green’s ansatz connections. Altogether, the paper broadens cohomological tools for tensor fields with mixed symmetries and lays groundwork for further geometric and algebraic generalizations, including extensions to $\bar{\partial}$-type $N$-complexes on complex manifolds.
Abstract
We construct $N$-complexes of non completely antisymmetric irreducible tensor fields on $\mathbb R^D$ which generalize the usual complex $(N=2)$ of differential forms. Although, for $N\geq 3$, the generalized cohomology of these $N$-complexes is non trivial, we prove a generalization of the Poincaré lemma. To that end we use a technique reminiscent of the Green ansatz for parastatistics. Several results which appeared in various contexts are shown to be particular cases of this generalized Poincaré lemma. We furthermore identify the nontrivial part of the generalized cohomology. Many of the results presented here were announced in [10].