Abstract
Sp(2M) invariant field equations in the space ℳ M with symmetric matrix coordinates are classified. Analogous results are obtained for Minkowski-like subspaces of ℳ M which include usual 4d Minkowski space as a particular case. The constructed equations are associated with the tensor products of the Fock (singleton) representation of Sp(2M) of any rank r. The infinite set of higher-spin conserved currents multilinear in rank-one fields in ℳ M is found. The associated conserved charges are supported by $$ \mathrm{r}M-\frac{\mathrm{r}\left(\mathrm{r}-1\right)}{2} $$ -dimensional differential forms in ℳ M , that are closed by virtue of the rank-2r field equations. The cohomology groups H p (σ − r ) with all p and r, which determine the form of appropriate gauge fields and their field equations, are found both for ℳ M and for its Minkowski-like subspace.
Highlights
Sp(2M ) invariant field equations in the space MM with symmetric matrix coordinates are classified
Analogous results are obtained for Minkowski-like subspaces of MM which include usual 4d Minkowski space as a particular case
All rank-two dynamical fields and field equations were found and it was shown that dynamical equations for most of the rank-two fields have the form of conservation conditions for conserved currents found in [6], which give rise to the full set of bilinear conserved charges in the rank-one theory
Summary
Belong to the space of antisymmetric tensor products of Y [1, 1] To show that they are described by almost symmetric diagrams we observe that anticommutativity of the differentials implies that the projection of ξABξCD to the window diagram is zero ξABξCD = 0. From here it follows that the product of n variables ξ antisymmetrized over n indices has the symmetry of the hook Y[n, 1, . Due to antisymmetrization in every column, this implies that any Young diagram associated with a differential form has the nested hook structure, i.e., is almost symmetric
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