Higher-Rank Fields and Currents

TL;DR

This paper classifies higher-rank invariant field equations in matrix coordinate spaces, constructs associated conserved currents, and analyzes their cohomology, extending understanding of higher-spin theories in generalized spacetime frameworks.

Contribution

It provides a classification of $Sp(2M)$ invariant field equations, constructs higher-spin conserved currents, and analyzes cohomology groups in matrix coordinate spaces.

Findings

Classified $Sp(2M)$ invariant field equations in ${ m{f M}}_M$ and Minkowski-like subspaces.

Constructed an infinite set of higher-spin conserved currents multilinear in rank-one fields.

Determined cohomology groups $H^p(\sigma^{f r}_-)$ for gauge fields and equations in these spaces.

Abstract

$Sp(2M)$ invariant field equations in the space ${\cal M}_M$ with symmetric matrix coordinates are classified. Analogous results are obtained for Minkowski-like subspaces of ${\cal M}_M$ 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 ${\mathbf{r }}$. The infinite set of higher-spin conserved currents multilinear in rank-one fields in ${\cal M}_M$ is found. The associated conserved charges are supported by $({\mathbf{r }} M-\frac{{\mathbf{r }} ({\mathbf{r }} -1)}{2})-$dimensional differential forms in ${\cal M}_M$, that are closed by virtue of the rank-$2{\mathbf{r }}$ field equations. The cohomology groups $H^p(\sigma^{\mathbf{r }}_-)$ with all $p$ and ${\mathbf{r }}$, which determine the form of appropriate gauge fields and their field equations, are found both for ${\cal M}_M$ and for its Minkowski-like subspace.