Extended diffeomorphism algebras and trajectories in jet space

TL;DR

This paper introduces an extended algebra called DRO(N), which combines diffeomorphisms, reparametrizations, and observer symmetries, providing new representations and a geometric understanding of modules related to higher-dimensional algebras.

Contribution

It constructs the DRO(N) algebra with Virasoro-like cocycles, develops its Fock modules, and relates reparametrization gauge fixing to known modules, offering a geometric perspective on toroidal Lie algebras.

Findings

DRO(N) extends diff(N) and diff(1) with Virasoro-like cocycles.

Fock modules are constructed for each p and gl(N) representation.

Reparametrization gauge fixing links DRO(N) modules to known modules.

Abstract

Let the DRO (Diffeomorphism, Reparametrization, Observer) algebra DRO(N) be the extension of $diff(N)\oplus diff(1)$ by its four inequivalent Virasoro-like cocycles. Here $diff(N)$ is the diffeomorphism algebra in $N$-dimensional spacetime and $diff(1)$ describes reparametrizations of trajectories in the space of tensor-valued $p$-jets. DRO(N) has a Fock module for each $p$ and each representation of $gl(N)$. Analogous representations for gauge algebras (higher-dimensional Kac-Moody algebras) are also given. The reparametrization symmetry can be eliminated by a gauge fixing procedure, resulting in previously discovered modules. In this process, two DRO(N) cocycles transmute into anisotropic cocycles for $diff(N)$. Thus the Fock modules of toroidal Lie algebras and their derivation algebras are geometrically explained.