Koszul-Tate Cohomology as Lowest-Energy Modules of Non-Centrally Extended Diffeomorphism Algebras

TL;DR

This paper constructs lowest-energy modules for multi-dimensional Virasoro algebras using antifield formalism, revealing finiteness constraints that restrict spacetime dimensions to four.

Contribution

It introduces a novel cohomological construction of modules for extended diffeomorphism algebras, linking finiteness to spacetime dimension constraints.

Findings

Fock modules are constructed via p-jet formalism.

Divergences in abelian charges restrict dimensions to N=4.

Finiteness requires specific order of EL equations for fermions and bosons.

Abstract

Fock modules for multi-dimensional Virasoro algebras (non-central extensions of the diffeomorphism algebra vect(N)) have recently been reported. Using ideas from the antifield formalism, I construct new classes of lowest-energy modules, as cohomology groups of a certain Fock complex. The Fock construction involves a passage to p-jets prior to normal ordering, but the abelian charges usually diverge in the limit p --> oo. The requirement of a finite limit imposes severe restrictions on the number of spacetime dimensions and on the order of the Euler-Lagrange (EL) equations. Under some natural assuptions (the EL equations are first order for fermions and second order for bosons, and no reducible gauge symmetries appear), finiteness is only possible when the number of spacetime dimensions N = 4.