Brane mechanics and gapped Lie n-algebroids

TL;DR

This paper explores the connection between higher-dimensional Hamiltonian mechanics, gauge theories, and advanced geometric structures called gapped almost Lie n-algebroids, revealing new insights into topological n-branes and their geometric underpinnings.

Contribution

It introduces the concept of gapped almost Lie n-algebroids and relates gauge system interactions to geometric tensors in these structures, advancing the understanding of higher-dimensional topological models.

Findings

Identification of covariant derivatives with geometric tensors

Relation of gauge interactions to polycurvature tensors

Extension of Lie n-algebroid structures to gapped cases

Abstract

We draw a parallel between the BV/BRST formalism for higher-dimensional ($\ge 2$) Hamiltonian mechanics and higher notions of torsion and basic curvature tensors for generalized connections in specific Lie $n$-algebroids based on homotopy Poisson structures. The gauge systems we consider include Poisson sigma models in any dimension and ``generalised R-flux'' deformations thereof, such as models with an $(n+2)$-form-twisted R-Poisson target space. Their BV/BRST action includes interaction terms among the fields, ghosts and antifields whose coefficients acquire a geometric meaning by considering twisted Koszul multibrackets that endow the target space with a structure that we call a gapped almost Lie $n$-algebroid. Studying covariant derivatives along $n$-forms, we define suitable polytorsion and basic polycurvature tensors and identify them with the interaction coefficients in the gauge theory, thus relating models for topological $n$-branes to differential geometry on Lie $n$-algebroids.