  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.
