Higher Affine Connections

Abstract

For a smooth manifold $M$, it was shown in \cite{BPH} that every affine connection on the tangent bundle $TM$ naturally gives rise to covariant differentiation of multivector fields (MVFs) and differential forms along MVFs. In this paper, we generalize the covariant derivative of \cite{BPH} and construct covariant derivatives along MVFs which are not induced by affine connections on $TM$. We call this more general class of covariant derivatives \textit{higher affine connections}. In addition, we also propose a framework which gives rise to non-induced higher connections; this framework is obtained by equipping the full exterior bundle $\wedge^\bullet TM$ with an associative bilinear form $\eta$. Since the latter can be shown to be equivalent to a set of differential forms of various degrees, this framework also provides a link between higher connections and multisymplectic geometry.Comment: 37 pages; main definition and results generalized; substantial changes after section

Twisted R-Poisson Sigma Models

“…. 𝐶 ∞ (𝑀) → 𝐶 ∞ (𝑀) which is totally antisymmetric and a 𝐶 ∞ -derivation in each of the arguments [24]. In addition, the space of multiderivations and the space of multivector fields of the same order are in one-to-one correspondence.…”

The BV action of 3D twisted R-Poisson sigma models

“…. C ∞ (M ) → C ∞ (M )which is totally antisymmetric and a C ∞ -derivation in each of the arguments[41]. In addition, the space of multiderivations and the space of multivector fields of the same order are in one-to-one correspondence.…”