Connection on the complex of vector bundles

Question

This is from the paper Representations up to homotopy of Lie algebroids by Camilo Arias Abad and Marius Crainic

Let $M$ be a smooth manifold.

Let $E\rightarrow M$ be a vector bundle. A connection on the vector bundle $E\rightarrow M$ is a map $$\nabla:\Gamma(M,TM)\times \Gamma(M,E)\rightarrow \Gamma(M,E)$$ satisfying certain conditions.

Let $A\rightarrow M$ be a Lie algebroid on $M$. An $A$-connection on the vector bundle $E\rightarrow M$ is a map $$\nabla:\Gamma(M,A)\times \Gamma(M,E)\rightarrow\Gamma(M,E)$$ satisfying the same conditions as mentioned before. :D

This notion seems to be introduced in the above paper, please correct me if I am wrong.

After Definition $2.9$ in the above paper, the authors mention the notion of an $A$-connection on the (adjoint) complex (of vector bundles). But, the authors do not even declare the meaning of the notion of connection on a chain/cochain complex of vector bundles.

Can someone suggest some reference where I can find a meaning to this notion?

I can make a guess but I am sure the exact notion is more than what I can guess.

Consider the adjoint complex $\rho:A\rightarrow TM$ (which, for me is just a nice morphism of vector bundles).

An $A$-connection on the complex $\rho:A\rightarrow TM$ should be just a pair $(\nabla_A,\nabla_{TM})$ where $\nabla_A$ is an $A$-connection on the vector bundle $A\rightarrow M$, and $\nabla_{TM}$ is an $A$-connection on the vector bundle $TM\rightarrow M$ such that, they are connected with each other with the help of the morphism $\rho:A\rightarrow TM$.

So, what exactly does it mean to refer to a connection on a ($2$-term) complex of vector bundles?

Answer 1

The concept of a linear $A$-connection on a vector bundle predates the paper of Arias Abad and Crainic. It goes back at least to

Sam Evens, Jiang-Hua Lu and Alan Weinstein, Transverse measures, the modular class and a cohomology pairing for Lie algebroids. Q. J. Math., Oxf. II. Ser. 50, No. 200, 417-436 (1999). Arxiv (see Remark 7.2)

Another paper where you can read about $A$-connections (also more general than linear ones) is

Rui Loja Fernandes, Lie algebroids, holonomy and characteristic classes. Adv. Math. 170, No. 1, 119-179 (2002). Arxiv

In pages 6, 7 there is a bit of discussion of the linear case, and pointers to other references. The special case of flat linear A-connections is older, and and those are also known as representations of the Lie algebroid $A$.

An $A$-connection on the vector bundle complex $(E^\bullet,\partial)$ is an $A$-connection on each $E^n$, commuting with $\partial$ (or to be precise, with the map induced by $\partial$ at the level of sections).

So in the case of the adjoint complex, the relevant sentence in the Arias Abad - Crainic paper is right after definition 2.9 where they mention

Note that $\nabla^\text{bas} \circ \rho = \rho \circ \nabla^\text{bas}$ i.e. $\nabla^\text{bas}$ is an $A$-connection on the adjoint complex.

where they are using the notation $\nabla^\text{bas}$ for the $A$-connections on both $A$ and on $TM$.

As for references, there are a lot, on representations up to homotopy both of Lie algebroids and Lie groupoids (and higher versions of those). Sticking to early ones, on algebroids, there is the paper you mention of Arias Abad and Crainic, and I think these two are good complementary sources:

Camilo Arias Abad, Florian Schätz, Deformations of Lie brackets and representations up to homotopy. Indag. Math., New Ser. 22, No. 1-2, 27-54 (2011). Arxiv

Alfonso Gracia-Saz, Rajan Amit Mehta, Lie algebroid structures on double vector bundles and representation theory of Lie algebroids. Adv. Math. 223, No. 4, 1236-1275 (2010). Arxiv (a different point of view, but section 4 relates to the one you are reading)

I think that's a good start (biased by the papers I've first learned from); for others, both early and more recent, you can follow papers that cite/ are cited by those, and find what's closer to your interests.