# Connection on the complex of vector bundles

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?

• As a side question, if some one can suggest another reference that discuss this idea of representation upto homotopy of Lie algebroids, I would be very thankful.. Jan 3, 2022 at 11:50
• A connection on a complex of bundles is a connection on each graded vector bundle. Most likely the differential is a map of connections. Jan 3, 2022 at 18:04
• Can you please see if you can think of some reference for this. @unknownymous Thanks for responding to most of my recent questions :) Jan 4, 2022 at 1:09

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.

• Thanks for the answer... I was reading Gracia-Saz and Mehta's paper.. They use the term super connections... Can you think of some PhD thesis or some video lectures (YouTube) that says some more details about representation upto homotopy.. I am still trying to look for a place which gives the definition of $A$-connection on chain complex... I am sure what you mentioned is the one the authors are mentioning, but, just for the sake of reference can you suggest some place which defines what it is... Jan 5, 2022 at 10:26
• I don't remember if I have seen explicitly the definition of $A$-connection on a chain complex singled out. Usually for A-connections on a complex, people are in the context of representations up to homotopy (ruths), and then simply present the connections on each bundle, the relations they satisfy are included in the conditions for being a ruth. You can find a good reference here (in particular points 1,2,8 of section 2 for what you want in the 2-term case): Frejlich, Pedro, Morita invariance of intrinsic characteristic classes of Lie algebroids link Jan 5, 2022 at 11:44
• Alternatively, the viewpoint usually taken for ruths: an $A$-connection on $E$ is the same as a degree 1 graded derivation $d$ on the $\Omega(A)$-module $\Omega^\bullet(A;E)$. It is flat (i.e. a representation of $A$) if and only if $d^2=0$. This also makes sense for a complex $E^\bullet$, it is just that the grading on $\Omega^\bullet(A;E^\bullet)$ now takes into account the grading on $E^\bullet$. And then $d^2=0$ if it is a ruth. That's the definition of A-superconnection in the paper of Gracia-Saz and Mehta, it is just that they are dealing with a 2-term complex. Jan 5, 2022 at 11:55
• As for PhD thesis with details, there are a few, I'll mention 1 early, 2 recent ones: the first one (groupoids, algebroids) would be the one of Camilo Arias Abad: Representations up to homotopy and cohomology of classifying spaces A recent one for algebroids, higher versions. Looking at it, the definition you wanted is on the bottom of page 17: Theocharis Papantonis, $\mathbb{Z}$-graded supergeometry: Differential graded modules, higher algebroid representations, and linear structures Jan 5, 2022 at 12:27
• A recent one for groupoids: Fernando Studzinski, On the cohomology of representations up to homotopy of Lie groupoids Jan 5, 2022 at 12:42