### L-infinity Associated Bundles, Sections and Covariant Derivatives

#### Posted by Urs Schreiber

Here is the alpha -version of a plugin for the article
*$L_\infty$-connections* (pdf, blog,
arXiv) which extends the functionality of the latter from principal $L_\infty$-connections to associated $L_\infty$-connections:

Sections and covariant derivatives of $L_\infty$-algebra connections(pdf, 8 pages)

Abstract.For every $L_\infty$-algebra $g$ there is a notion of $g$-bundles with connection, according to [SSS]. Here I discuss how to describe

$\;\;$ - associated $g$-bundles;

$\;\;$ - their spaces of sections;

$\;\;$ - and the corresponding covariant derivatives

in this context.

**Introduction**. Representations of $n$-groups are usually thought of as $n$-functors from the $n$-group into the
$n$-category of representing objects. In the program [BaezDolanTrimble]
one sees that possibly a more fundamental perspective on representations is
in terms of the corresponding action groupoids sitting *over* the given group.

This is the perspective I will adopt here and find to be fruitful.

The definition of $L_\infty$-modules which I proposed in *$L_\infty$-modules and the BV-complex* (pdf, blog)
can be seen to actually comply with this perspective. Here I further develop this
by showing that this perspective also helps to understand associated
$L_\infty$-connections, their sections and covariant derivatives.

As you may have noticed, many of the concepts I used to discuss here at the $n$-Café do appear in our article in their Lie $\infty$-algebraic incarnation: $n$-transport, its $n$-curvature with values in the tangent category $INN(G)$, the charged $n$-particle, transgression, etc.

One concept which I liked to discuss a lot, however, does not appear at the moment: sections of $n$-bundles and their covariant derivatives.

One reason is that it turned out to be not quite straightforward to move the definition of that which I am so fond of ($n$-sections and their covariant derivative as morphisms *into* the $n$-curvature, as described here) to the Lie $\infty$-algebraic world.

There is a good reason for why that’s non-straightforward: this description of sections makes crucial use of non-invertible morphisms in the $n$-category of $n$-vector spaces. This means it falls out of the realm of $\infty$-groupoids. So our map from Lie $\infty$-groupoids to Lie $\infty$-algebroids fails and hence this concept does not internalize properly in the differential realm.

I was pretty upset about that. quantization of the $n$-particle is supposed to be all about taking $n$-spaces of sections of the background field $n$-bundle. And the Lie $\infty$-algebraic formulation is supposed to be the powerful tool to handle this $n$-bundle. So it’s too bad that this tool doesn’t admit taking sections.

I thought for a while that it just means that before taking sections I simply need to send everything Lie $\infty$-algebraic back to the integral world by hitting everything in sight with $\Pi_\infty(Hom(--,\Omega^\bullet(--)))$ and then proceed there.

While that might be quite an interesting thing to do, it seems comparatively cumbersome for just taking $n$-sections, compared to how nicely everything else goes throu on the Lie $\infty$-algebraic level.

But now I think I understood how to get the best of both worlds.

There is a reformulation of the concept of a morphism *into* the $n$-curvature of a $G_{(n)}$-bundle with connection in terms of a $V//G_{(n)}$-$n$-groupoid bundle, where $V$ is an $n$-representation and $V//G_{(n)}$ the corresponding action $n$-groupoid. And that reformulation does fit nicely into the Lie $\infty$-algebraic world.

There it looks like this:

as we describe in the article, for $g$ an $L_\infty$-algebra a corresponding bundle with connection can be represented by a diagram which involves, among other things, a morphism of the kind

$\Omega^\bullet(Y) \stackrel{(A,F_A)}{\leftarrow} \mathrm{W}(g) \,,$

where $Y \to X$ is some surjective submersion over base space $X$ and $\mathrm{W}(g)$ is the Weil algebra of the $L_\infty$-algebra $g$.

Now, pick a representation $V$ of $g$ and form the corresponding action Lie $\infty$-algebroid which comes with its Chevalley-Eilenberg algebra $CE(g,V)$ and Weil algebra $\mathrm{W}(g,V)$.

We have a canonical injection $\array{ \mathrm{W}(g,V) \\ \uparrow \\ \mathrm{W}(g) } \,.$

A *section* $\sigma$ of the given $g$-bundle is then a completion of
$\array{
&&\mathrm{W}(g,V)
\\
&&\uparrow
\\
\Omega^\bullet(Y)&\stackrel{(A,F_A)}{\leftarrow}&\mathrm{W}(g)
}$

to

$\array{ &&&\mathrm{W}(g,V) \\ &\multiscripts{^{(\sigma,\nabla_A \sigma,A,F_A)}}{\swarrow}{}&&\uparrow \\ &\Omega^\bullet(Y)&\stackrel{(A,F_A)}{\leftarrow}&\mathrm{W}(g) } \,.$

The “curvature” part of that is, automatically, $\nabla_A \sigma$, the covariant derivative of the section $\sigma$.

## an alpha-version idea to be patched by the next service pack

Andrew Stacey rightly writes in to point out that I have stolen the idea of an

alpha-releaseof mathematical thought from the first line of the abstract of his notes on Comparative Smootheology.