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January 30, 2008

L-infinity Associated Bundles, Sections and Covariant Derivatives

Posted by Urs Schreiber

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

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

Abstract. For every L L_\infty-algebra gg there is a notion of gg-bundles with connection, according to [SSS]. Here I discuss how to describe
\;\; - associated gg-bundles;
\;\; - their spaces of sections;
\;\; - and the corresponding covariant derivatives
in this context.

Introduction. Representations of nn-groups are usually thought of as nn-functors from the nn-group into the nn-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 L_\infty-modules which I proposed in L 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 L_\infty-connections, their sections and covariant derivatives.

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

One concept which I liked to discuss a lot, however, does not appear at the moment: sections of nn-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 (nn-sections and their covariant derivative as morphisms into the nn-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 nn-category of nn-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 nn-particle is supposed to be all about taking nn-spaces of sections of the background field nn-bundle. And the Lie \infty-algebraic formulation is supposed to be the powerful tool to handle this nn-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 Π (Hom(,Ω ()))\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 nn-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 nn-curvature of a G (n)G_{(n)}-bundle with connection in terms of a V//G (n)V//G_{(n)}-nn-groupoid bundle, where VV is an nn-representation and V//G (n)V//G_{(n)} the corresponding action nn-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 gg an L L_\infty-algebra a corresponding bundle with connection can be represented by a diagram which involves, among other things, a morphism of the kind

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

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

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

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

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

to

W(g,V) (σ, Aσ,A,F A) Ω (Y) (A,F A) W(g). \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, Aσ\nabla_A \sigma, the covariant derivative of the section σ\sigma.

Posted at January 30, 2008 8:38 PM UTC

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6 Comments & 6 Trackbacks

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

Posted by: Urs Schreiber on January 31, 2008 2:50 PM | Permalink | Reply to this

Re: L-infinity Associated Bundles, Sections and Covariant Derivatives

I think I have now figured out how the adjoint representation of an L L_\infty-algebra on itself works. (currently def. 2).

(When I say this, I am saying this in the context that I am looking at here. Actions of L L_\infty-algebras on complexes and on themselves have of course been considered before, noticably at the end of Lada-Markl. I think what I am doing here is the same, but from a different point of view.)

So this now allows me to say in particular what an adjoint String 2-bundle is. And what a section of that is: locally such a section is a gg-valued 0-form together with an ordinary 1-form.

I can compute the covariant derivative of that section, with respect to the given String 2-connection, but need to better understand what the result is telling me.

Maybe nothing much. What we really want to understand are sections of Chern-Simons 3-bundles and their relation to WZW theory.

So on with that…

Posted by: Urs Schreiber on January 31, 2008 4:23 PM | Permalink | Reply to this

Chern-Simons states

I wrote:

What we really want to understand are sections of Chern-Simons 3-bundles

The states of Chern-Simons theory should be obtained like this:

- start with the Chern-Simons 3-bundle (2-gerbe) with connection on BGB G;

- pick a 2-dimensional manifold Σ\Sigma, form the space of maps ΣBG\Sigma \to BG;

- transgress the Chern-Simons 3-bundle to that space of maps;

- form the vector space of sections of the resulting line bundle.

That should be the space of states.

Using the technology we’ve been setting up, in fact all of these steps now have a well-defined meaning:

- the Chern-Simons 3-bundle with connection on BGB G is represented by the b 2u(1)b^2 u(1)-connection descent object in prop. 35, p. 64;

- the space of maps is the presheaf maps(inv(g),Ω (Σ))\mathrm{maps}(\mathrm{inv}(g),\Omega^\bullet(\Sigma)) from def. 4, p. 17;

- the transgression of the Chern-Simons bundle to the space of maps is the image of the original b 2u(1)b^2 u(1)-connection descent object under the functor Ω (maps(,Ω (Σ)))\Omega^\bullet(\mathrm{maps}(--,\Omega^\bullet(\Sigma))) from def. 5, p. 18 as described for the case we are interested in on p. 87;

- the vector space of sections, finally, we form as described here in this entry, (after quotienting out the irrelevant symmetries, thereby performing the integration without integration of the Chern-Simons action functional, as in section 4);

So this gives a well-defined, effectively computable construction of the space of states of Chern-Simons theory.

And so I sat down and effectively computed the well-defined result.

The result which I found tonight is this:

after unwrapping the definitions here, a state is a \mathbb{C}-valued function ψΩ 0(Ω 1(Σ,g)) \psi \in \Omega^0(\Omega^1(\Sigma,g)) on the space Ω 1(Σ,g)\Omega^1(\Sigma,g) of gg-valued 1-forms on Σ\Sigma which, when pulled back to the sub-space of flat gg-valued 1-forms satisfies the descent condition

dψ+(tg Σμ)ψ=0 d \psi + (tg_\Sigma \mu)\wedge \psi = 0

where tg Σμtg_\Sigma \mu is the 1-form on the space of gg-valued 1-forms which we obtain from transgressing the connection 3-form μ\mu of the Chern-Simons 3-bundle over BGB G.

To compare this expression with the existing physics literature one can rephrase it in terms of the kind of component-based functional derivative equations used there (which actually receive a rigorous interpretation thereby). Doing so, I found that the above is equivalently expressed as

C c ab(A μ a(x)δδA bν(x)+A μ a(x)A ν b(x))ψ=0, C^c{}_{ab} \left( A^a_\mu(x) \frac{\delta}{\delta A_{b\nu}(x)} + A^a_\mu(x) A^b_\nu(x) \right) \psi = 0 \,,

for all cc ranging over a chosen basis {t a}\{t_a\} of gg, for all xΣx \in \Sigma and for all μ,ν\mu,\nu ranging over a basis of T xΣT_x \Sigma.

Here A μ a(x)Ω 0(Ω 1(Σ,g))A^a_\mu(x) \in \Omega^0(\Omega^1(\Sigma,g)) is the function (the “field”) on the space of gg-valued forms, as in section 9.3.1, whose value on a given gg-valued connection AΩ 1(Σ,g)A \in \Omega^1(\Sigma,g) is the a,μa,\mu-component of that 1-form at xΣx \in \Sigma in the chosen basis, and {C ab c}\{C^c_{ab}\} are the structure constant of the Lie algebra gg in that basis.

A description of this computation is now in section 4, p. 8.

I am hoping I didn’t make a mistake. It is getting late here. Take this with a grain of salt for the moment.

What I need to do next is to figure out how to bring a complex structure into the game. The sections we really want are holomorphic sections with respect to some complex structure.

Posted by: Urs Schreiber on January 31, 2008 10:52 PM | Permalink | Reply to this

Re: Chern-Simons states

This looks great, congratulations! Now I can try to understand all this L-infinity stuff… there is something concrete to calculate, the space of states of Chern-Simons theory.

Posted by: Bruce Bartlett on February 4, 2008 5:54 PM | Permalink | Reply to this

Re: Chern-Simons states

Now I can try to understand all this L-infinity stuff… there is something concrete to calculate, the space of states of Chern-Simons theory.

Yes. There are more concrete things to calculate, but this is one of the main goals of where we are headed, according to

slide 6 of String and Chern-Simons nn-Transport (pdf, blog).

(We want to eventually get all the way from the \infty-geometric data here, via the quantization edge of the cube, to the 3-functorial description of Chern-Simons/WZW theory).

A minute ago I have posted more details here, this time looking at the states over the circle.

This is clearly just dipping a toe into the water. But it looks promising.

Posted by: Urs Schreiber on February 4, 2008 6:24 PM | Permalink | Reply to this
Read the post States of Chern-Simons Theory
Weblog: The n-Category Café
Excerpt: A list of selected literature discussing Chern-Simons theory and its space of states.
Tracked: February 1, 2008 5:50 PM
Read the post Chern-Simons States from L-infinity Bundles, III: States over the Circle
Weblog: The n-Category Café
Excerpt: On computing the states of Chern-Simons theory over the circle from the L-infinity algebraic model of the Chern-Simons 3-bundle over BG.
Tracked: February 4, 2008 6:10 PM
Read the post Smooth 2-Functors and Differential Forms
Weblog: The n-Category Café
Excerpt: An article on the relation between smooth 2-functors with values in strict 2-groups, and an outline of the big picture that this sits in.
Tracked: February 6, 2008 12:05 PM

loop Lie algebroids

Following up on the discussion here on how transgression of String 2-bundles and the corresponding obstructing Chern-Simons 3-bundles yields Kac-Moody centrally extended loop bundles and the corresponding obstructing line 2-bundle (lifting gerbes) I have started writing out the theory of the loop Lie \infty-algebroids in more detail.

In the new section 2.3 I do the following:

a) quickly review Simon Willerton’s neat observation about the loop groupoid of a finite group, which can be found summarized in The baby version of Freed-Hopkins-Teleman;

b) then generalize the discussion from finite groups to Lie groups using the result of Parallel transport and functors (arXiv, blog);

c) then use that to guide the study of the corresponding loop Lie \infty-algebroids using the transgression operation from section 9 of L L_\infty-connections and applications (pdf, blog).

I show that

d) the loop Lie algebra of an ordinary Lie algebra gg is, as you’d expect, just the standard loop Lie algebra Ωg\Omega g

e) that the loop Lie algebra of the (skeletal version of the) String Lie 2-algebra g μg_\mu is the Kac-Moody central extension Ω^g \hat \Omega g of Ωg\Omega g.

This is to be read as a reply to my remark here.

Clearly, this is closely related to the step From loop groups to Lie 2-groups (arXiv, blog).

But the precise relation I won’t start spelling out before I had some sleep and took care of other things I am supposed to be doing.

Posted by: Urs Schreiber on February 7, 2008 12:30 AM | Permalink | Reply to this
Read the post Slides: L-Infinity Connections and Applications
Weblog: The n-Category Café
Excerpt: Talks on L-infinity algebra connections.
Tracked: February 15, 2008 5:47 PM
Read the post Charges and Twisted n-Bundles, I
Weblog: The n-Category Café
Excerpt: Generalized charges are very well understood using generalized differential cohomology. Here I relate that to the nonabelian differential cohomology of n-bundles with connection.
Tracked: February 29, 2008 4:14 PM
Read the post Sections of Bundles and Question on Inner Homs in Comma Categories
Weblog: The n-Category Café
Excerpt: On inner homs in comma categories, motivated from a description of spaces of sections of bundles in terms of such.
Tracked: March 4, 2008 10:14 PM

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