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May 31, 2007

The Second Edge of the Cube

Posted by Urs Schreiber

What is a Cartan-Ehresmann connection, really?

If we know what a Cartan-Ehresmann connection with values in a Lie algebra gg really is, we should immediately know what an nn-Cartan-Ehresmann connection with values in any Lie nn-algebra is.

Cubism

As described in The first Edge of the Cube, connections are conveniently conceived in terms of their parallel transport functor.

Given such a transport functor, there are various natural things one can do with it, like locally trivializing it (this takes us along a first edge of a cube of concepts), or categorifying it or quantizing it (by which we proceed along other edges of this cube).


Yet another edge of The Cube: Differentiation

For practical purposes of somewhat more urgent relevance, however, is yet another operation: locally trivializing and differentiating the transport functor – which is a morphism of Lie nn-groupoids satisfying certain conditions – to obtain a morphism of Lie nn-algebras satisfying certain conditions.


Interlude: Curvature

The appearance of the inner automorphisms and inner derivations here is due to the fact that the Lie algebra morphisms on the right really are the differentials of pseudofunctors from the pair groupoid of XX. That’s because the tangent algebroid Vect(X)\mathrm{Vect}(X) is the 1-algebroid obtained from differentiating the pair groupoid Vect(X):=Lie(X×X). \mathrm{Vect}(X) := \mathrm{Lie}(X \times X) \,. But locally, say for XX convex, a pseudofunctor on the pair groupoid is essentially the same as an nn-functor on the fundamental nn-groupoid Π n(X)\Pi_n(X). But these describe flat nn-transport.

As a result, the morphisms here don’t just describe the parallel nn-transport, but actually the corresponding nn-curvature. And that, as described in part in

nn-Curvature

is an (n+1)(n+1)-functor which takes values in inner automorphisms of the structure group. One way to understand this is by pairing Schreier theory of groupoid extensions with the Atiyah sequence of groupoids associated with principal bundles. This observation goes back to Danny Stevenson. I discussed it in nn-Transport and Higher Schreier Theory.

So

A connection with values in a Lie nn-algebra g (n)g_{(n)} is not, in general, a morphism Vect(X)g (n) \mathrm{Vect}(X) \to g_{(n)} (only if all its kk-form curvatures vanish), rather, it is a morphism Vect(X)inn(g (n)) \mathrm{Vect}(X) \to \mathrm{inn}(g_{(n)}) to the Lie (n+1)(n+1)-algebra of inner derivations of g (n)g_{(n)}.

This actually proves to clarify a lot of things that otherwise remain mysterious or require awkward workarounds.

In a sense

Curvature is more fundamental than connection.

Since Lie nn-algebras may be tricky to integrate to Lie nn-groups, it would be very useful to have a notion of connection with values in a Lie nn-algebra which doesn’t require making the integrated parallel transport manifest, but realizing just its differential part, which takes values just in the Lie nn-algebra.

This is to some degree complementary to the approach by Anders Kock: where Kock uses synthetic differential geometry, this here is supposed to use ordinary differential goemetry.


Descent at the Lie level: how to do it right

I had begun thinking about the issue of putting the right gluing data/descent conditions on (dual) morphisms of qDGCAs f *:( sg (n) *,d inn(g (n)))Ω (X) f^* : (\wedge^\bullet sg_{(n)}^*, d_{\mathrm{inn}(g_{(n)})}) \to \Omega^\bullet(X) here, by trying to formulate the right descent condition for arbitrary covers of base space. But this has the problem that the linearization in general misses crucial data.

But then from Jim Stasheff I learned what is the right way to do this:

A) Don’t consider connection forms on arbitrary covers, but restrict attention to the cover given by the total space of the bundle itself. (This avoids the above problem.)

B) Pass to the differentiated version of the usual statement of the Cartan-Ehresmann conditions on a connection 1-form. This makes their true nature manifest: these two conditions really say that the connection commutes with certain derivations.


The program

Thinking about this for a while, it turns out that there is a nice story to be told here. It deserves to be an edge of the cube by itself. So now the cube is spanned by four edges. The projection to three dimensions of relevance for the present case is this:

With Jim I am trying to go all the way along the main diagonal and formulate, for any Lie nn-algebra g (n)g_{(n)} the theory of nn-Cartan-Ehresmann connections with values in g (n)g_{(n)}. This is the goal of

U.S. & J. Stasheff
\;\; Structure of Lie nn-Algebras
\;\; Connections with Values in Lie nn-Algebras
\;\; Zoo of Lie nn-Algebras

aspects of which I had mentioned before (I, II, III, IV).


Interlude: Ehresmann versus Cartan.

I am saying “Cartan-Ehresmann connection” where, strictly speaking, I mean “Ehresmann connection”:

a Lie algebra valued 1-form AΩ 1(P,g)A \in \Omega^1(P,g) on the total space of a principal GG-bundle PP satisfying two compatibility conditions.

If we understand that, we’ll easily understand the Cartan version, too, which is about G-bundles whose structure group may be reduced. Derek Wise in Gravity and Cartan Geometry gives a nice description of the two concepts and their relation.


What Cartan-Ehresmann really means: differentiating the Universal Transition

When trying to understand what statements made in terms of differential forms really mean, it is usually helpful to find the corresponding integrated statements.

In a way, this is the very motivation of the synthetic differential geometry approach of Lawvere, Kock, and others: reformulate differential geometry in a way that tangent vectors literally appear like little paths.

Anyway: what do the Ehresmann conditions on a connection 1-form on the total space of a principal bundle correspond to when integrated?

John Baez and Derek Wise emphasized the nice geometric picture behind it (TWF 243).

That’s cool. But here we need the “arrow-theoretic” meaning of Cartan-Ehresmann. It should be this:

In

U.S. & Konrad Waldorf
Parallel Transport and Functors
arXiv:0705.0452v1 [math.DG]

we describe in detail how the local trivialization of a parallel transport functor gives rise to a functor on paths of the cover, which extends to a functor on the Universal Transition (called the “path puhsout” in that paper).

This universal transition C(Y) C(Y) is a groupoid whose morphisms consist of

- paths in the cover YY

- jumps between patches

such that the obvious compatibility condition is satisfied. The corresponding functor C(Y)ΣG C(Y) \to \Sigma G is actually the smooth anafunctor corresponding to the local trivialization of the parallel transport.

In a sense we have

A Cartan-Ehresmann connection is the differential of this anafunctor.

To make this morally true statement precisely true, we need to follow Jim Stasheff’s advice and locally trivialize the bundle not over an arbitrary cover – but over itself.


The crucial step: Canonically trivialize the bundle over itself

Every principal GG-bundle p:PXp : P \to X canonically trivializes when pulled back to its own total space PP P× XPP×G. P \times_X P \simeq P \times G \,. Fix this canonicaly trivialization and then turn the crank on the universal transition anafunctor described above. One finds that our anafunctor now will be such that

- It sends jumps between patches – which now are labeled by group elements gGg \in G – to precisely that group element. This is the integrated version of the first Cartan-Ehresmann condition.

- It sends paths γg\gamma \cdot g that arise from translation with gGg \in G to the image of the original path γ\gamma, conjugated by gg. This is the integrated version of the second Cartan-Ehresmann condition.


Arrow theory and Implementation

Of course there are much more pedestrian ways to understand the Cartan-Ehresmann condition from the point of view of locally trivialized bundles with connection. We give a detailed description of a bunch of elementary but useful facts concerning this in section 3.3.2 of Connections with Values in Lie nn-algebras.

The real point of the above exercise is the following:

- On the one hand we discover the arrow theory that underlies the concept of a Cartan-Ehresmann connection. While for ordinary bundles the above functorial description may feel like overkill, its great advantage is that once we have understood this, we easily understand its generalization to nn-bundles with connection.

- On the other hand, as the above table indicates, we see how the arrow theory implements into the world of quasi-free differential graded algebras. That’s good, because while arrow-theory is useful for understanding what’s going on, implementation in qDGCAs allows us to efficiently compute.

In particular, we see that, on the qDGCA level, the consistency conditions on our connection are a bunch of commutator relations. That’s very handy.

But here there is actually also a good arrow-theoretic way to understand these commutators as intertwiners of arrow-theoretic Lie derivatives.

This will be useful when passing from 1-connections to nn-connections.

Posted at May 31, 2007 6:37 PM UTC

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5 Comments & 7 Trackbacks

Re: The second Edge of the Cube

Again, you are way beyond the realm of anything I could possibly ever understand, but you present it in a way that I can’t help but find compelling or “neat”. Fascinating stuff. It “feels” like you are doing something right.

Posted by: Eric on June 1, 2007 3:13 PM | Permalink | Reply to this
Read the post Connections on String-2-Bundles
Weblog: The n-Category Café
Excerpt: On connections on String 2-bundles.
Tracked: June 3, 2007 4:05 PM

Re: The Second Edge of the Cube

Bruce wrote:

Could you explain a bit more about what cartan connections are, and what they’re good for? I tried to read the wikipedia article but it’s quite a long one and I’m not sure which is the way you would encourage us to think of them.

Sure!

From some discusssion going on behind the scenes, I gather that there is an issue here with different people assigning different meanings to the words “Cartan connection”.

So I’ll say what exactly it is we are looking at here, and then we might discuss what the best terminology for this actually is.

For GG any Lie group we are talking about GG-principal bundles PX P \to X with connection.

For definiteness, let the action be from the right and denote it by R:P×GP. R : P \times G \to P \,.

There are many ways to say what a connection on PP is. They are all equivalent, and all useful for particular purposes.

Here, we are after a formulation which lives as much as possible in the world of the Lie algebra g:=Lie(G). g := \mathrm{Lie}(G) \,.

The reason is that we have powerful computational techniques for dealing with Lie nn-algebras. So the strategy is to formulate a connection as much as possible Lie algebraically and then categorifying that.

One way to define a connection on PP is in terms of a 1-form AΩ 1(P,g) A \in \Omega^1(P,g) on the total space of PP with values in the Lie algebra of GG, such that this satisfies two conditions:

1) For X ξ:=R *(,ξ) X_\xi := R_*(\cdot,\xi) a vertical vector field on PP induced by the Lie algebra element ξg\xi \in g, we have A(X ξ)=ξ. A(X_\xi) = \xi \,.

2) For all hGh \in G we have A=Ad h(R h *A). A = \mathrm{Ad}_h (R_h^* A) \,.

I’d think this definition is what is called an Ehresmann connection, though it is not clear to me whether people want this term to be used in a way that it distinguishes between the definition which involves the 1-form, and the definition which talks about the horizontal subspaces induced by this 1-form.

By the way, I found the following fact very useful, but have never seen it emphasized (probably just my ignorance):

The above Ehresmann connection is precisely the differential cocycle condition π 2 *A=Ad g(π 1 *A)+gdg 1 \pi_2^* A = \mathrm{Ad}_g(\pi_1^* A) + gdg^{-1} for the case that the transition function gg is that obtained from the canonical trivialization of any principal bundle over its own total space.

That’s a very useful fact for connecting what we are talking about here, with all those other things we have been talking about so much, in the context of connections on bundles.

Anyway, as Jim Stasheff emphasized, what is so good about the above definition is that it may be differentiated with respect to the transition parameter. Condition 2) above then becomes L X ξA=ad ξA, L_{X_\xi} A = \mathrm{ad}_\xi \circ A \,, where L vL_v denotes the Lie derivative on differential forms along the vector field vv.

That’s already nice, but it gets much better. Jim pointed out to me that we should be thinking of this relation as featuring a Lie derivative on both sides of the equality sign.

Namely it makes very good sense to think of the dual of the adjoint action of a Lie algebra on itself as being a kind of Lie derivative, too. To emphasize this, we first pass to the dual formulation of everything in sight.

Instead of considering the linear map A:TXg A : T X \to g we take the dual morphism f *:g *Ω (X). f^* : g^* \to \Omega^\bullet(X) \,. (I write f *f^* instead of A *A^* just for fun. Freeing us from the AA will become useful when we categorify.)

If on top of that we declare the notation L ξ:=(ad ξ) *. L_\xi := (\mathrm{ad}_\xi)^* \,. condition 2) above becomes L X ξf *(ω)=f *(L ξω) L_{X_\xi} f^*(\omega) = f^*(L_\xi \omega) for all ω\omega in the dual space g *g^* of the Lie algebra.

This is finally approaching the “purely algebraic” formulation that we are after: condition 2) on a connection may be read as a compatibility with a natural notion of Lie derivative. Think of the above as the commutator relation [L ξ,f *]=0. [L_\xi, f^*] = 0 \,.

As I tried to indicate in What is a Lie derivative, really? this is more than just playing with notation.

In particular, it makes good sense to massage condition 1) such that it similarly appears as a commutator expression:

let i vi_v denote the usual contraction (“interior product”) of differential forms with vector fields. The condition 1) can equivalently be rewritten as i X ξf *(ω)=f *(ω(i ξ)), i_{X_\xi} f^*(\omega) = f^*(\omega(i_\xi)) \,, for all ωg *\omega \in g^*. Here I am adopting the convention that f *f^* applied to a number yields the corresponding constant 0-form on XX. (This, like all other ingredients appearing here, will nicely follow from one unique structure once we have identified that.)

Again, it makes very good sense to abbreviate this as [i ξ,f *]=0. [i_\xi , f^* ] = 0 \,.

Okay, so, in summary, this is what Jim taught me to call a Cartan connection on a principal GG-bundle PP:

a linear map f *:g *Ω 1(P) f^* : g^* \to \Omega^1(P) such that 1) [i ξ,f *]=0 [i_\xi, f^*] = 0 and

2) [L ξ,f *]=0. [L_\xi,f^*] = 0 \,.

Okay, so far this is in principle standard stuff, if maybe in unusual clothes.

What we can now add to this structure is the following observation:

The above already begins to involve the Weyl algebra ( (sg *),d g)(\wedge^\bullet (s g^*), d_g), which is the free graded commutative algebra generated from the dual space g *g^*, regarded to be in degree 1, equipped with a differential d gd_g of degree +1 which encodes precisely the Lie bracket on gg.

The morphism f *f^* above will respect the differential [d,f *](ω)=d deRhamf *(ω)f *(dω) [d,f^*](\omega) = d_{\mathrm{deRham}}f^*(\omega) - f^*(d\omega) if and only if the curvature F AF_A of AA vanishes.

As I have discussed in great detail elsewhere, we may hence really want to pass to morphisms not to the Lie algebra gg, but to its inner derivation Lie 2-algebra inn(g)\mathrm{inn}(g).

So let’s instead look at morphisms of differential algebras f *:( inn(g) *,d inn(g))(Ω (P),d). f^* : (\wedge^\bullet \mathrm{inn}(g)^*, d_{\mathrm{inn}(g)}) \to (\Omega^\bullet (P), d) \,. By definition, these are required to commute with the differential [d,f *]=0. [d,f^*] = 0 \,. (This now no longer says that the 1-connection AA is flat, but that the corresponding curvature 2-connection F AF_A is flat: this is now the Bianchi identity).

Since L v:=[d,i v]L_v := [d,i_v], it should now come as no surprise that in terms of this new generalized connection morphism f *f^*, the two conditions from before reduce to a single one: [i ξ,f *]=0. [i_\xi, f^*] = 0 \,. This formulation is then supposed to be the starting point for lifting all these from Lie 1-algebras gg to Lie nn-algebras g (n)g_{(n)}.

Posted by: urs on June 11, 2007 2:03 PM | Permalink | Reply to this

Re: The Second Edge of the Cube

Commenting just on nomenclature:
There are two Cartan’s involved: Elie and Henri.
My understanding (enhanced by Derek Wise)
is that Elie Cartan generalized the Levi-Civita connection, Ehresmann went beyond that to the case of a general G-principal bundle for a (finite dimensional?) Lie group. Henri Cartan further abstracted to a purely differential algebra situation which includes (crucially) the Weil algebra,
which played the role of forms on a universal principal bundle, which didn’t yet exist in those days.

I prefer the following distinction in the geometric setting:

connection as the collection of `horizontal’ subspaces

connection form as a 1-form on the principal bundle or the collection of local 1-forms on the base

covariant derivative d+[A, ]

parallel tranport as a lift of the
manifest map Maps(I, M) to M x M

Posted by: jim stasheff on June 11, 2007 2:53 PM | Permalink | Reply to this

Re: The Second Edge of the Cube

connection form as a 1-form on the principal bundle or the collection of local 1-forms on the base

I like to think of “the collection of local 1-forms on the base” as a single 1-form on a cover YXY \to X of the base. If that cover has severaly connected components (like when YY is the disjoint union of open subsets of a good cover) these look like “many 1-forms”.

A 1-form on such a cover, satisfying the relevant conditions I’d call a differential cocycle.

But the nice thing is, we can take the bundle PXP \to X itself as a cover Y:=P Y := P and then see that the single 1-form on PP is nothing but a differential cocycle for the choice Y=PY = P.

Posted by: urs on June 11, 2007 3:21 PM | Permalink | Reply to this

Re: The Second Edge of the Cube

Thanks for this explanation. Some beautiful geometrical ideas here.

Posted by: Bruce Bartlett on June 12, 2007 12:18 AM | Permalink | Reply to this
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