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October 16, 2007

n-Curvature, Part III

Posted by Urs Schreiber

The context of weak cokernels within obstruction theory seems to be the best way to think of nn-curvature (I II, III).

Consider the statement

Curvature is the obstruction to flatness.

For n=1n=1 this sounds pretty obvious and trivial. But I claim that we should really read it as

(n+1)(n+1)-Curvature is the obstruction to nn-flatness.

This is now a statement about nn-bundles with connection and (n+1)(n+1)-bundles with connection (or rather about the correspodinng (n+1)(n+1)-transport and nn-transport). And it is not all that trivial anymore. There is a general notion of obstruction theory for nn-bundles with connection, I think, and it applies here and produces a statement about (n+1)(n+1)-curvature which is at least non-obvious enough to have occupied me for quite a while.

But what was non-obvious once may become obvious as we refine our senses.

There is something non-trivial to be understood here, but we want to understand it in a natural way.

The main thing to be understood is why (n+1)(n+1)-curvature of a G (n)G_{(n)}-valued nn-transport takes values in the (n+1)(n+1)-group of inner automporphisms INN 0(G (n)). \mathrm{INN}_0(G_{(n)}) \,.

The full answer to this involves three main insights:

a) Obstruction theory.

b) (n+1)(n+1)-Curvature is the obstruction to lifting a trivial nn-transport to a flat nn-transport

c) Inner automorphisms and weak cokernels of identities on nn-groups

General remark: I’ll say nn-group and nn-bundle throughout, but all statements in the world of Lie nn-groupoids I can and have made precise so far only up to n=2n=2 with everything strict. But not so in the differential picture, which is mentioned in the remarks at the end: the great advantage of that differential picture is that we obtain it from the integral picture and then extend it to arbitrary nn, since Lie nn-algebras are handled so much more easily than Lie nn-groups.

a) Obstruction theory.

To measure the obstruction to lifting a G (n)G_{(n)}-transport tra:PΣG (n) \mathrm{tra} : P \to \Sigma G_{(n)} through a sequence

KBG (n) K \to B \to G_{(n)}

we form the weak cokernel

wcoker(KB)G (n) \mathrm{wcoker}(K \to B) \simeq G_{(n)}

to first puff up the nn-group G (n)G_{(n)} to an equivalent (n+1)(n+1)-group wcoker(KB)\mathrm{wcoker}(K \to B), such that the projection BG (n)B \to G_{(n)} becomes an inclusion Biwcoker(KB)G (n) B \stackrel{i}{\hookrightarrow} \mathrm{wcoker}(K \to B) \simeq G_{(n)}

which then allows us to form, in turn, the cokernel of that inclusion

coker(i) wcoker(KB) i K B G (n). \array{ &&&&&& \mathrm{coker}(i) \\ &&&&&\nearrow \\ &&&& \mathrm{wcoker}(K \to B) \\ &&&\multiscripts{^i}{\nearrow}{} & \uparrow^{\simeq} \\ K &\to& B &\to& G_{(n)} } \,.

The composition PtraΣG (n)wcoker(KB)coker(i) P \stackrel{\mathrm{tra}}{\to} \Sigma G_{(n)} \stackrel{\simeq}{\to} \mathrm{wcoker}(K \to B) \to \mathrm{coker}(i) then gives a coker(i)\mathrm{coker}(i)-transport which measures the failure of tra\mathrm{tra} to lift to a BB-transport.

(More details and examples are discussed in String- and Chern-Simons nn-Transport – see the section of the same name – and in the BIG diagram. )

b) Curvature is the obstruction to lifting a trivial transport to a flat transport

A G (n)G_{(n)}-nn-bundle without connection on XX is a transport nn-functor P:Π 0(X)ΣG (n) P : \Pi_0(X) \to \Sigma G_{(n)} equipped with a smooth local G (n)G_{(n)}-trivialization. (See The first Edge of the Cube if that sounds strange.)

A G (n)G_{(n)}-nn-bundle on XX with flat connection is a transport nn-functor mathrntra:Π n(X)ΣG (n). \mathrn{tra} : \Pi_n(X) \to \Sigma G_{(n)} \,.

Given a G (n)G_{(n)}-bundle with connection, we may ask if we can extend it to a G (n)G_{(n)}-bundle with flat connection

Π 0(X) Π n(X) * tra flat ΣG (n). \array{ \Pi_0(X) &\hookrightarrow& \Pi_n(X) \\ \downarrow^* & \swarrow_{\mathrm{tra}_{\mathrm{flat}}} \\ \Sigma G_{(n)} } \,.

In general we cannot. The obstruction is given by a wcoker(Id G (n))\mathrm{wcoker}(\mathrm{Id}_{G_{(n)}})-transport.

To see this more clearly, we need a little bit of local data:

A possibly nontrivial G (n)G_{(n)}-bundle without connection on XX is a surjective submersion FYXF \to Y \to X with connected fibers, together with a flat ΣG (n)\Sigma G_{(n)}-transport on the fibers

P:Π n(F)ΣG (n). P : \Pi_n(F) \to \Sigma G_{(n)} \,.

A flat G (n)G_{(n)}-connection, on this, is an extension of this to a functor on all of YY:

tra flat:Π n(Y)ΣG (n). \mathrm{tra}_{\mathrm{flat}} : \Pi_n(Y) \to \Sigma G_{(n)} \,.

In general, this does not exist. What always exists, though, is the completely trivial bundle with connection

tra 0:Π n(Y){}, \mathrm{tra}_0 : \Pi_n(Y) \to \{\bullet\} \,,

i.e. the principal bundle for the trivial structure group.

Hence the question that we are asking when asking for curvature is:

Can we lift the connection for the trivial group through the exact sequence G (n)IdG (n){} G_{(n)} \stackrel{\mathrm{Id}}{\to} G_{(n)} \to \{\bullet\} ?

Curvature is a very degenerate case of general obstruction theory: we are asking for obstructions to extending the trivial structure group.

More precisely, we want to find a lift of tra\mathrm{tra} which does restrict to the fixed functor P:Π n(F)ΣG (n) P : \Pi_n(F) \to \Sigma G_{(n)} on the fibers of the surjective submersion, meaning we want to lift to

Π n(F) Π n(Y) tra ΣG (n) Id ΣG (n). \array{ \Pi_n(F) &\hookrightarrow& \Pi_n(Y) \\ \downarrow && \downarrow^{\mathrm{tra}} \\ \Sigma G_{(n)} &\stackrel{\mathrm{Id}}{\to}& \Sigma G_{(n)} } \,.

In general this will not work. But we have obstruction theory as above to figure out what the obstructing (n+1)(n+1)-bundle with connection will be: it will be an (n+1)(n+1)-transport with values in wcoker(i) \mathrm{wcoker}(i) obtained by first lifting the {}\{\bullet\}-transport tra 0\mathrm{tra}_0 to an equivalent wcoker(Id G (n)) \mathrm{wcoker}(\mathrm{Id}_{G_{(n)}})-transport and then checking which mistake in wcoker(i)\mathrm{wcoker}(i) we make thereby:

Π n(F) Π n(Y) Π n(X) tra curv ΣG (n) Id wcoker(Id G (n)) Id wcoker(i) cocycle attemptedflatlift failureoflift cocycle nEhresmannconnection curvature. \array{ \Pi_n(F) &\hookrightarrow& \Pi_n(Y) &\to& \Pi_n(X) \\ \downarrow && \downarrow^{\mathrm{tra}} && \downarrow^{\mathrm{curv}} \\ \Sigma G_{(n)} &\stackrel{\mathrm{Id}}{\to}& \mathrm{wcoker}(\mathrm{Id}_{G_{(n)}}) &\stackrel{\mathrm{Id}}{\to}& \mathrm{wcoker}(i) \\ \\ cocycle && attempted flat lift && failure of lift \\ \\ cocycle && n-Ehresmann connection && curvature } \,.

It remains to compute these weak cokernels.

c) Inner automorphisms and weak cokernels of identities on nn-groups

As I reported in Detecting higher order necklaces Enrico Vitale discusses that the inner automorphism (n+1)(n+1)-group on the nn-group G (n)G_{(n)} that I discussed with David Roberts is the weak cokernel of the identity on G (n)G_{(n)}: wcoker(Id (G (n)))=INN(G (n)). \mathrm{wcoker}(\mathrm{Id}_{(G_{(n)})}) = \mathrm{INN}(G_{(n)}) \,.

So that’s why we see these inner automorphism (n+1)(n+1)-groups appearing in the theory of nn-curvature.

This is the story behind the slogan

(n+1)(n+1)-Curvature is the obstruction to nn-flatness.


a) Notice how the obstructions here are not just classes, but really full-fledged (n+1)(n+1)-transports. I am claiming that this is related to the phenomenon called holography, which also exhibits That shift in dimension – essentially the image of the shift we see here under quantization. There would be more to say about this. But it is, unfortunately, top secret.

b) Everything I said applies directly also to the differential Lie-nn-algebraic picture. There the inner automorphism (n+1)(n+1)-group INN(G (n))\mathrm{INN}(G_{(n)}) becomes the inner derivation Lie (n+1)(n+1)-algebra inn(g (n))\mathrm{inn}(g_{(n)}), as described in section: Bundles with Lie nn-algebra connection (see also, for instance, Lie nn-algebra cohomology).

Recall from More on tangent categories that forming the inner derivation Lie (n+1)(n+1)-algebra corresponds, Koszul-dually, to forming the shifted tangent bundle of the dg-manifold corresponding to g (n)g_{(n)}.

c) There is more going on than meets the eye. See the blockbuster diagram movie (original with subtitles) in section: nn-Categorical background, subsection G (n)G_{(n)}-bundles with connection, based on the bestselling novel Tangent categories to get an impression.

Posted at October 16, 2007 9:20 PM UTC

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8 Comments & 5 Trackbacks

Re: n-Curvature, Part III

(n+1)-Curvature is the obstruction to n-flatness.

This was of course my main motivation to do n-gauge theory all along. On the lattice, 1-flatness means that horizontal and vertical links commute (if the corresponding matrices only depend on orientation and not on position). Similarly, 2-flatness becomes the Yang-Baxter equation, and n-flatness the n-simplex equation, which is necessary condition for lattice integrability in n dimensions.

This is the useful definition of (n+1)-curvature on the lattice, because flatness reduces to an important condition. Unfortunately, no good solutions to the n-simplex equations beyond n=2 appear to be known, so maybe (n>2)-flatness is impossible.

Posted by: Thomas Larsson on October 17, 2007 9:50 AM | Permalink | Reply to this

Re: n-Curvature, Part III

Thanks. I was unaware of the relation to the n-simplex eqn.
I am troubled by the indexing of connection and curvature. Since ordinary curvature is the obstruction to ordinary flatness, I would have thought that should read:
1-curvature is the obstruction to 1-flatness

Posted by: jim stasheff on October 17, 2007 2:36 PM | Permalink | Reply to this

Re: n-Curvature, Part III

Several years ago, I have had a long and rather heated discussion with Urs about the merits of different notions of n-transport, which ended with us agreeing to disagree. My observation is simply this:

In lattice 1-gauge theory, flatness is the 1-simplex equation (in the case that links only depend on separation and not on position).

In the correct version of lattice n-gauge theory, flatness should therefore be the n-simplex equation (in the same limit).

The model described in math-ph/0205017 has this desirable property. This is not surprising, because I build on work on the n-simplex equations by Maillet and Nijhoff in the late 1980s. Don’t read the confused section about gerbes, though.

Posted by: Thomas Larsson on October 17, 2007 4:43 PM | Permalink | Reply to this

Re: n-Curvature, Part III

ended with us agreeing to disagree

That’s still not how I remember it. Rather, it ended with me saying that you are looking at a certain nn-transport with values in an nn-monoid instead of an nn-group, whose precise definition might need more work.

I guess now I am disagreeing that we agreed to disagree. But you migth disagree with that. In that case I would then indeed opt that we agree to disagree that we agreed to disagree.

Posted by: Urs Schreiber on October 17, 2007 5:02 PM | Permalink | Reply to this

Re: n-Curvature, Part III

I am troubled by the indexing of connection and curvature. Since ordinary curvature is the obstruction to ordinary flatness, I would have thought that should read: 1-curvature is the obstruction to 1-flatness

I agree. I am abusing the indexing conventions here.

I think I wanted to emphasize that the curvature is really an (n+1)(n+1)-structure, a fact that is kind of important.

I need to make up my mind what to do. Either I switch to saying “nn-transport has nn-curvature”, which would be reasonable — or I will try to argue that we should have been calling ordinary curvature “2-curvature” all along.

And to some extent that would even be right: ordinary curvature is 2-form curvature.

Posted by: Urs Schreiber on October 17, 2007 5:34 PM | Permalink | Reply to this

Re: n-Curvature, Part III

Too many twos. Beware of coincidences which are only notational.

Posted by: jim stasheff on October 18, 2007 2:16 AM | Permalink | Reply to this

Re: n-Curvature, Part III

Beware of coincidences which are only notational.

Right. But here I am after emphasizing that standard notation hides a certain important fact:

the curvature of a line bundle, being a global 2-form, is really to be thought of as being itself a 2-thing: namely a (trivial) gerbe with connection!

This sounds like overkill, but I think it is important because it is this pattern which generalizes:

the curvature of an nn-thing is itself an (n+1)(n+1)-thing.

You may turn this around: a trivializable gerbe is tivialized by the bundle whose curvature it is.

A non-trivializable gerbe, however, which has a torsion class, is “weakly trivialized” by the twisted bundle whose twisting curvature it is.

So that’s my point here: like the obstruction to lifting a Spin-bundle to a String 2-bundle is not just a 4-class in cohomology but really a 2-gerbe realizing that class – the Chern-Simons 2-gerbe – we may want to keep track of how the curvature of a bundle – the thing obstructing its flatness – is really a 2-thing itself.

But I think I will follow your advice and say that nn-transport has nn-curvature. The statement then is – and that fits both your remark while at the same time conveying the point I like to convey:

The nn-curvature of nn-transport is itself an (n+1)(n+1)-transport. Its (n+1)(n+1)-curvature in turn, which is an (n+1)(n+1)-transoport, is always trivial. This fact is the generalized Bianchi identity.

ntransport (n+1)transport (n+2)transport curvature curvatureofcurvature tra curv tra curv curv tra arbitrary flat trivial \array{ n-transport & (n+1)-transport & (n+2)-transport \\ & curvature & curvature of curvature \\ \mathrm{tra} & \mathrm{curv}_{\mathrm{tra}} & \mathrm{curv}_{\mathrm{curv}_{\mathrm{tra}}} \\ arbitrary & flat & trivial }

Posted by: Urs Schreiber on October 18, 2007 2:05 PM | Permalink | Reply to this

Re: n-Curvature, Part III

maybe (n>2n \gt 2)-flatness is impossible.

We need to be careful with what we mean:

Take any space XX and any nn-group G (n)G_{(n)}, form the trivial nn-bundle P=X×G (n) P = X \times G_{(n)} and equip it with the trivial nn-transport Π n(X)Atiyah(P) \Pi_n(X) \to Atiyah(P) which sends every kk-path to the identity kk-morphism on the… on the identity 1-morphism on the fibers over the endpoints.

That’s a flat nn-connection.

It seems to me what you have in mind is rather this:

given a certain nn-monoidal target built from matrices, and given some nn-transport with values in that, figuring out whether or not it is flat amounts to looking at the corresponding nn-simplex equation.

Posted by: Urs Schreiber on October 17, 2007 3:55 PM | Permalink | Reply to this
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