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October 11, 2005

Synthetic Differential Geometry and Surface Holonomy

Posted by Urs Schreiber

L. Breen and W. Messing in their famous math.AG/0106083 had noted that what is called synthetic differential geometry with its use of combinatorial differential forms is naturally suited for talking about connections on higher order structures such as gerbes in terms of ‘finite’ morphisms between these structures.

Synthetic differential geometry goes back to category-theoretic ideas by Lawvere and was developed mainly by Anders Kock. There is a textbook

Anders Kock
Synthetic Differential Geometry
London Mathematical Society Lecture Notes Series 51
Cambridge University Press (1981)

as well as a series of more recent papers which discuss things like gauge theory

A. Kock
Combinatorics of Curvature, and the Bianchi Identity
Theory and Application of Categories 2 (1996), 69-89

and distribution theory

A. Kock
Categorical Distribution Theory; Heat Equation
to appear in Cahiers de Top. et Geom. Diff. Categorique.
preprint available here

from the synthetic point of view.

L. Breen and W. Messing have reformulated and generalized this framework to a scheme-theoretic context in

L. Breen, W. Messing
Combinatorial Differential Forms
math.AG/0005087 .

One can safely include synthetic/combinatorial differential geometry in the list of concepts which are very simple and easy to handle in their pedestrian version, but which are powerful and far-reaching enough to admit mind-bogglingly complex generalizations. Breen and Messing discuss the generalized setup. Kock mostly cares about the more pedestrian version.

Combinatorial differential forms make again an appearance in a recent paper by Jussi Kalkkinen which further investigates aspects of Breen&Messing’s work on gerbes with connection:

Jussi Kalkkinen
Topological Quantum Field Theory on Non-Abelian Gerbes
hep-th/0510069 .

The paper discusses, motivated by similar construction in physics, how to enlarge the ‘field content’ of local data of a (nonabelian) gerbe by odd-graded ‘ghost’ fields such that odd graded BRST-like nilpotent operators generate the infinitesimal version of gauge transformations on this data.

The approach used is different but not totally unrelated to the construction presented in section 13 of hep-th/0509163.

Incidentally, I have recently been thinking about how to use synthetic differential forms in order neatly relate smooth pp-holonomy pp-functors to their associated pp-forms. This is a kind of technical issue with probably little interest for physically inclined people, but it seems that there is a conceptually nice mechanism at work which relates ‘macroscopic’ pp-functors to their ‘infinitesimal’ parts.

Some preliminary notes which review material from the theory of smooth (‘diffeological’) spaces as well as synthetic differential geometry and uses them in order to analyse smooth pp-holonomy pp-functors can be found here:

Notes: Holonomy on Smooth Path Spaces .

I imagine that much more can be done with synthetic differential geometry in the context of pp-holonomy, but it should be a first step.

Posted at October 11, 2005 11:57 AM UTC

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4 Comments & 5 Trackbacks

Re: Synthetic Differential Geometry and Surface Holonomy

Hi Urs!

It’s been a long time. I’m glad I have one machine that still automatically loads this web site when I launch Mozilla :)

Just glanced at your notes. I think you can guess that I found the section “Combinatorial Differential Forms for Mortals” to be very interesting. Some of those formulas looked very familiar though. That is the language Dimakis and Hoissen like to use when discussing their version of noncommutative geometry. I am still waiting for you to make the connection (pun?) between all this and what we were playing around with :)

Best wishes!
Eric

PS: I hate to be the one to break it to you, but you stopped being mortal a LONG time ago ;)

Posted by: Eric on October 16, 2005 12:19 AM | Permalink | Reply to this

Re: Synthetic Differential Geometry and Surface Holonomy

Hi Eric,

nice to hear from you again!

Some of those formulas looked very familiar though.

Yes, indeed, I did notice that too. This is very closely related to what we were doing. In a sense it is about rigorously taking the continuum limit of what we played around with.

I still need to absorb this stuff more, though. It is of the interesting kind that much of it looks exremely elementary – up to some subtle subtleties.

If you have the time and leisure, I can recommend Anders Kock’s writings to you. It’s nice to read. He gives two simple axioms and everything follows elementarily from that. Think of an element (x,y)(x,y) of what they call the first neighborhood of the diagonal as a tiny little edge x→γyx \overset{\gamma}{\to} y and there you go. It’s all very neat.

The best thing is, which is essentially why Breen and Messing are using this synthetic stuff successfully for their gerbe connections, this provides a language to talk about differential geometry functorially. Very convenient.

I was about to say more about synthetic geometry, but now I am absorbed with working out how 2-connections describe strings on orbifolds and orientifolds, discrete torsion and things like that.

Here is a little puzzle which I tinkered with a little while on the train to Hamburg this morning. I expect the solution is well-known, but I seem to have some signs wrong or something.

Pick two tiny edges which span a 2-simplex. Compute the holonomy of some connection around the boundary of that simplex. The result is the exponential of the curvature of the connection evaluated on the two ‘vectors’ which span the simplex.

Fine. My question is: how does this generalize?

Pick three tiny edges which span a 3-simplex. Pick some path along the edges of that 3-simplex. Shouldn’t its holonomy be something like the exponential of the Chern-Simons 3-form evaluated on that 3-simplex?

Is this true? How precisely do I have to formulate the setup to make it true?

Of course what I am asking for is the combinatorial interpretation of the Chern-Simons 3-form. This must be well known, but I didn’t quite get it this morning.

Understanding this should help understanding the relation between Chern-Simons 2-gerbes (3-bundles) and WZW gerbes, I expect.

Posted by: Urs on October 17, 2005 12:59 PM | Permalink | Reply to this

Re: Synthetic Differential Geometry and Surface Holonomy

Hello!

Yes, I am very much aware of Anders Kock’s stuff on synthetic differential geometry. That is why I was happy to see you taking an interest in it. Like always, I get only far enough to see that something is important and then I rely on you to make everything make sense :)

Regarding the combinatorics of the (abelian) Chern-Simons 3-form, my friend Robert Kotiuga noted some of its properties in the context of computational electromagnetics.

(1)⟨A⌣dA|i 0i 1i 2i 3⟩=⟨A|i 0i 1⟩⟨dA|i 1i 2i 3⟩=⟨A|i 0i 1⟩[⟨A|i 2i 3⟩−⟨A|i 1i 3⟩+⟨A|i 1i 2⟩] \langle A\smile dA|i_0i_1i_2i_3\rangle = \langle A|i_0i_1\rangle \langle dA|i_1i_2i_3\rangle = \langle A|i_0i_1\rangle \left[\langle A|i_2i_3\rangle-\langle A|i_1i_3\rangle+\langle A|i_1i_2\rangle\right]

Then you antisymmetrize and get something that has a kind of neat geometrical interpretation. Kotiuga related this to some special kind of determinant that escapes me (surprise).

See Week 107.

By the way, it looks like Scott Wilson just finished his thesis in August. I described a little bit about what he was doing when I visited Harrison at that math conference.

On the Algebra and Geometry of a Manifold’s Chains and Cochains

From what I understood, this is an alternative to what we did that I suspect might be equivalent. It is the commutative vs associative thing (and the mapping between them). He develops a theory of chains and cochains that maintains commutativity at the expense of associativity (whereas we maintained associativity at the expense of commutativity). Personally, I thought that the lack of commutativity was a feature, not a bug :)

I’m happy to check in once in a while.

Take care!
Eric

PS: Speaking of Chern-Simons, do you know what Simons is up to these days? Just in case you might be tempted to consider a career change ;)

Posted by: Eric on October 19, 2005 4:20 AM | Permalink | Reply to this

Re: Synthetic Differential Geometry and Surface Holonomy

Yes, I am very much aware of Anders Kock’s stuff on synthetic differential geometry.

Oh, ok. It took Breen&Messing to make me aware of this jewel.

my friend Robert Kotiuga noted some of its properties in the context of computational electromagnetics.

Cool. Thanks for the link to TWF 107. So you say I should take the time and dig out Kotiuga’s papers? :-)

Let’s see, I want AA to be a ‘group valued’ synthetic form, i.e. something that assigns to an edge |i 0i 1⟩|i_0 i_1\rangle its ‘holonomy to first order’. Then I guess the formula you give should read

(1)⟨A⌣dA|i 0i 1i 2i 3⟩=⟨A|i 0i 1⟩⟨A|i 1i 2⟩⟨A|i 2i 3⟩⟨A|i 1i 3⟩ −1 \langle A \smile dA | i_0 i_1 i_2 i_3 \rangle = \langle A| i_0 i_1 \rangle \langle A| i_1 i_2 \rangle \langle A| i_2 i_3 \rangle \langle A| i_1 i_3 \rangle^{-1}

where all products are products in the group.

Probably the entire synthetic CS form would then amount to multiplying this with

(2)⟨A⌣A⌣A|i 0i 1i 2i 3⟩=⟨A|i 0i 1⟩⟨A|i 1i 2⟩⟨A|i 2i 3⟩⟨A|i 0i 3⟩ −1?? \langle A \smile A \smile A | i_0 i_1 i_2 i_3 \rangle = \langle A| i_0 i_1 \rangle \langle A| i_1 i_2 \rangle \langle A| i_2 i_3 \rangle \langle A| i_0 i_3 \rangle^{-1} \; ??

Hm. I need to think more about this stuff. But no time right now…

He develops a theory of chains and cochains that maintains commutativity at the expense of associativity (whereas we maintained associativity at the expense of commutativity).

Interesting.

Personally, I thought that the lack of commutativity was a feature, not a bug :)

Yes, certainly. So has anyone thought about Hodge star operators in the context of Kock’s synthetic DG? I bet everything we did translates directly into statements about synthetic DG. I need to think more about that. But no time right now… :-)

Speaking of Chern-Simons, do you know what Simons is up to these days? Just in case you might be tempted to consider a career change ;)

I didn’t know. But now I looked it up. Pretty amazing. Me, I still have some way to go towards my first billion bucks.

Posted by: Urs Schreiber on October 19, 2005 11:44 AM | Permalink | Reply to this
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