Integration Without Integration

January 24, 2008

Posted by Urs Schreiber

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In some comments to On Lie $N$ -tegration and Rational Homotopy Theory, starting with this one, I began thinking about defining integration of forms over a manifold in terms of a mere passage to equivalence classes.

There is a big motivation here coming from the observation in Transgression of $n$ -Transport and $n$ -Connection, that fiber integration is automatically induced by hitting transport functors with inner homs.

We want the Lie $∞$ -algebraic version of this, in order to possibly understand how to perform the path integral of a charged $n$ -particle coupled to a Lie $∞$ -algebraic connection as in the last section of $L_{∞}$ -connections and applications to String- and Chern-Simons $n$ -transport (arXiv:0801.3480).

I think I made some progress with understanding this in more detail. I talk about that here:

Integration without Integration (pdf, 6 pages)

Abstract: On how transgression and integration of forms comes from internal homs applied on transport $n$ -functors, on what that looks like after passing to a Lie $∞$ -algebraic description and how it realizes the notion of integration without integration.

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While that is nice, I’d be grateful for further pointers to existing literature on “integration without integration”. I understand that there exist monographs on how to use that within the variational bicomplex of classical mechanics and hence, I suppose, say something about the path integral. But I haven’t yet managed to get my hands on these texts.

Posted at January 24, 2008 9:05 PM UTC

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16 Comments & 3 Trackbacks

Re: Integration without Integration

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Urs wrote:

I’d be grateful for further pointers to existing literature on “integration without integration”.

Two thoughts. First, your slogan reminds me of motivic integration; have you looked that up? There’s a nice paper by Thomas Hales, which as I remember is called “What is motivic measure?” and in the Bulletin of hte AMS.

Second, I found a small result that you might call integration for free (well, not quite free, but at a knock-down bargain price). There might be some connection.

Posted by: Tom Leinster on January 24, 2008 9:44 PM | Permalink | Reply to this

Re: Integration without Integration

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Tom,

thanks a lot for these two pointers. I have looked at them quickly to get an impression.

The talk by you which you point to I like a lot. While I cannot quite see it in detail yet, this clearly has some reseamblance with what I tried to talk about.

The thing about motivic measure leaves me a bit puzzled for the moment, but we’ll see if that changes as time goes by…

It is maybe a sign that you were so quick to post a comment to my entry here (for which I am grateful): it was you and your theory of category cardinality which was very much in the back of my mind when I wrote the above.

You might recall that ever since I understood how your category cardinality is related to colimits ( $\sim$ integrals!) of functors over that category , as you explained here, I was trying to see if that helps to understand “path integrals” in quantum mechanics.

And in fact, apparently it does: it turns out that the the path integral measures both of Dijkgraaf-Witten theory # and of its higher dimensional analog # are precisely the “Leinster measure”, and also one gets the right lattice approximation to the quantum propagator of the free point particle by applying the Leinster measure idea to that situation (#).

While that is nice, it is a long way from understanding true path integrals in non-discrete setups.

But in the continuous cases, there turns out to be lots of nice ( $n$ -)category theory at work, too. In a way, it seems that what physicists invented under the name of BV-formalism is, secretly, a way to talk about measures on Lie $∞$ -groupoids.

After thinking about this for quite a bit, I began seeing a rather nice $n$ -strcutural interpretation for this. I am still lacking one or two puzzle pieces, but the hints in the last section of arXiv:0801.3480 together with the remarks in those slides might give an idea.

Anyway, in that context it became ever more clear that the “path integral” is best understood as a a kind of transgression of Lie $∞$ -algebraic structures.

The discussion of integration of 1-forms in my above notes is supposed to be a warm-up for the full thing.

So while writing these notes, I was thinking to myself: “Very well, that looks nice. But is this still related to your other ideas about path integrals using the Leinster measure, or is this different?”

It looks very different. But maybe there is a relation to be found.

Hopefully.

Posted by: Urs Schreiber on January 24, 2008 10:26 PM | Permalink | Reply to this

Re: Integration without Integration

Here lectures on motivic integration I found very usefull.

Posted by: Thomas Riepe on January 25, 2008 1:53 PM | Permalink | Reply to this

Re: Integration without Integration

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Thanks.

Would anyone like to share his personal way of thinking about what motivic integration is?

Posted by: Urs Schreiber on January 25, 2008 2:22 PM | Permalink | Reply to this

Re: Integration without Integration

I see it his way: Usually, formulas are perceived as algorithms which produce sets, so one wants to integrate those sets by reducing them with disections and recollections to standart shapes. In motivic integration, such sets are perceived as only interpretations of the formulas, so one wants to integrate the formulas themself in an interpretation independent way. It is this interpretation independence, which makes motivic integration interesting. The old way to disect and recollect sets is formalized within suitable scissors relations, so that the integration in question produces data in the universal ring defined by such relations. In case one integrates over varieties, that universal ring, whose elements are the results of integrations, is related to motives, which explains the term “motivic” integration.

Posted by: Thomas Riepe on January 26, 2008 4:55 PM | Permalink | Reply to this

Re: Integration without Integration

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Thomas,

thanks. So it all looks to me that, while very interesting in itself, motovic integration is actually not related to the idea of defining integrals in terms of equivalence classes on differential forms? This is what I had in mind when talking about “integration without integration”.

Or is there a secret connection which I am missing?

(Of course there need not be one. I understand that Tom originally just said that the slogan “integration without integration” reminded him of motivic integration. I am just trying to understand if there is a closer connection than it might seem.)

Posted by: Urs Schreiber on January 28, 2008 9:35 AM | Permalink | Reply to this

Re: Integration without Integration

The only vague idea of how motivic integration could perhaps be connected to integration as eq.-classes of differential forms coming into my mind is that motives resulting from motivic integration have a crystalline representation. But motivic integration produces infinite sums of quotients of motives, because motives are building blocks like unit cubes, with which one can measure point sets only after one allows divided cubes and infinite sums of them.

Posted by: Thomas Riepe on January 28, 2008 7:50 PM | Permalink | Reply to this

Re: Integration without Integration

Just to complete this, here a mention of the very interesting:

Denef, Jan; Loeser, François On some rational generating series occurring in arithmetic geometry.

from the MathSciNet-review:
“..the authors illustrate the following heuristic principle: “Rational generating series occurring in arithmetic geometry are motivic in nature.” What does “motivic in nature” mean in this setting? Philosophically, a series is motivic in nature if we can attach canonically to F another power series, with coefficients in some Grothendieck ring of varieties, or motives, such that every an is the number of rational points of An in some (fixed) finite field…”

Posted by: Thomas Riepe on February 23, 2008 1:34 PM | Permalink | Reply to this

Re: Integration without Integration

All the mental connections that I had between motivic integration and Urs’s integration without integration can be found in Thomas’s answer (though his description is better than anything I could have written). So the similarity is that in both contexts, integration is defined via equivalence relations on the class of things you’re interested in integrating. But that seems to be as far as it goes.

Posted by: Tom Leinster on January 28, 2008 11:36 AM | Permalink | Reply to this

Re: Integration without Integration

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In a different thread, Jim Stasheff asks:

1)Once we start talking about BG I wonder why we need inverses? G a monoid works fine as should a monoidoid (yuch!).

Could we not ask the same question here? In particular if you remember that discussion of homotopy as a kind of truth-valued path integral in changing the rig, does your $G$ need to be a group?

Posted by: David Corfield on January 25, 2008 12:33 PM | Permalink | Reply to this

Re: Integration without Integration

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does your $G$ need to be a group?

It need not in the integral picture. But I don’t really know (and have the impression that nobody does, but see below) how to describe the differential version of smooth $n$ -categories which are not $n$ -groupoids.

Say you have a space $X$ and consider not its path groupoid, but its path category: objects are points, morphisms are paths modulo orientation-preserving diffeomorphisms. So composition of paths is associative, but no nontrivial path has an inverse.

Then hand me a smooth 1-functor from that path category to some smooth category (not necessarily a groupoid) and by applying the internal hom construction which I described, now using $B ℕ$ where I used $B ℤ$ before, I can transgress the thing to the category of loops in $X$ and thereby integrate it (compute its holonomy, really).

So everything goes through as before. Same for higher $n$ .

But if you then ask me to hit all this with some $Lie$ -operation such as will send Lie $∞$ -groupoids to Lie $∞$ -algebras (described by Ševera in arXiv:math/0612349v1) then I don’t know what such an operation would send the non-groupoid Lie $n$ -categories to.

(Well, I have a speculation, but not more, at this point.)

Posted by: Urs Schreiber on January 25, 2008 12:51 PM | Permalink | Reply to this

Re: Integration without Integration

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I have now added a discussion of how to do the Chern-Simons functional by “integration without integration”.

See the new section 3 here.

Couple the 3-particle (membrane) to the canonical Chern-Simons 3-bundle over the classifying space of $G$ , transgress to the configuration space, there look at the characteristic 0-forms. This spits out the Chern-Simons integral automatically.

Posted by: Urs Schreiber on January 25, 2008 10:07 PM | Permalink | Reply to this

Re: Integration without Integration

From a different perspective, integration is the unique nonlocal map between modules of tensor fields which commutes with diffeomorphisms.

Consider for simplicity the algebra vect(n) of vector fields in d dimensions, say acting on formal Laurent series. For each gl(n) module V, we have the coinduced vect(n) module T(V) - tensor fields of type V. T(V) is irreducible, except if it is a form, because then the exterior derivative d is an intertwining operator.

d is the only local homomorphism. However, if we give up locality, there is one more map, namely integration. Since it is a map Ωⁿ -> Ω⁰, it makes the de Rham complex cyclic.

Posted by: Thomas Larsson on January 26, 2008 7:00 AM | Permalink | Reply to this

Re: Integration without Integration

Quick comment about de Rham: if I is the integration operator, then

Id = 0 (Stokes thm)
dI = 0 (gradient of constant fcn = 0)

Posted by: Thomas Larsson on January 26, 2008 9:31 AM | Permalink | Reply to this

Re: Integration without Integration

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Thanks, Thomas. This looks like it is indeed closely related to the mechanism I was talking about.

What I am observing is: one sees that the integration functional $I : Ω^{n} \to Ω^{0}$ arises as constituting that particular 0-form on the space of $n$ -forms which is “basic” (a “characteristic” 0-form, “invariant under vertical derivations”) in the sense of section 5.3.

I think what makes this work is a known mechanism, like you indicate. But it is nice to know that this can be understood as arising from such general principles.

Next I’d like to understand path integrals along these lines. But I am not there yet…

Posted by: Urs Schreiber on January 26, 2008 11:07 AM | Permalink | Reply to this

Re: Integration without Integration

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Quick question from the sidelines:

Integrating a 0-form $ϕ \in Ω^{0} (M)$ over a 0-dimensional domain, i.e. a point, $p$ is just evaluating the function at that point

(1)

\int_{p} ϕ = ϕ (p)

As we discussed here (or was it the Coffee Table?) ages ago, evaluating a 0-form $ϕ \in Ω^{0} (LM)$ over a point $p \in LM$ on loop space (which is a loop $L_{*} p$ in $M$ ) is again just evaluation, but

(2)

\int_{p} ϕ = ϕ (p) = \int_{L_{*} p} α,

where $α$ is a 1-form on $M$ and we can think of the 0-form $ϕ$ as the pull-back of $α$ (or something like that) to loop space

(3)

ϕ = L^{*} α .

Is that at all related to what is going on here?

As an aside, I love this discussion. I think of integration as more than just a nice toy that lets you compute stuff, but is rather quite fundamental to physical measurement. We can talk about fields, currents, or whatever, all day long, but in the end, something has to be measured. That measurement involves projecting, or integrating/measuring, those fields to a number. In other words, integration and measurement are somewhat interchangable concepts.

Cheers,

Eric

Posted by: Eric on January 26, 2008 7:35 PM | Permalink | Reply to this

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