## March 8, 2007

### Canonical Measures on Configuration Spaces

#### Posted by Urs Schreiber

Quantum theory crucially involves - and maybe is all about - subtle summations over all “histories” – or all “configurations” – of a physical system: the path integral.

The spaces of all these histories, or these configurations, usually carry a natural category structure on them:

Distinct but physically indistinguishable configurations are related by isomorphisms addressed as gauge transformations.

Even though the categorical (or groupoid) nature of configuration spaces is often not made explicit, dealing with the path integral in the presence of such gauge isomorphisms is a topic that occupies physicists a lot, as is evidenced by the large number of surname initials that go into the names of the theories handling these: BRST, FV, BV, BFV-BRST.

The entire kinematics and dynamics of quantum theory can nicely be conceived in terms of natural push-forward or colimit operations involving the category of configurations – an observation apparently going back to Dan Freed.

Or almost. Curiously, it appears as if the highest dimensional level of a quantum theory always requires a non-canonical choice: a measure on the space of histories.

While on the one hand this measure seems to stand out from the rest of the structure in its non-canonicalness, it is at the same time the source of much of the subtlety – and the richness – of quantum theory. Not the least, the lack of a rigorous handle on this measure, in general, is what kept and keeps large parts of physics from being accessible, and being accessed, by mathematically acceptable methods.

One may try to accept this as a sad fact of life – or one might take it as a hint for a hidden structure that still needs to be properly identified.

This post here is for those daring $n$-Café readers who are prepared to not shy away from attempting to chase and hunt down natural abstract structures that might possibly, and secretly, govern our current theories of this world. Or that might not.

I’ll try to summarize and wrap up some of the things that we talked about in the long discussion going on in Isham on Arrow Fields. It’s not that I think this discussion has reached a saturation point where everything interesting has been mentioned and just needs to be archived. Quite the opposite. I expect that there might be some really interesting aspects lurking right around the corner. But I feel that it would be helpful to draw a coherent picture of what we were discussing so far.

The obvious first step is to be serious about configuration spaces, and spaces of histories, as really being categories, whose morphisms encode gauge equivalences or other relations.

Chris Isham coined the phrase quantization on a category for this perspective. I want to emphasize, though, that we don’t want to talk about another notion of quantization. Rather this is supposed to refine our understanding of ordinary quantization. The refinement lies in the fact that we make explicit a structure that is otherwise often swept under the carpet, but is present nonetheless: a morphism space over the space of configurations, or of histories.

We had been discussing this point of view from various angles here at $n$-Café. Parallel to that, and seemingly unrelated, a lot of interest revolved around a Café table where discussion was about a curious function, called a weighting on a category, that Tom Leinster found was canonically associated to the spaces of objects of certain (finite) categories: its sum over all objects gives the Euler characteristic of a category.

It was Bruce Bartlett who suggested that this weighting is in fact the canonical measure to use on categories of configurations of physical systems when doing the path integral.

Bruce wrote:

[Chris Isham] mentions that “it is necessary to develop a proper measure on the set [of histories]”.

Indeed! That is precisely what all of us at the $n$-cafe were talking about in the discussion of Tom Leinster’s Euler characteristic of categories.

I am not sure if anybody else from “all of us” actually had made that connection. But Bruce is an expert on Dijkgraaf-Witten theory. So he knew the following fact:

How Dijkgraaf-Witten theory is governed by the Leinster-measure

In Dijkgraaf-Witten theory, the “worldvolume” is the fundamental groupoid $\mathrm{worldvol} = \Pi_1(M) \,,$ of some manifold $M$.

Target space is a finite group $G$, regarded as a category $\mathrm{tar} = \Sigma G$ with a single object.

The space of histories $\mathrm{hist} = [\mathrm{worldvol},\mathrm{tar}]$ is the category of functors from the fundamental groupoid into the gauge group. Each such functor can be thought of as representing a $G$-bundle with connection on $M$. Since, for $G$ finite, there is really only a unique connection on every such bundle, this is the same here as just $G$ bundles on $M$.

The morphisms in $\mathrm{hist}$ are natural transformations between these functors. These describe gauge transformations relating the respective bundles.

In particular, the the space of histories, $\mathrm{hist}$, here happens to be a groupoid!

And it is equivalent to a finite groupoid. For instance, we can simply replace the fundamental groupoid of $M$ with its skeleton, which is then nothing but a disjoint union of (suspended) fundamental (finite) groups, one for each connected component. (This works here, but not in general gauge theory, because there is no smoothness condition or anything like that. All categories and functors are internal just to $\mathrm{Set}$ and so every connected groupoid is equivalent to any of its vertex groups.)

As a finite groupoid, its Leinster measure $d\mu$ is defined and coincides with the Baez-Dolan prescription: $d\mu : x \mapsto \frac{1}{|s^{-1}(x)|} \,,$ where the denominator is the number of morphism with source $d\mu$.

Moreover, our groupoid is equivalent to its skeleton, which is just a disjoint union of 1-object groupoids. On this skeleton, the above measure reduces to $d\mu : x \mapsto \frac{1}{|\mathrm{Aut}(x)|} \,,$

This measure is indeed precisely the measure that enters the path integral (which here is just a finite sum) in Dijkgraaf-Witten theory.

You can find this, for instance, in equation (2.1) on p. 9 of

Daniel S. Freed, Frank Quinn, Chern-Simons Theory with Finite Gauge Group

or in equation (4.25) and (4.31) on p. 68 of

Bruce H. Bartlett, Categorical Aspects of Topological Quantum Field Theories.

How integrals of set-valued functors are governed by the Leinster measure

Bruce went on to claim that

When [the category of histories] is a poset, it’s also well known what measure to use (see Tom Leinster’s paper).

This clearly involves a generalized notion of “well known”, it seems. In any case, when I first read this statement I was sceptical. Similar reservations were later also voiced by Eric Forgy # and Tim Silverman #.

But then I was converted from a sceptic to a true believer. This happened after I went back to Tom Leinster’s explanation of the way he found his weighting. Suddenly everything seemed to fall into place, and and rather beautiful general picture suggested itself.

So consider this:

While summation of numbers is something invented by man, direct summation of sets is something given to us from the gods.

Better yet, while integration of functions is something invented by man, integrals (colimits) of set-valued functors is something given us from the gods.

Given a category $X$, and a functor $F : X \to \mathrm{FinSet}$, the colimit $\mathrm{colim}_X F$ of $F$ over $X$ is, heuristically, the disjoint union of all the sets associated by $F$ to all the objects of $X$, divided out by all the identitfications implied by the value of $F$ on the morphisms of $X$.

One way to think of the natural numbers is as names of isomorphism classes of finite sets. In fact, in many texts on foundations, this is the way to think of natural numbers.

So let’s do that. If we have a finite set $S$, its equivalence class is its cardinality $|S|$. What is the equivalence class of the colimit of a set-valued functor, i.e. what is $\int_X F := |\mathrm{colim}_X F| \;\;\;?$

There is maybe no general answer to that. But when $F$ acts freely in a precise sense (and if $X$ is finite) then, Tom Leinster found, there is (see his paper for the detailed conditions and qualifications) a function $d\mu : \mathrm{Obj}(X) \to \mathbb{R}$ such that $\int_X F = \sum_{x \in \mathrm{Obj}(X)} |F(x)| \; d\mu(x) \,.$

This function is precisely that weighting on the category $X$.

In other words: if we were only willing to accept that (natural) numbers are really labels of sets (and really they are!), and that number-valued functions are then really decategorifications of set-valued functors – then we are lead to regard the Leinster measure on a category as a canonical measure induced on a space of objects by a space of morphisms over these.

In fact, there are other indications that we would ultimately need to do precisely that: regard the numbers that appear in quantum theory as placeholders for sets. While I don’t want to get into that right now, it does give me additional impetus to take this connection serious.

How to test if the Leinster measure on configuration categories possibly governs the path integral

Given that, I am prepared to speculate: maybe the correct identification of the spaces of morphisms over the spaces of configurations that we encounter in physics will induce, via the weighting on that category, the right measure for the path integral over these.

We already know that this is the case for Dijkgraaf-Witten theory.

It would be good to test this in other examples.

One problem is that, at this point, the Leinster measure is only understood, and defined, for finite categories.

This means that if we want to check our speculation on ordinary quantum mechanical systems, like the free quantum particle – and this is what I would like to do – we need to find suitable discrete approximations to their spaces of configurations.

So I posed myself the following

Excercise: Find a finite category, $\mathrm{tar}$, that decently models a $d$-dimensional pseudo-Riemannian spacetime. Find another finite category, $\mathrm{par}$, that decently models the abstract worldline of a relativistic particle. Then determine the Leinster measure on the category of histories $\mathrm{hist} = [\mathrm{par},\mathrm{tar}] \,.$ thought of as a model for the category of paths traced out by the particle in target space.

This measure – or rather a suitable limit of it as we take the size of our categories to tend to infinity – would be the canonical candidate for the measure that enters the path integral of the relativistic particle. I would like to check if it comes anywhere close to the expected result.

First observations on possible models of pseudo-Riemannian spacetimes by finite categories

The exercise posed is vague. It is one of those typical vague problems one encounters in physics, of the kind: “find a good mathematical model for a given physical situation”.

On the other hand, some people have thought hard about pretty much this exercise before, in a very related context, but without the light of the Leinster measure to guide them.

Pseudo-Riemannian structures of signature $(-++\cdots+)$ have a very notable property: they are, under mild conditions, equivalent to light cone structure with measure:

if, on some manifold, you know at every point all tangent vectors that are lightlike, and at the same time know the volume density at that point, then you can reconstruct the corresponding peudo-Riemannian metric.

In the general context of efforts of coming to grips with quantum gravity, concerned with understanding the nature of the general relativistic theory of gravity in light of the fact that physics is, fundamentally, quantum physics, it was mainly Rafael Sorkin who emphasized that this equivalent reformulation of pseudo-Riemannian structures might be helpful:

Rafael Sorkin, Causal Sets: Discrete Gravity .

The fact that a light cone structure introduces the structure of a poset on the set of spacetime points brought categories into the picture. On the other hand, my impression is that the suggestion that we think of spacetime – in its incarnation as the configuration space of the single particle – as a category, has not been seriously thought to its end yet.

In any case, motivated by these ideas, I suggested (after some trial and error) to approach the above exercise as follows:

Suppose we model a fixed pseudo-Riemannian spacetime in terms a finite category by interpreting the space of objects as a finite approximation to the collection of spacetime points, by interpreting the morphisms as future-pointing light-like paths between these points (thus as defining a light-cone structure) and by interpreting the Leinster-measure on this category as the volume density on this spacetime: which categories will correspond to which pseudo-Riemannian structures?

A large class of examples of suitable finite categories comes from categories freely generated from finite directed graphs $(V,E)$ without loops. The vertices of these graphs would model the spacetime points, while the edges would model the infinitesimal lightlike paths between these.

It seems that I finally convinced myself of the following #

Proposition: The Leinster measure $d\mu$ on the category $X$ freely generated from a finite directed graph $(V,E)$ without loops is $d\mu : x \mapsto 1 - |s^{-1}(x)| \,.$ Here the second term denotes the number of edges in $E$ emanating from the vertex $x$.

Proof: Use induction. The statement is clearly true when the vertex has no outgoing edges. So suppose the vertex has a positive number of outgoing edges, and for all vertices reachable by following paths of emanating edges the statement has already been shown. Then it follows that the entire tree of edges sitting above each of the edges in $s^{-1}(x)$ contributes one unit. This implies the statement for the vertex $x$.

This is potentially helpful, in that the classifying spaces of these categories are nothing but (the union of the edges of) the graphs they come from. See the last paragraph on “interpretation” for a comment on that.

A couple of simple examples

The Leinster measure on the category freely generated from the graph $\array{ (n,m) &\to& (n+1,m) &\to& (n+2,m) &\to& (n+3,m) &\to& (n+4,m) \\ \downarrow && \downarrow && \downarrow && \downarrow \\ (n,m+1) &\to& (n+1,m+1) &\to& (n+2,m+1) &\to& (n+3,m+1) \\ \downarrow && \downarrow && \downarrow \\ (n,m+2) &\to& (n+1,m+2) &\to& (n+2,m+2) \\ \downarrow && \downarrow \\ (n,m+3) &\to& (n+1,m+3) \\ \downarrow \\ (n,m+4) }$ is $\array{ -1 &\to& -1 &\to& -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow \\ -1 &\to& 1 \\ \downarrow \\ 1 }$

Maybe more interesting than the graph of the above shape, for the applications that we (that is: myself and whoever wants to join in) have in mind, is a Lorentzian cylinder, i.e. piece of a cylindrical 2-dimensional Minkowski spacetime:

$\array{ && && (n,m) &\to& \\ && && \downarrow && \downarrow \\ && (n+1,m-1) &\to& (n+2,m-1) &\to& (n+1,m+1) &\to& \\ && \downarrow && \downarrow && \downarrow \\ (n,m) &\to& (n+1,m) &\to& (n+2,m) &\to& (n+3,m) &\to& (n+2,m+2) \\ \downarrow && \downarrow && \downarrow && \downarrow \\ &\to& (n+1,m+1) &\to& (n+2,m+1) &\to& (n+3,m+1) \\ && \downarrow && \downarrow \\ && && (n+2,m+2) }$ Notice the periodic identification. The Leinster-measure of (the category generated by) this is still of the now familiar form, -1 everywhere except for the future boundary: $\array{ && && -1 &\to& \\ && && \downarrow && \downarrow \\ && -1 &\to& -1 &\to& -1 &\to& \\ && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -1 &\to& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \downarrow \\ &\to& -1 &\to& -1 &\to& 1 \\ && \downarrow && \downarrow \\ && && 1 } \,.$

So this Minkowski 2-cylinder has proper volume -6.

Next, consider our 2-diamond cylinder from before #, but now at one spot I have added in just one more edge (the two edges at this point now indicated by a double arrow $\Downarrow$). Here is the resulting graph with its Leinster measure $\array{ && && -1 &\to& \\ && && \downarrow && \downarrow \\ && -1 &\to& -1 &\to& -1 &\to& \\ && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -1 &\to& -2 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \Downarrow \\ &\to& -1 &\to& -1 &\to& 1 \\ && \downarrow && \downarrow \\ && && 1 } \,.$

This remains true for arbitrary insertions of an additional edge in the interior: it only affects the Leinster meausure at its source vertex, nowhere else: $\array{ && && -1 &\to& \\ && && \downarrow && \downarrow \\ && -1 &\to& -1 &\to& -1 &\to& \\ && \downarrow && \downarrow && \downarrow \\ -1 &\to& -1 &\to& -2 &\Rightarrow& -1 &\to& 1 \\ \downarrow && \downarrow && \downarrow && \downarrow \\ &\to& -1 &\to& -1 &\to& 1 \\ && \downarrow && \downarrow \\ && && 1 } \,.$

And now, a gravitational Leinster-Sorkin wave propagating on a 2-dimensional cylindrical spacetime:

$\array{ && && -2 &\to& \\ && && \Downarrow && \downarrow \\ && -1 &\to& -2 &\to& -1 &\to& \\ && \downarrow && \Downarrow && \downarrow \\ -2 &\Rightarrow & -2 &\Rightarrow& -3 &\Rightarrow& -2 &\Rightarrow& 1 \\ \Downarrow && \downarrow && \Downarrow && \downarrow \\ &\to& -1 &\to& -2 &\to& 1 \\ && \downarrow && \Downarrow \\ && && 1 } \,.$ It originates in a pointlike perturbation at the incoming boundary, that produces two wave crests which propagate at the speed of light in both spatial directions. Both run around the spacelike circle once, pass right through each other once and then hit the outgoing boundary.

Apart from toying around with such examples that can explicitly written down on the back of an envelope, it would be interesting to have a general procedure that takes a globally hyperbolic spacetime of the form $M = \Sigma \times \mathbb{R}$ with $\Sigma$ a Riemannian space.

Eric Forgy has thought about this problem. He presents a way to obtain from a triangulation of $\Sigma$ a light-cone graph on $\Sigma \times \mathbb{R}$ in a systematic way.

Eric Forgy: Light Cone Graphs from Spatial Triangulations.

We should eventually see if we can better understand the volume measure that this procedure induces on $M$.

Interpretation of these models

One might want to pause for a moment and reflect on precisely how these examples model what we would like them to model.

I think they do give a consistent picture, if we accept that for these categories to model what we want them to model, the above is telling us that what looks like pseudo-Riemannian volume on large scales is the accumulation of lots of topological defects on tiny scales.

In a way, this is an old quantum-gravitist’s dream: realize flat space(time) on small scales as a very non-classical object which lots of curvature and lots of topological funny stuff going on, such that on sufficiently large scales it looks just like ordinary space.

The fun thing is: I am not dreaming this up! I am just pointing out that given the single one proviso that we accept that it is good to think of integer-valued functions on a graph as actually set-valued functors, then this interpretation forces itself upon us. This is just how such large graphs do appear to us on small and on large scales.

Now try to complete the above exercise

Yes, let’s try to do that next…

(But not right now. Local time here is after 2 AM, and counting. In three hours a taxi will (hopefully) pick me up and bring me to the train which will bring me to the bus which will bring me to the plane which will bring me to the train that will bring me back home. Hopefully the above account is not too affected by my level of bleary-eyedness. Any polishing will have to wait until that journey is over.)

Posted at March 8, 2007 12:55 AM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1194

### Re: Canonical Measures on Configuration Spaces

Man, Urs. Bleary-eyed or not, this is the most beautiful piece of scientific thinking I have ever seen. I’m happy just to be able to say I know you :) In some wishful fantasy world, I would like to think I nudged you in one or two right directions along the way even though I never possessed the skills to go where you have gone. Time spent batting ideas around with you has been the highlight of my humble scientific career. It is certainly goofy, but my eyes are a bit moist just thinking about the beauty of this. And to think, I remember back when you were just learning differential forms :)

Ok. Enough gushing :)

It would be good to test this in other examples.

Quantum mechanics is certainly an obvious place to start, but there is perhaps an even easier testing group that might be a natural next step. That is “Wick rotated” QM, a.k.a. Brownian motion. We already know that a binary tree provides a nice approximation to Brownian motion (and stochastic calculus via discrete calculus of the binary tree).

I could tell you were onto something deep when I spewed the statement here. I know the steps haven’t been completed, but the path seems pretty clear to me (although I could never take the steps myself, at least I can see them).

Fantastic!

Posted by: Eric on March 8, 2007 5:30 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Wow - you guys have been busy! Somehow I missed all of this stuff where you perform all these explicit calculations on diamond graphs.

I’m very frustrated right now - I can’t join in the fun since my head refuses to grasp the concept of what precisely a “diamond graph” is!

I looked at Eric’s notes… lovely pictures! But right at the end it says “I claim that we have just bounded a 3-diamond”. Can you guys explain what a diamond graph is one more time?

I think my confusion is intertwined with the following : the principle is that objects are spacetime points, and morphisms are paths which lie on the light cone… not just future pointing paths. Is that right?

Anyhow, I like the “gravitational Leinster-Sorkin” wave :-)

(a) Amusingly, just yesterday I got embroiled in some Euler characteristic discussions - someone explained to me about the way Witten calculates the Euler characteristic of an orbifold. You can find it in these papers from the 80’s : Strings on Orbifolds I and Strings on Orbifolds II . By the way, recall that the Dijkgraaf-Witten model is an orbifold model, where $G$ is a finite group.

Anyhow, Simon then explained to me that Witten’s Euler characteristic of an orbifold $M/G$ (is that how you write it?) was explained by Segal and Atiyah to just be the Euler characteristic of equivariant $K$-theory:

(1)$\chi(M/G) = \sum_i (-1)^i K_G^i (M).$

Anyhow, I’m bringing this up in the hope that n-cafe patrons can incorporate this into the grand scheme, i.e. it might provide a way to extend this stuff from finite category type things like the Dijkgraaf-Witten model to more conventional quantum field theories. At the very least it involves the words “Euler characteristic” and “quantum field theory” - thus being a prime target for n-cafe patrons.

(b) Let me see if I understand the bigger picture : We want to complete Urs’s exercise above. So far it seems one can construct the “target space” side of things. How are we going to get a measure on $hist = [par, tar]$… I’m worried about that problem of the presence of terminal objects in $hist$ making things trivial.

(c) I’m confused about the analogy between finite models for spacetime, and continuous spacetime. In the latter, you need the poset structure and a measure, in order to recover the spacetime metric. These two pieces of data are independent. But in finite models, somehow one can derive the measure from the poset structure… what’s going on?

Posted by: Bruce Bartlett on March 8, 2007 4:38 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Woops… I mucked up what I said above about orbifolds and the Euler characteristic (that’s because what I know about orbifolds can be written on a postage stamp!). Anyway, I think its nice, so let me say it again.

Witten, Harvey, Vafa and Dixon, in their “Strings on Orbifolds I and II “papers in the 80’s, defined the Euler characteristic of an orbifold (i.e. of a pair $(M, G)$ where $G$ is a finite group which acts on $M$). It’s easy : you let $M_{g,h} \subset M$ be equal to the intersection of the fixed points of $g, h \in G$:

(1)$M_{g,h} := Fix(g) \cap Fix(h)$

And then you just take the sum of the ordinary Euler characteristics of the $M_{g,h}$ over the commuting pairs in the group, weighted by the order of the group:

(2)$\chi (M, G) := \frac{1}{|G|} \sum_{gh = hg} \chi(M_{g,h}).$

Then Atiyah and Segal came along and showed that this is ‘just’ the Euler characteristic of equivariant K-theory:

(3)$\chi(M, G) = rank K_G^0 (M) - rank K_G^1 (M).$

Then the 90’s came and went… and the 00’s came, and John Baez and James Dolan taught us how to calculate the cardinality of groupoids. Then Tom Leinster came along and taught us how to calculate the cardinality of a general finite category - which one calls taking the Euler characteristic of the category.

What would be nice would be if someone could show how what Witten et. al were calling the “Euler characteristic of an orbifold” in the 80’s is a special case of Tom and John and James’ paradigm. Possibly one of these last three have already done this.

Posted by: Bruce Bartlett on March 8, 2007 5:17 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Oops, one more thing. Urs wrote above that “Bruce is an expert on Dijkgraaf-Witten theory”. Trust me, its not the case. Everything I wrote down, in my master’s thesis, for example, was a shameless steal from John Baez’s quantum gravity seminar, where I learnt almost everything I know now. Luckily for me, John’s unique viewpoint on TQFT’s and higher categories was relatively unknown to many people, which allowed me to get away with just “explaining it again”, in some sense.

Many other people have also been inspired by the QG seminar - Jeffrey Morton, for instance, also works with all this stuff.

He “cheated”, though… he actually attended the QG seminar!

Right now, I see some interesting things are happening in the course on “Quantization and Cohomology Week 17”. Alas! Its impossible to keep up with all these ideas .

Posted by: Bruce Bartlett on March 8, 2007 6:25 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Bruce wrote:

He “cheated”, though… he actually attended the QG seminar!

Yes, I suppose that gave him an unfair advantage. But you did very nicely yourself without cheating.

More seriously:

Jeffrey Morton is doing a wonderful thesis, which will construct the Dijkgraaf–Witten model as an extended TQFT, or more precisely, a 2-functor

$Z: n Cob_2 \to 2Vect$

where $nCob_2$ is the weak 2-category of

• compact $(n-2)$-manifolds
• compact $(n-1)$-manifolds with boundary going between these
• compact $n$-manifolds with corners going between those.

and $2Vect$ is essentially Kapranov and Voevodsky’s 2-category of

• 2-vector spaces
• linear functors between these
• natural transformations between those.

And, the really cool thing — as far as the current discussion goes — is that the path integrals defining this field theory rely heavily on groupoid cardinality, spans of groupoids, and a ‘pull-push’ construction. Everything that Urs loves!

All this is a natural followup of Jeff’s sadly neglected paper on categorified algebra and Feynman diagrams, which uses groupoid cardinality to understand the symmetry factors showing up in Feynman diagrams.

So, it’s very fascinating to imagine groupoid cardinality — or more generally the Euler–Leinster characteristic of a category — as the basis of all measures in path integrals.

Certainly we need some revolutionary insight to understand what’s really doing on with path integrals. It would be nice if this were part of that needed insight.

Right now, I see some interesting things are happening in the course on “Quantization and Cohomology Week 17”. Alas! Its impossible to keep up with all these ideas .

Yes, it’d be a full-time job for anyone to really keep up with all the ideas on this blog!

I know this by experience. I’m so busy with teaching 3 courses, serving on 2 hiring committees, selecting grad students for next year, and meeting with 5 grad students, James Dolan, Danny Stevenson and Alissa Crans that I don’t have time to keep up with the ideas Urs is generating!

Luckily, there’s a huge overlap between what he’s doing and what I’m doing. In fact, these days, when it comes to quantum field theory, my ideas seem to be a proper subset of his. So, you can just read his stuff. But if you read mine, you might see some things explained a bit differently. There’s a big elephant here, and it will take lots of wise men to describe it.

What would be nice would be if someone could show how what Witten et al were calling the “Euler characteristic of an orbifold” in the 80’s is a special case of Tom and John and James’ paradigm. Possibly one of these last three have already done this.

Yeah — check out the abstract of Tom’s paper.

Posted by: John Baez on March 9, 2007 1:36 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

and a ‘pull-push’ construction. Everything that Urs loves!

Is that the same beloved pull-push as is going on here? If they can have push-pull too, why can’t we?

Posted by: David Corfield on March 9, 2007 9:01 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Hi Bruce,

Urs is much better for explaining this, but I’ll give it a shot. Perhaps I can save him some typing with my efforts, so he’ll only need to correct my errors :)

A couple years ago I somehow managed to entice some interest from Urs in some stuff I was working on by telling him that I could derive stochastic calculus from commutative relations between zero forms and one forms. I knew Urs had some interest in stochastic calculus from some of Nelson’s stuff he had read.

After sending him some reference to material by Dimakis and Mueller-Hoissen, we learned that given a directed graph, it induces special commutative relations. Since a special commutative relation, namely

[dx,x] = dt

gives rise to stochastic calculus, we asked what kind of directed graph would give rise to that commutative relation. He subsequently left on a bike ride and when he came back, we had both independently found the answer. A binary tree gives rise to the commutative relations that give rise to stochastic calculus. A binary tree is essentially a square grid turned at 45 degrees so that two edges emanate from each node and two edge are incident into each node.

In hindsight, it was obvious. A binary tree is known to represent a random walk and a random walk is what drives stochastic processes.

Encouraged by the success, we went on to try to build a complete discrete differential geometry. It didn’t take Urs long to discover that if we wanted our directed graph to somehow represent an n-dimensional manifold, it would have to have exactly n edges emanating from and into each node.

At first, I didn’t like this result. For example, if we wanted to represent $R^3$, then we would need 3 edges emanating from and directed into each node. If you “discretize” $R^3$ via a cubic grid, then since it is a directed graph, this implies a preferred direction. I didn’t like this at all. Imagine a cubic grid and assign all arrows in the direction of increasing x, y, and z. This grid has a preference toward the positive direction along along three axes.

That is when we (or at least I, Urs probably knew already) that there really was a preferred direction. Only it wasn’t a spatial dimension. It was time. We can think of the time direction as directed along the diagonal of the cube. This way, there is no preferred spatial direction and some symmetry is restored.

From then, instead of think of the manifolds as n-dimensional, we started thinking of them as (n+1)-dimension, i.e. the notation implies that n dimensions were spatial and one was temporal.

In other words, a directed graph of this sort with n directed edges away from and into each node represented a manifold of the form $R\times M$ where $M$ is an (n-1)-dimensional manifold.

The edges can now be thought of as being light like. It is as if each node has n rays of light emanating from it and n rays of light coming into it.

If you take one node that is in the future of another node and intersect the future light cone (pyramid really in 3d) of the past node and the past light cone (pyramid in 3d) of the future node, you get an example of something that we call an “n-diamond” (or “3-diamond”).

An n-diamond graph is any such directed graph such that each node has exactly n edges directed into and away from each node. I’d probably also through in something about the fact the the directed edges must define some “flow of time”.

The highest dimensional cell of an n-diamond graph is obvious referred to as an n-diamond, but also each m-cell for each m < n is also an m-diamond.

The simplest example of an n-diamond graph is a n-cubic graph tilted up along its diagonal. The diagonal direction then represents time.

Ok. I probably confused you more than helped, but hopefully Urs can clear up thing :O

In our paper, Urs presented just the simplest example of a directed n-cubic grid and called it an n-diamond graph. This made me cringe a little because anyone attempting to read the paper might think the whole machinery was constrained to such n-cubic graphs, but my attempts to rework the material in a way even I could understand got continually delayed until we decided to just submit what we had.

Those little pdf notes of mine that Urs just posted represent a recent attempt I made at proving a hunch I’ve had all along that any manifold of the form $R\times M$ can be “diamonated”. Proving this would show that any such manifold is amenable to a study via discrete differential geometry.

By the very nature of the construction of a diamond graph via light rays, the connection to Sorkin’s poset stuff seems obvious to me (which is why I called it “Discrete Differential Geometry on Causal Grpahs”). Now that Urs is making progress with stuff like “canonical measures”, this, to a certain extent justifies the assumption the poset guys make that the measure is just “counting”.

Now when you throw in the quantization of categories, the fact that a diamond is a category, and the relation to causet stuff, it all paints a really pretty picture that Urs is in the process of working out.

Then again, I’m probably just confused :)

Gotta run! Apologies for typos and bad grammar! :)

Posted by: Eric on March 9, 2007 12:38 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Hi Eric,

Thanks for this explanation. I think I better understand now what a $n$-diamond graph is! So an $n$-diamond graph is a directed graph where each node has $n$ edges coming and out of it, and we should think of the “direction” as time. I’m confused about one thing : in an $n$-cubic graph, when you tilt it forty five degrees, it seems that the “top” and “bottom” points have only edges going in or out of them, and not vice versa. What’s going on?

Posted by: Bruce Bartlett on March 11, 2007 12:45 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Hi Bruce,

I’m glad that you seem to have similar remarkable skills at making some sense out of my incoherent explanations :)

If I understood your question, it might help to think of a simple example. Maybe imagine a square grid consisting of 3 rows and 3 columns for a total of 9 squares aligned along an x and y axes. Now, assign a direction to each edge along the positive x and y directions. Then time will be flowing at a 45 degree angle along the diagonal of the squares.

As usual, I wasn’t very precise in my description because only the center square truly satisfies the condition that there are 2 edges directed into and out of each edge.

Still, I would call this a 2-diamond graph.

How can I fixed the description? Urs help! :)

The thing about this example is that it has a “beginning” and an “end” and contains a “boundary” in between. I’m tempted to describe these as the beginning, i.e. big bang, the end, i.e. big crush, and the boundary of this toy universe.

Also, we can say that this 2-diamond graph represents the intersection of the future of the earliest node with the past of the latest node.

To come up with a better definition of an n-diamond graph, it might require borrowing something from the definition of a manifold with boundary. I think what I described is the equivalent of a “manifold without boundary”.

Anyway, I think you understand what an n-diamond graph is now regardless of how poorly I defined it :)

The neat thing, in my opinion, of having an n-diamond graph laying around is that it allows you to define a discrete calculus on it. If my wild conjecture is true (which I still lack the time and skills to determine one way or another) about any manifold of the form $R\times M$ allowing a diamond approximation, then that would mean that what Urs and I did is more general than maybe we originally thought (although my gut always told me it should be so).

I hope this helps.

Eric

Posted by: Eric on March 11, 2007 2:25 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

the principle is that objects are spacetime points, and morphisms are paths which lie on the light cone… not just future pointing paths. Is that right?

Yes. Somewhere in the discussion we decided that posets whose objects are spacetime points and whose morphism encode the relation “is in the future of” are not quite what we are looking for. The reason is that they have a “pathological” Leinster measure, from the point of view of interpreting that measure as that induced by a pseudo-Riemannian metric on the space of objects.

Then we discovered that “diamond graphs” (those whose interior looks like the obvious directed graph associated with a lattice of the form $\mathbb{Z}^n$) don’t have this problem: they come with a nicely homogeneous Leinster measure.

These graphs can still be used to encode (on the average over many objects) a pseudo-Riemannian structure, but now, as you say, a single edge (of the graph) would always be interpreted as light-like.

A typical morphism in the category freely generated from these graphs would be, microscopically, always light-like, but with many kinks where the spatial direction of propagation changes. On a somewhat coarse-grained level, we’d see time-like paths. But we retain more information than in a mere causal poset: the category does not just remember which points are in the future of which other points, but it also remembers all possible paths realizing that relation.

Euler characteristic of an orbifold

Thanks a lot for making this connection! We should try to see if there are any hints what kind of structure generalizes the Leinster measure from finite to non-finite categories.

We want to complete Urs’s exercise above.

Yes, let’s do that! It may involve more trial-and-error. As with the passage from posets to free graph categories, our initial guess about what the right categories involved are may be wrong. The formlism will tell us if so.

So far it seems one can construct the “target space” side of things. How are we going to get a measure on $\mathrm{hist} = [\mathrm{par},\mathrm{tar}]$.

Just a slight remark on terminology: a “history” (path in spacetime in the present case) should be a map from a “worldvolume” (the worldline, in our case) into “target space” (a graph category in our case). The (incoming and outgoing) boundary components of the worldvolume we address as “parameter spaces” – at least that’s the convention I was using (hopefully consistently).

So, I would write $\mathrm{hist} = [\mathrm{worldvol},\mathrm{tar}] \,.$

I’m worried about that problem of the presence of terminal objects in hist making things trivial.

That problem has vanished, now that target space is no longer a poset!

In fact, I think I can now say what this measure is, at least for one obvious choice of $\mathrm{worldvol}$.

Proposition (measure on $\mathrm{hist}$):

Let $1 \to 2 \to \cdots \to n$ be the category freely generated from the indicated graph.

Let $\mathrm{tar}$ be a category freely generated from a finite circuit-free graph and let $\mathrm{hist}^x := [\mathrm{worldvol}_n,\mathrm{tar}]$ be the full sub-category of $[\mathrm{worldvol}_n,\mathrm{tar}]$ whose objects are such that $n \in \mathrm{wordlvol}_n$ maps to $x \in \mathrm{tar}$.

Then the Leinster-measure on a given path $\gamma : \mathrm{worldvol}_n \to \mathrm{tar}$ is $1 - (n-n_{\mathrm{const}}(\gamma)) \,,$ where $n_{\mathrm{const}}(\gamma))$ is the number of objects of $\mathrm{worldvol}$ that have the same image under $\gamma$ as their successor does.

Proof: Well, I haven’t yet written that down in a clean way, but here is how I think it works:

Since target is a graph category, the only natural transformations between paths in that target consist of translations of points on that path along that path.

In fact, I think the category of these functors and their transformations is itself a category freely generated from a circuit-free graph: the generating edges emanating from a given functor are those natural transformations which have the identity component everywhere except at one object $m \in \mathrm{worldvol}_n$, where the component is one of the edges generating $\mathrm{tar}$.

Such transformations can only exists if the successor $m+1$ of $m$ has an image in thge future of $m$.

Together with the general formula for Leinster measures on categories generated from finite circuit-free graphs that I stated in the above entry, this implies the above statement.

Now, what is that telling us? Does this reproduce the measure that we expect to see?

1)

Let’s first recall that our choice for $\mathrm{worldvol}_n$ may be unsuitable. The category $1 \to \cdots \to n$ does have a terminal object. Hence, as a model for a measure space, it does not behave like $[0,n] \subset \mathbb{R}$, with the standard measure on it.

I am unsure about this point. The worldline of a relativistic particle does not really need any measure on it. So maybe we should just ignore the fact that $\mathrm{worldvol}_n$ has a “pathological” Leinster measure.

Or maybe this is telling us that we need another model for $\mathrm{worldvol}$.

2)

Before deciding whether the Leinster measure on $\mathrm{hist}$, as found above (and assuming that I determined that correctly), does or does not reproduce – in a suitable approximative sense – the measure on relativistic trajectories that we would like to see, we need to make sure we understand to which degree we need to average and coarse-grain.

As I mentioned above, at the microscopic level all trajectories in $\mathrm{tar}$ are lightlike. So there is no point in trying to weight them by their proper length at this microscopic level.

Or, put the other way around, on the microscopic level the above-found Leinster measure on $\mathrm{hist}$ – which is essentially constant on all paths, and in any case makes no reference to proper length – is quite consistent with what we expect in terms of our model.

We need to look at some macroscopic averaging.

Assume that two consecutive objects in $\mathrm{worldvol}_n$ are mapped to two points in $\mathrm{tar}$ that lie on each others light cone. Then there is exactly one path of edges between them.

In general, for two points on a 2-dimensional diamond graph whose spatial distance is $\Delta x$ and whose temporal distance is $\Delta t$, there will be $\Delta t \mathrm{choose} \Delta x$ different microscopic paths between them.

Possibly we want to coarse-grain over those.

If so, we would want to invoke a suitable asymptotic formula for the binomial coefficient.

I’m confused about the analogy between finite models for spacetime, and continuous spacetime. In the latter, you need the poset structure and a measure, in order to recover the spacetime metric. These two pieces of data are independent. But in finite models, somehow one can derive the measure from the poset structure… what’s going on?

Well, that’s what we are trying to find out! :-)

Reformulate this question for the path integral: in general we have a category whose objects are histories and whose morphisms are gauge isomorphisms, and we have as additional data a measure on that. But in Dijkgraaf-Witten theory, that measure is already determined to be the Leinster measure on that category… what’s going on?

One obvious guess is: that measure should in fact always be thought of as the Leinster measure on the category of histories. Only that

a) we don’t know what “Leinster measure” would mean for non-finite categories

b) we haven’t checked this yet for more cases than Dijkgraaf-Witten.

But that’s what we are now trying to do!

Posted by: urs on March 11, 2007 10:42 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

This question probably makes no sense, but in a way, are we seeing geometry emerge from the topology of a graph?

Posted by: Eric on March 12, 2007 12:00 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

are we seeing geometry emerge from the topology of a graph?

Yes, in a way. We see some kind of geometry emerge from the topology of a space together with the topology of a space of morphisms over that space.

Posted by: urs on March 12, 2007 7:43 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

I wrote:

I think the category of these functors and their transformations is itself a category freely generated from a circuit-free graph

Oh, sorry, that’s not quite right. The category is not freely generated, of course.

Posted by: urs on March 12, 2007 6:44 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Hi Eric,

it certainly feels nice to see your enthusiasm!

I am currently at the airport and have some time to kill. Realized that there are a couple of small mistakes in what I wrote above and some omissions: I really wanted to link to your description of how to obtain a spacetime graph from a triangulation of a spatial slice – that was really part of my original motivation for writing up a summary entry in the first place!

I’ll see if I can polish that up now, using this funny public terminal here.

Posted by: urs on March 8, 2007 9:28 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

All right, I now fixed a couple of minor things and included the link to Eric’s pdf.

Posted by: urs on March 8, 2007 10:04 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Hi, Urs:

What you’re discussing is closely related to things that I’ve been talking about with John and Jim recently. It’s rather related to the comment that I tried to make a little while ago about spans, so I’ve got some observations.

The Dijkgraaf-Witten model lives on a cobordism category that can be seen in terms of cospans in a category of topological spaces, say $Top$ (of topological spaces and continuous maps), or maybe $Man$ of manifolds with boundary/corners, and smooth maps. I don’t care too much exactly which category this is in for now, so call it $C$. A cobordism $S: X \rightarrow Y$ can be interpreted as a special cospan $X \rightarrow S \leftarrow Y$ with two inclusion maps. One then builds, for each of the objects in the cospan, a path groupoid, so we have $\Pi_1(X) \rightarrow \Pi_1{S} \leftarrow \Pi_1(Y)$. Then connections on the cobordism (or one of the manifolds on the boundary) is a functor from one of these path groupoids to the gauge group $G$ thought of as a category (groupoid) with one object.

The functor categories you get, $hom(\Pi_1(X),G)$ and so forth. For shorthand, I call this $[\Pi_1(X),G]$ etc. and these categories are all groupoids. The gauge transformations are natural transformations between the functors into $G$. These functor categories are now the configuration spaces you are talking about. Since the process of taking $hom$ into something - the operation $[ - , G ]$, is a contravariant functor, we have a span of the resulting groupoids, where the inclusions are replaced with restriction maps:

(1)$[\Pi_1(X),G] \leftarrow [\Pi_1(S),G] \rightarrow [\Pi_1(Y),G]$

This is a span where we have a groupoid of all “histories” in the middle, and of “configurations” at the ends, with projection maps from the histories to the configurations. These are source and target maps, when we think of this as a representative morphism in the category $Span(Top)$, or the more restricted $n Cob$. This represents something like a classical configuration space for some system, whose individual states are connections. So the interesting point then is that this really can be thought of as a groupoid - since the worldvolume (or indeed space) should be thought of as a path groupoid to allow it to support parallel transport. Turning space into a groupoid forces us to categorify the gauge group (the DW model has it as a group seen as a one-object groupoid, though one might generalize to a 2-group, for example, which is something Martins and Porter talk about.)

Quantizing a configuration space like this might normally involve taking an $L^2$ space of certain functions into $\mathbb{C}$, but since we’ve categorified, we have to do something like take functors into $Vect$. The result should be a 2-vectorspace which - modulo possibly some refinements if the configuration space isn’t finite - is some kind of additive category. Anyway, the point is then to use the functor we were applying in $C$ to yield some corresponding map in $2Vect$ as the representation of that cobordism.

So then we can a cospan again (because the functor $[ -, Vect]$ is contravariant again) like

(2)$[ [ \Pi_1(X) , G ] , Vect ] \rightarrow [[\Pi_1(S),G], Vect ] \leftarrow [[\Pi_1(Y),G], Vect ]$

but we really want something else - we’d actually like to have a 2-linear map from the source to the target of this. This is what you want to get - via the fact that these are functor categories on our configuration space groupoids - and to do it you want to to a combined pullback-pushforward. Pullback is easy enough (take functors on the centre groupoid that restrict to the specified functor on the source), but pushforward is what requires the extra structure (a direct integral, a colimit - the details depend on what concept of $2Vect$ we’re using, for example). To do this, you need some kind of measure.

I thought about an example of this idea a while ago - the categorified harmonic oscillator. Similarly, you can start a span of groupoids which represent the classical configuration space of some system. For the oscillator, this just consists of the energy levels of the harmonic oscillator (the spectrum of its energy operator). The groupoid representing it is $FinSet_0$, which has finite sets for objects and bijections for morphisms. It’s equivalent to a direct sum of all symmetric groups (automorphism groups on $n$-sets). These configuration spaces are quantized (to give a 2-Hilbert space) by taking all functors of groupoids into that configuration space. Though you can turn the functor around and turn it into a functor out of the groupoid, as in the way of talking about the DW model. Maps between the 2-Hilbert spaces are represented by spans of groupoids. There’s a cardinality operation there which comes into play when decategorifying the setup to get the usual Hilbert spaces and operators. The cardinality operation is “groupoid cardinality”, which is a special case of this Leinster measure.

It’s interesting to make this observation - groupoid cardinality deals well with finite sets with group action on them, by weighting according to the size of the group nontrivially acting on a given point. It then could be seen as a generalization of just regular counting measure on finite sets. On the other hand, in the DW model, where the congfiguration space consists of connections, if you have a finite group, you again would be looking at a finite set with a group action (by gauge transformation). If the group were a compact Lie group, you’d also have some measure on the configuration space - inherited from the Haar measure. If a is group acts on this (i.e. you turn it into a groupoid in a good way), I would look for a parallel generalization of that measure. I suppose this to be the Leinster measure you’re talking about.

In both cases, it seems the natural measure to use comes from some good “set-level” measure (counting measure in one case, and a measure inherited from a measure on $G$, whether counting or Haar). However, we are adding an extra layer of categorical structure, so the measure also involves some weighting factor that’s related to the Leinster measure, groupoid cardinality, Euler characteristic, etc…

It seems to me this is the starting point for another story involving Crane & Yetter’s measurable categories (seen as 2-vector spaces), and some curious things involving cardinalities, measures, and more, but I don’t really want to carry this further just at the moment.

Anyway - it seems you have made plenty progress on this idea, which is great! It has been puzzling me for the last little while.

Posted by: Jeffrey Morton on March 9, 2007 9:57 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Great introduction to your thesis, Jeff! As usual, I will find a tiny mistake in what you wrote, and only discuss that:

If the group were a compact Lie group, you’d also have some measure on the configuration space - inherited from the Haar measure. If a group acts on this (i.e. you turn it into a groupoid in a good way), I would look for a parallel generalization of that measure. I suppose this to be the Leinster measure you’re talking about.

Not quite, I think. Tom Leinster showed that any sufficiently nice category has a god-given measure on its set of objects. This is what Urs is calling ‘Leinster measure’.

You seem to be talking about something slightly different. Given a measure space $X$ and a Lie group $G$ acting on $X$ via measure-preserving transformations, there should — at least under some conditions! —- be god-given measure on the set of objects of the weak quotient $X//G$.

Both these ideas are special cases of some poorly understood blend of measure theory and category theory.

There’s an important case that fits in the intersection of these ideas: if we take a set $X$ with counting measure and let a discrete group $G$ act on it, the Leinster measure on the groupoid $X//G$ matches the one obtained from the weak quotient procedure.

By the way, I don’t think Tom Leinster said much about ‘Leinster measure’. He mainly discussed the total Leinster measure of a sufficiently nice category. He called this the Euler characteristic of the category. As I explained in week244, this generalizes the cardinality of a groupoid, which in turn generalizes the cardinality of a set.

But, we can also talk about the Leinster measure of any subset of the objects of our (sufficiently nice) category.

Posted by: John Baez on March 9, 2007 7:58 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Under what circumstances is the zeta operator non-invertible? I guess these are the same circumstances under which the weightings/co-weightings are non-unique? This is probably in Tom’s paper somewhere but maybe I’ll understand it better if I just ask.

As far as I can tell, a skeletal, circuit-free category gives rise to an invertable zeta operator and a unique weighting and co-weighting. The only way I can see this not happening is if there are several isomorphic objects. In this case the zeta operator is clearly non-invertible. Also, a set of several isomorphic objects needs to share out among themselves the contribution to the Euler characteristic that would be contributed by a single object in the skeletal case, and there isn’t a unique way to do this.

Is this true? Or am I completely off track?

Posted by: Tim Silverman on March 9, 2007 9:45 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

D’oh! That was a case of my somehow not seeing an entire section of the paper because I wasn’t ready to ask the right question of it. Not only that, but after thinking about the determinants of zeta operators here, I think I could even make a stab at characterising categories with non-invertible zeta operators. Oh well.

Posted by: Tim Silverman on March 10, 2007 1:06 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

The relation between this canonical measure and the one from the Haar measure seems to have to do with both of them being defined so as to be invariant under conjugation/translation/etc on this groupoid built from the Lie group.

There’s something else in what I said that doesn’t seem right - I was describing certain spans of groupoids as “configuration spaces”, but in the case of the oscillator, for example, it’s not the config. space of the classical oscillator. It’s just something that contains all the “pure” classical states the system can be in - which is not every possible energy level the classical one can be in. I don’t see that that’s the case for the space of connections (unless it’s that the flat ones are the only classical states the quantum system can be in, just like the oscillator can only have integer energies - that seems possible).

Posted by: Jeffrey Morton on March 9, 2007 10:09 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Hi guys!

Here’s a few things that come to my mind when I read these ideas.

(a) John mentioned how Tom’s paper has a section relating his definition of Euler characteristic to “the” Euler characteristic of an orbifold. One should be careful here though : this whole thing is very confusing ! There seem to be a number of ways of defining the Euler characteristic of an orbifold - and they all give different answers.

It seems that Tom is talking about the Euler characteristic computed from (equivariant) simplicial cohomology of orbifolds - is this right?

There’s also Thurston’s definition of Euler characteristic of an orbifold, and then also Witten’s. And there’s others too.

Atiyah and Segal showed that Witten’s definition is computing the Euler characteristic of equivariant $K$-theory . Its not the same as the one obtained from (equivariant) simplicial cohomology (it gives different answers). See this paper by Jim Bryan and Jason Fulman for more about comparing the various notions of “Euler characteristic for orbifolds”. Anyhow, it seems there remains some stuff here to be understood more clearly.

(b) Jeffrey wrote about his philosophy regarding the finite group model, groupoids, cospans, etc. I agree with this entirely! This is essentially the way Freed thought about this stuff too.

There are some caveats though:

(i) The picture needs to smoothly accommodate the twisted finite group model. This is not so easy as it seems. You need to set up line bundles and things, and make sense of integrating singular cocycles over manifolds with boundary. Freed and Quinn showed a beautiful way to do this (in Appendix B of their 1993 paper), making nice use of groupoids and connections and (secretly) 2-morphisms and all sorts of nice things.

Nevertheless, I think their construction can be made even more elegant, using n-cafe language. Anyone care to help have a go?

(ii) Jeffrey (I think, since he uses cospans, etc.) and I tend to prefer working in the “cobordism” or “Segal” picture where a TQFT is thought of as a functor thingy, rather than in the “Atiyah” picture where a TQFT is a “gluing rules” thingy.

Something which has been annoying me recently though is that in all the cases that I know, its always easier to write things down in the “Atiyah” picture first. You can convert to the “Segal” picture by letting the bent cylinder be a metric, and then using this metric to “raise” the indexes. See this interesting paper which deforms the finite group model and applies it to Gromov-Witten theory. Not that I understand it, but they too first do “Aityah”, then “Segal”. Even Freed and Quinn first did “Atiyah”, then “Segal”. This is annoying : it would be nice if the cobordism picture were not only more pleasing conceptually, but also easier to write down.

Sometimes, the “Atiyah” picture is not just a technical tool - it can be the more relevant picture. For example, suppose there are $n$ marked points on the sphere. and we want to count the number of $d$-fold coverings of the sphere which are ramified over the marked points, with monodromies in certain conjugacy classes $(C_1, \ldots, C_n)$ in the symmetric group. These are classical things called the Hurwitz numbers , and Andrei Okounkov, for instance, showed how they’re related to Gromov-Witten theory. Not that I understand any of this stuff, but anyway!

These numbers are nothing but the numbers coming out from the finite group TQFT where $G$ is the symmetric group $S_d$!

A “sphere with n-marked points and having presecribed monodromy” is just a convoluted way of thinking of a pair of pants with $n$ inputs and no outputs, and counting the number of $G$-bundles which restrict to prescribed $G$-bundles on the boundary .

When we think of things using “marked points” we are in the Atiyah picture : cobordisms don’t seem to feature (although they secretly do of course!).

Posted by: Bruce Bartlett on March 11, 2007 2:04 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Jeffrey Morton listed a couple of structures that go into the formulation of Dijkgraaf-Witten field theory.

Since a while ago John kind of complained that my $\mathrm{par}$-$\mathrm{tar}$-$\mathrm{conf}$-terminology is hard to digest (but the trick is to read it out explicitly: “parameter space”, “target space”, “configuration space” – that’s supposed to be helpful), I’ll quickly set up a dictionary between Jeffrey’s notation and the one I have been using in a series of posts.

So:

These fundamental groupoids of a 2-manifold $X$ or $Y$ are the parameter spaces $\mathrm{par}_X = \Pi_1(X)$ of a 3-particle (a membrane).

The fundamental groupoid of the cobordim $S$ is the worldvolume of shape $S$: $\mathrm{worldvol}_S = \Pi_1(S) \,.$

These parameter spaces and worldvolumes are mapped, by the fields of the theory, into target space, which for Dijkgraaf-Witten theory is a finite group: $\mathrm{tar} = \Sigma(G) \,.$

Accordingly (as Jeffrey mentioned), configuration space is the space of all maps from parameter space into target space: $\mathrm{conf}_X = [\mathrm{par}_x,\mathrm{tar}] = [\Pi_1(X),\Sigma(G)] \,.$ Accordingly, the space of histories is the space of maps from the worldvolume into target space: $\mathrm{hist}_S = [\mathrm{worldvol}_S,\mathrm{tar}] = [\Pi_1(S),\Sigma(G)] \,.$

In addition, there should be a background gauge field on target space. And there is. More on that below.

Jeffrey wrote:

the worldvolume (or indeed space) should be thought of as a path groupoid to allow it to support parallel transport.

I think that the worldvolume should even be thought of as a 3-groupoid, and the configuration spaces as 2-groupoids.

For Dijkgraaf-Witten theory, due to the finiteness of target space, much of the general structure of the charged $n$-particle (here: 3-particle) collapses and becomes invisible. So it does indeed look as if there are just 1-groupoids in the game – except for the twisting (as Bruce mentioned): for DW we may regard this as a group cocycle on $G$. But secretly that means it is a pseudofunctor $\mathrm{tra} : \Sigma(G) \to \Sigma^3(U(1)) \,,$ which is to be thought of as the parallel transport of a 2-gerbe on $B G$.

For DW it may be overkill to make this explicit. But for Chern-Simons (with $G$ now a Lie group) I think it becomes essential.

Quantizing a configuration space like this might normally involve taking an $L^2$ space of certain functions into $\mathbb{C}$, but since we’ve categorified, […]

I think that, in general, the $n$-particle will couple to an $n$-bundle with connection on target space. Trangressing that to the $(n-1)$-dimensional configuration space always produces some kind of $(n-(n-1)=1)$-bundle. The space of states is something like square integrable sections of that 1-bundle. This will be $L^2$-functions when the background gauge field is trivial. But for DW theory at non-trivial level, this won’t be the case.

For DW theory, these details have been made very explicit in Simon Willerton’s paper.

So then we can a cospan again like

$[[\Pi_1(X),G],\mathrm{Vect}] \to [[\Pi_1(S),G],\mathrm{Vect}] \leftarrow [[\Pi_1(Y),G],\mathrm{Vect}]$

This I am not really sure about, mainly for the reasons just stated. But probably I am just not fully aware of the details of your description.

we’d actually like to have a 2-linear map from the source to the target of this

This I also don’t quite understand: the map from the source to the target of this should in particular reproduce the ordinary linear map that people in ordinary Segal-like QFT would assign to the cobordism $S$. It should be a linear map that takes square integrable sections of some line bundle on $\mathrm{conf}_X$ to square integrable sections of some line bundle over $\mathrm{conf}_Y$. I’d think.

you can start a span of groupoids which represent the classical configuration space of some system. For the oscillator, this just consists of the energy levels of the harmonic oscillator

But the collection of these energy levels is not the classical configuration space, is it? That should rather be the real line. No?

Posted by: urs on March 12, 2007 7:40 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Jeff said: I thought about an example of this idea a while ago - the categorified harmonic oscillator. Similarly, you can start a span of groupoids which represent the classical configuration space of some system. For the oscillator, this just consists of the energy levels of the harmonic oscillator (the spectrum of its energy operator). The groupoid representing it is FinSet 0 , which has finite sets for objects and bijections for morphisms. It’s equivalent to a direct sum of all symmetric groups

This sounds very interesting and potentially useful. (I haven’t quite figured out all the math-display goodies yet, so sorry about the typesetting). Often when studying things involving the symmetric groups Sn - say, their characters - its easier to work with them all at once rather than one at a time. In particular, if we consider the direct sum of all of them, we get something that is a representation of the heisenberg algebra.

An example I’m interested in: let X be some projective surface, and let Xn be either the hilbert scheme of n points on X, or the nth symmetric product of X (as an orbifold). Then, if we want to study the cohomology ring of Xn, one forms the space H=sum {over all n >=0} H^*(Xn), and constructs an action of a heisenberg algebra on H.

It seems like your harmonic oscillator example could be one way of understanding why these heisenberg representations arise. Even better, it seems to be saying that these representations are very natural and might have an analog at some level higher than cohomology, which could be used in studying finer properties of Xn, such as their quantum cohomology.

That was all rather imprecise, and I haven’t had time to more than glance at your paper, but I was just wondering if you’d thought about these Heisenberg algebra actions at all.

Posted by: Paul Johnson on March 13, 2007 8:48 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

I seem to have persistent technical problems when trying to express that thought. A superstitious person might take this as evidence it isn’t a good idea… ;)

Posted by: Jeffrey Morton on March 9, 2007 10:11 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

There seems to be a problem posting comments at the moment. I’ve removed four of the five copies of your comment. I trust they were identical.

Posted by: David Corfield on March 9, 2007 11:09 AM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Superstition or no, I’m sure it’s a good idea. In particular I’m sure because my whole project is about recasting knot theory as the 1-dimensional version of this picture. The “topological” invariants of knots are “really” functors between categories of cospans, and I’m getting more and more sure that “combinatorial” invariants are decategorifications of such functors.

Posted by: John Armstrong on March 9, 2007 1:57 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

Speaking of which, I never got back to you about that business about groupoid cardinalities and categorifying higher order operations than + and multiplication. Tom Leinster pointed out to me that groupoid cardinality doesn’t get along with exponentiation in the obvious way, anyway - i.e. the groupoid cardinality of the category of functors between two groupoids doesn’t match the exponential you might like… Consider two finite groupoids with cardinality 1/2 for example.

Posted by: Jeffrey Morton on March 10, 2007 12:58 AM | Permalink | Reply to this

### Random supersets and emergence; Re: Canonical Measures on Configuration Spaces

In the same sense as Feynman diagrams (however weighted and summed) are a subset of syntactically correct but nonphysical edge-labelled vertex-labelled digraphs, it seems to me that interesting n-diamond graphs are an asymptotically small subset of random Z^n graphs with a random unit vector parallel to one axis, in each cell. I’ve been working on enumeration of distinct (to isomorphism) trees and loops in such “directed n-dimensional lattice animals.” Results are well known for the Ising, X-Y and related models, and probability distributions for loops and knots in Z^2 and Z^3 versions.

Dynamics on these are interesting, and in Physics yieled the great Ken Wilson et al Renormalization Group results and statistical mechanics with associated temperatures of phase transitions.

In the Euclidean Z^2 lattice, this is simply putting a random element from {N,E,S,W} in each cell and interpreting this as a possible step in a self-avoiding random walk on the lattice.

Causal digraphs emerging from a random soup is Quantum Loop Gravity-like in style, to my amateur eyes.

http://arxiv.org/pdf/math.DS/0703167
From: Tom Meyerovitch
Entropy of cellular automata

Other regular and semiregular lattices are stylish for Cellular Automata, as with http://www.cse.sc.edu/%7Ebays/CAhomePage

Of course, the lattice need not be Euclidean. There are more papers emerging on cellular automata in hyperbolic spaces, including in arXiv, and my unpublished results on this shown last year at ICCS-2006 suddenly look clumsy but on the right track.

Animal enumerations on regular tilings in Spherical,
Euclidean, and Hyperbolic 2-dimensional spaces
http://www.ieeta.pt/~tos/animals.html

Relating to my earlier (ICCS-2006)
Game of Life in a Hyperbolic Tiling:

Carter Bays has a page of unusual cellular automata.
One of his discoveries is a slow pentagonal glider.
http://www.mathpuzzle.com/
http://www.cse.sc.edu/%7Ebays/CAhomePage

Computer Science, abstract
http://arxiv.org/pdf/cs.DM/0702155
From: Maurice Margenstern

On a characterization of cellular automata in tilings of the hyperbolic plane

In this paper, we look at the extention of Hedlund’s
characterization of cellular automata to the case of
cellular automata in the hyperbolic plane. This
requires an additionnal condition. The new theorem is
proved with full details in the case of the pentagrid
and in the case of the ternary heptagrid and enough
indications to show that it holds also on the grids
$\{p,q\}$ of the hyperbolic plane.

I suspect that t6his categorifies and n-categorifies nicely.

Posted by: Jonathan Vos Post on March 9, 2007 7:31 PM | Permalink | Reply to this

### Re: Canonical Measures on Configuration Spaces

I was off line for two days. Relaxing, family, etc. It’s still week end here, and I am not supposed to be hanging around on the web. I’ll see if I can drop a quick comment or two right now. But any serious response will have to wait until tomorrow (evening).

Posted by: urs on March 11, 2007 7:06 PM | Permalink | Reply to this
Read the post QFT of Charged n-Particle: The Canonical 1-Particle
Weblog: The n-Category Café
Excerpt: On the category of paths whose canonical Leinster measure reproduces the path integral measure appearing in the quantization of the charged particle.
Tracked: March 19, 2007 9:08 PM
Read the post The Canonical 1-Particle, Part II
Weblog: The n-Category Café
Excerpt: More on the canonical quantization of the charged n-particle for the case of a 1-particle propagating on a lattice.
Tracked: August 15, 2007 11:47 AM
Read the post BV-Formalism, Part IV
Weblog: The n-Category Café
Excerpt: Lie algebroids of action groupoids and their relation to BRST formalism.
Tracked: October 11, 2007 9:47 PM
Read the post Dijkgraaf-Witten and its Categorification by Martins and Porter
Weblog: The n-Category Café
Excerpt: On Dijkgraaf-Witten theory as a sigma mode, and its categorification by Martns and porter.
Tracked: January 5, 2008 3:23 AM

### Re: Canonical Measures on Configuration Spaces

In the above entry I discussed how the path integral measure of Dijkgraaf-Witten theory is nothing but the Leinster measure for the corresponding configuration groupoid.

As I see now, the same is true for the kind of categorified Dijkgraaf-Witten theory which goes back to D. Yetter and has been further discussed by Tim Porter, João Martins and others, like by Girelli-Pfeiffer-Popescu in their recent article.

A discussion of how the path integral measure for Yetter’s categorified DW theory is once again just the Leinster measure on its configuraton groupoid I have posted here.

Posted by: Urs Schreiber on January 5, 2008 4:36 PM | Permalink | Reply to this
Read the post Teleman on Topological Construction of Chern-Simons Theory
Weblog: The n-Category Café
Excerpt: A talk by Constant Teleman on extended Chern-Simons QFT and what to assign to the point.
Tracked: June 18, 2008 9:51 AM
Read the post News on Measures on Groupoids?
Weblog: The n-Category Café
Excerpt: Benjamin Bahr apparently thought about measures on groupoids of connections.
Tracked: July 17, 2008 10:03 AM
Read the post Alm on Quantization as a Kan Extension
Weblog: The n-Category Café
Excerpt: An observation on a relation between Kan extensions and path integral quantization -- by Johan Alm.
Tracked: May 26, 2009 7:45 PM

Post a New Comment