Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

April 1, 2007

Oberwolfach CFT, Arrival Night

Posted by Urs Schreiber

After supper I went jogging along the Wolf river through the tiny village Oberwolfach, from where one can peer up the black forest mountains and see the illuminated MFO library shining through the fir trees, with no other light source except for a bright full moon on a starlit sky. That’s probably about as romantic as math can get.

On the eve of the CFT workshop starting tomorrow, I am struggling with understanding how…

… how the Araki-Haag-Kastler-Axioms should fit into the grand scheme of things – in particular, how they relate to Segal’s axioms, and their refinement to extended QFTs.

I expect that the local in “Haag-Kastler local QFT” is the extended in “Stolz-Teichner/Freed-Hopkins extended QFT”.

I also expect that the passage

extended Segal functorial QFT \to Haag-Kastler algebraic QFT

is, for nn-dimensional QFT, essentially the nn-fold categorification of the functor

Schrödinger picture \to Heisenberg picture

which is essentially induced by postcomposition with the equivalence

Hilbert spaces with cyclic vacuum vector \simeq C *C^*-algebras with pure normal state

along the lines discussed in QFT of Charged n-Particle: Sheaves of Observables.


There are various indications for how this should work, but I clearly don’t get the full picture yet.

My Guru says: When in doubt, formulate the arrow theory – then follow the Dao.

I’ll try to do that, and I am sure all I need to do is to meditate over these mandalas a little further.

Starting with an extended Segal-like functorial QFT, what does “forming the algebra of observables”, really mean? It turns out that when one writes it out, a nice pattern emerges: QFT of Charged nn-Particle: Algebra of Observables.

This gives a “2-monoid of observables”, which, for n=1n=1 reproduces the ordinary Weyl algebras coming from canonical positions and momenta on target space, while for n>1n\gt 1 one can see on heuristic grounds that it produces the corresponding algebra, now for loop space, double loop space, and so on.

By taking these loops, double loops, etc, to really be functorial images γ:partar \gamma : \mathrm{par} \to \mathrm{tar} of a parameter space par\mathrm{par} which is not just a single loop, say, but something resolving this into a category of paths in the loop, etc, one finds, at least heuristically again, that this Weyl algebra on loop space decomposes into lots of local algebras living over all these intervals.

Then feed all this into the general arrow-theory for disk correlators, as described in D-Branes from Tin Cans, III: Homs of Homs to find what the Dao wants to assign to boundaries.

As always, I’d just have to follow where the formalism leads me here, but for this step I have not managed to do so yet.

This is what my thoughts were revolving around as I jogged by the bank of the river Wolf.

It does not seem implausible that on a boundary interval II this procedure produces the chiral algebra A(I)A(I) living there, arising essentially as the restriction of the above bulk algebras BB to the constant paths starting at that boundary, and that the canonical inclusion ABA \to B appears at the level of morphisms, thus reproducing the subfactor inclusion/Q-system/Frobenius algebra of DHR endomorphisms that one would expect here, along the lines that I mentioned in Some Notes on Local QFT.

But as yet I fail to see this in detail.

I realize that one reason for this is that, while I think I know what the nn-algebra of observables itself is like, arrow theoretically, I have no good understanding yet of what the nn-categorical Heisenberg picture evolution functor would actually look like on morphisms. This, in turn, is at least partly due to the fact that I have not yet fully used, in the present context, all information available about the categorification of the equivalence Hilb cycC stat * \mathrm{Hilb}_{\mathrm{cyc}} \simeq C^*_{\mathrm{stat}} of Hilbert spaces with cyclic states and C *C^*-algebras with pure normal states on them. This will involve the categorified Gelfand-Naimark theorem.

One problem with not getting confused in this context here is that algebras are making an appearance in various different guises. Not every algebra one encounters here plays the same kind of role, and that entities of different nature appear to us as just algebras makes some important structures hard to discern.

For instance, I am wondering: is the phenomenon that AQFT works so very well for 2-dimensional QFTs (where it is a very powerful tool), but has as yet found no real application to physically nontrivial QFTs in higher dimensions (as far as I am aware), just a simple consequence of the fact that people study the simpler examples first, or is there maybe a connection to the fact that the von Neumann algebras playing such a prominent role naturally live precisely in a 2-category, namely the 2-category whose morphisms are their bimodules?

While authors of standard AQFT literature rarely think of algebras as living in the 2-category of bimodules, this fact is actually implicitly of utmost importance for most of the crucial constructions revolving around DHR endomorphisms: morphisms of morphisms of algebras of observables appear all over the place, being addressed as “intertwiners”. These intertwiners are really morphisms of the left modules induced by the respective algebra morphisms, hence really 2-morphisms in bimodules of von Neumann algebras.

I am being thrown back and forth between believing that this is just a coincidence of low dimensions which I’d better ignore in order not to confuse myself, or if it is rather a crucial hint that I should take serious and think through to the end.

Maybe the next days will help crystallize answers to these puzzling issues.


By the way, concerning Segal’s axioms of QFT, I discovered this interesting paper:

Doug Pickrell
P(ϕ) 2P(\phi)_2 quantum field theories and Segal’s axioms
math-ph/0702077

Posted at April 1, 2007 11:13 PM UTC

TrackBack URL for this Entry:   http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1222

21 Comments & 2 Trackbacks

Re: Oberwolfach CFT, Arrival Night

So the obvious question then:

Vector
Hilbert Space
Von Neumann Algebra
?????????

Posted by: A.J. on April 2, 2007 6:00 AM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

So the obvious question then:

Vector

Hilbert Space

Von Neumann Algebra

?????????

Yes, that’s actually a good question.

By the way, I would start not with “vector”, but with “complex numbers”.

The complex numbers are the complex 0-vector space, with each number being a 0-vector.

Then, it is true that each algebra may be regarded as a 2-vector space (roughly: a 2-vector space with a chosen basis).

What I am not quite sure about yet is what it means, from this point of view, that the algebra is von Neumann.

And it gets more confusing:

in application to QFT, it turns out that there are other algebras that play the role of bases of 2-vector spaces, namely algebras internal to the category of endomorphisms of a given vN algebra AA.

Gotta run now…

Posted by: urs on April 2, 2007 8:00 AM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

There seems to be a pretty obvious analogy between the norm on the complex numbers, the inner product on a Hilbert space, and the “non-commutative measure theory” description that we sometimes give vN algebra. I’m not sure if this is really what’s wanted though…

Posted by: A.J. on April 2, 2007 5:39 PM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

There seems to be a pretty obvious analogy between the norm on the complex numbers, the inner product on a Hilbert space, and the “non-commutative measure theory” description that we sometimes give vN algebra.

This is an intersting point of view. I will have to think about this. It would be great if we could understand von Neumann algebras and their bimodules as 2-vector spaces with a certain special property, like maybe a norm, and morphisms respecting that.

Right now I am too tired to think about that with the required care, but I’ll keep this in mind,

Posted by: urs on April 3, 2007 11:52 PM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

I wrote:

The complex numbers are the complex 0-vector space, with each number being a 0-vector.

Sorry, I meant to say:

The complex numbers are the 0-category of 0-vector spaces 0Vect \mathbb{C} \simeq 0\mathrm{Vect}_{\mathbb{C}} with each complex number being one 0-vector space.

Posted by: urs on April 2, 2007 8:53 PM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

Urs wrote:

… is the phenomenon that AQFT works so very well for 2-dimensional QFTs (where it is a very powerful tool), but has as yet found no real application to physically nontrivial QFTs in higher dimensions (as far as I am aware), just a simple consequence of the fact that people study the simpler examples first … ?

I think it’s a consequence of the fact that 2-dimensional interacting quantum field theories are easily to construct in a rigorous way, while higher-dimensional ones are darn near impossible.

One reason for this is that the operator

1/(Δ+m 2)1/(\Delta + m^2)

has finite trace on L 2L^2 of a compact 1-dimensional manifold, but not on L 2L^2 of a compact nn-dimensional manifold for nn > 1.

If you think about it, this follows from the fact that

k1/(k 2+m 2) \sum_k 1/(k^2 + m^2)

converges but

k 1,k 21/(k 1 2+k 2 2+m 2) \sum_{k_1, k_2} 1/(k_1^2 + k_2^2 + m^2)

and similar sums in higher dimensions do not. (These sums are traces of 1/(Δ+m 2)1/(\Delta + m^2) on tori, but a torus is a sufficiently general example.)

But, this phenomenon, which sounds like it’s all about analysis, is also related to categorification! This is explained in Jouko Mickelsson’s work! Certain nn-cocycles only converge when nn is sufficiently large compared to the dimension of space. That’s why we get interesting central extensions of the gauge groups C (M,G)C^\infty(M,G) when MM is 1-dimensional, but not when it’s higher-dimensional.

In dimension 1, we get these central extensions from certain 2-cocycles. In higher dimensions, we get well-defined nn-cocycles only for higher nn, which give — not interesting group extensions, but interesting nn-group extensions of C (M,G)C^\infty(M,G).

So, there’s something big going on here, which may overlap with your own guesses.

Posted by: John Baez on April 2, 2007 7:13 AM | Permalink | Reply to this

2D

For instance, I am wondering: is the phenomenon that AQFT works so very well for 2-dimensional QFTs (where it is a very powerful tool), but has as yet found no real application to physically nontrivial QFTs in higher dimensions (as far as I am aware), just a simple consequence of the fact that people study the simpler examples first…

I should point out that the bootstrap program never went anywhere in D3D\geq 3, either (aside from inspiring string theory). But, through the work of the Zamolodchikov’s and their successors, it has proven spectacularly successful in D=2D=2.

I suspect that this, and your remark about the successes of AQFT in D=2D=2 are not unrelated. Philosophically (if not technically), there are great similarities between AQFT and the bootstrap program.

Posted by: Jacques Distler on April 2, 2007 8:59 AM | Permalink | PGP Sig | Reply to this

Re: Oberwolfach CFT, Arrival Night

I think it’s a consequence of the fact that 2-dimensional interacting quantum field theories are easily to construct in a rigorous way

[…]

But, this phenomenon, which sounds like it’s all about analysis, is also related to categorification!

Yes, okay. So 2-dimensions are tractable and higher dimensions ought to involve higher categorical structures.

But maybe it is noteworthy that the standard AQFT constructions crucially involve at many places the fact that vN algebras naturally live in a 2-category.

Few people in that business ever admit that of course. The only bimodules of vN algebras that are ever considered – implicitly – in the standard literature are those induced from algebra homomorphisms, and the 2-morphisms, i.e. the morphisms between these bimodules are then addressed as “intertwiners”.

Adjunctions built from these are not called adjunctions but “conjugations” and the monads obtained from those are not called monads but “Q-systems”.

Therefore I am wondering: if we properly make all this secretly used 2-categorical structure explicit, will we maybe find that local nets of vN algebras are the right thing to look at only in 2 dimensions?

Will we, maybe need something beyond mere nets of vN algebras in higher dimensions to have any chance at all to describe nontrivial QFTs?

The formalism does suggest so, at least vaguely.

There are other indications that making the 2-category of vN algebras explicit in AQFT would be helpful. For instance this:

at the moment, AQFT cannot describe topological QFTs. That’s a pity, because many of the interesting and tractable examples of QFTs in higher dimensions are topological in one way or another.

But let’s look at the reason that AQFT cannot handle these: the algebra of local observables of a TQFT is just the trivial algebra \mathbb{C}. So if you look for Q-systems of DHR-reps of these, namely Frobenius algebra objects in endomorphisms of \mathbb{C} you find just the trivial one.

But this problem is immediately alleviated if you allow not just \mathbb{C}-bimodules induced from endomorphisms of \mathbb{C}, but arbitrary \mathbb{C}-bimodules.

These are of course nothing but vector spaces. Looking for “Q-systems” of these simply leads to ordinary Frobenius algebras internal to vector spaces.

And indeed, this is exactly the structure that defined a local TQFT (following the Fukuma-Hosono-Kawai construction).

Posted by: urs on April 2, 2007 9:46 AM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

I should add that when thinking of DHR reps as bimodules instead of mere modules, I have the following observation in mind, which seems to me to address the structure that is “really” underlying this business.

Let AAI A \stackrel{A}{\to} I be the algebra AA regarded as a module over itself, and write A ρAA A \stackrel{{}_\rho A}{\to} A for AA regarded as a bimodule over itself, with the obvious right action and the left action twisted by some AA-endomorphism ρ\rho.

Then a DHR rep induced by an endomorphism ρ:AA \rho : A \to A and equipped (as a Q-system) with a “unit”, namely an “intertwiner” i:Id Aρ i : \mathrm{Id}_A \to \rho should in fact be thought of as the component of 2-transformation that is the following 2-cell in bimdoules: I Id I A i A A ρA A. \array{ I &\stackrel{\mathrm{Id}}{\to}& I \\ A\downarrow\;\; &\; \Downarrow i & \;\;\downarrow A \\ A &\stackrel{{}_\rho A}{\to}& A } \,.

This picture explains a couple of otherwise ad-hoc-looking things.

For instance, even though this lives in bimodules, the composition of two such squares is clearly just horizontal juxtaposition which does indeed boil down to nothing but the ordinar composition of the endomorphisms that induce these bimodules I Id I Id I A i A i A A ρA A ρA A. \array{ I &\stackrel{\mathrm{Id}}{\to}& I &\stackrel{\mathrm{Id}}{\to}& I \\ A\downarrow\;\; &\; \Downarrow i & \;\;\downarrow A &\; \Downarrow i' & \;\;\downarrow A \\ A &\stackrel{{}_\rho A}{\to}& A &\stackrel{{}_{\rho'} A}{\to}& A } \,.

There are more helpful pictures like that, all of them following the general principle that also underlies the arrow theoretic exegesis of the theory of amplimorphisms.

And I’d think this is not a coincidence. But nevertheless I don’t fully see the big picture yet, at this moment.

Posted by: urs on April 2, 2007 11:21 AM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

This is explained in Jouko Mickelsson’s work! Certain n-cocycles only converge when n is sufficiently large compared to the dimension of space. That’s why we get interesting central extensions of the gauge groups C(M,G) when M is 1-dimensional, but not when it’s higher-dimensional.

It might be worth decomposing this statement into irreducible components:

1. These gauge groups do have central extensions independent of the dimension of M.

2. When dim M > 1, these extensions are not interesting from your point of view at this time.

However, I find these central extensions very interesting, because they do arise in QJT. The whole idea is to avoid the above-mentioned infinitites, which make it impossible to construct the Hilbert space of a higher-dimensional QFT, by replacing fields by trajectories in jet space. At the very least, this does give representations of C(Rd,G) as well-defined operators acting on a linear space, free of infinities. Considering that nobody has succeeded in constructing an interacting higher-dimensional QFT, I think that this is quite cool.

Posted by: Thomas Larsson on April 2, 2007 3:36 PM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

Do you find that jogging helps your thinking? I tend to reserve mathematical problems for uphill stretches and philosophical ones for downhill. Many a painful ascent has been unnoticed by the device of a visit to the platonic realm.

Posted by: David Corfield on April 2, 2007 8:47 AM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

Do you find that jogging helps your thinking?

Yes. But I also find that when I begin thinking too much, I can no longer run but start to walk. Maybe I find walking even more helpful for thinking. But it’s less helpful with regard to the typical lack of physical exercise…

Posted by: urs on April 2, 2007 11:05 AM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

As someone who has run several marathons and an ultramarathon or two, I can reassure you that walking is just as helpful (maybe moreso) to your health than jogging, so no need to feel guilty about walking :)

Posted by: Eric on April 2, 2007 2:44 PM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

I looked at the paper

Doug Pickrell

P(ϕ) 2P(\phi)_2 quantum field theories and Segal’s axioms

math-ph/0702077

The functional analysis goes way over my head, but its interesting to see how he treats “Segal’s axioms for quantum field theory”. Namely, he really works in the Atiyah picture, since the fundamental construct is to associate a state vector Z(Σ)Z(Σ)Z(\Sigma) \in Z(\partial \Sigma) in the vector space associated to the boundary of Σ\Sigma.

This is the standard way of doing things, and it ties in with a post I made earlier. Namely, one first constructs ‘Atiyah-style data’, i.e. vector spaces Z(S)Z(S) for closed (n1)(n-1) manifolds SS and vectors Z(Σ)Z(Σ)Z(\Sigma) \in Z(\partial \Sigma) for nn-manifolds (with boundary) Σ\Sigma), and then one (purely formally) passes to the ‘Segal picture’, i.e. a functor

(1)Z:nCobVect Z : nCob \rightarrow Vect

by using the “metric” to raise indices.

I find it a bit annoying that, although the Segal picture is the most pleasing conceptually (a qft is a functor from this to that), the Atiyah picture always turns out to be more “fundamental” in that all the formulas, etc. are much more naturally expressed in the Atiyah picture than in the Segal picture.

That’s the case for the Dijkgraaf-Witten finite group model, for instance. You first define Atiyah data using very natural looking formulas, and then you pass to the Segal picture by using the “metric” to raise indices. Its kind of a bit artificial, and you get the feeling you’re just putting a “man-made spin” on something which the underlying maths itself was never calling out for.

I was very impressed when I first found out that “a QFT is a functor”. I would like that to be a deep fact, i.e. “a QFT is really urging us to think of it as a functor” instead of the more cynical “ we are urging the QFT to be a functor”. If that’s the case, then its a bit of a shame, as far as I’m concerned. Does anyone have any philosophical comments to make about this?

Posted by: Bruce Bartlett on April 2, 2007 7:38 PM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

Hi Bruce!

I find it a bit annoying that, although the Segal picture is the most pleasing conceptually (a qft is a functor from this to that), the Atiyah picture always turns out to be more “fundamental” in that all the formulas, etc. are much more naturally expressed in the Atiyah picture than in the Segal picture.

I think I get your point.

But I think it is important to keep in mind that in the “Atiyah picture” (thanks for emphasizing that this is indeed one more flavor of kinds of descriptions of QFT) there are also a bunch of constraints in the game, the “sewing constraints”, which ensure that the assignment of vectors is consistent with composition.

It is these “sewing constraints” that encode the functoriality present in the Segal-picture, of course.

I once tried to go through the exercise of spelling out how this looks like in abstract functorial terms. I did this here for a slightly different situation, namely one where we don’t just assign a vector to a cobordism, but an entire vector space, which is supposed to be the vector space of all consistent assignments of vectors parameterized by some extra structure on the cobordism. This is the situation as one encounters it in the FFRS approach, where the extra conformal structure has first been forgotten.

But I’d think the general strategy discussed at the above link should also more or less apply to the Atiyah-picture in general.

I must admit, though, that it is very late here and I am a little short of time. Please take a look at what I write at that link, think about it from your perspective, and then tell me what you think. Possibly – in as far as this is worthwhile at all – this can be improved on. (I think I just founjd a silly typo at a crucial place in that entry, for instance. But I am too tired to try to fix it now. Sorry.)

Posted by: urs on April 3, 2007 12:10 AM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

Hi Urs, your post is very interesting and the ‘holography principle’ it alludes to is really cool, and seems to be a very general phenomenon. I certainly agree that the picture is:

(1)Atiyahpicture Segalpicture sewingconstraints functoriality \begin{matrix} Atiyah picture & Segal picture \\ sewing constraints & functoriality \end{matrix}

Right now I’m just complaining pic that I haven’t yet found a nice formula which implements the Segal picture directly , without first doing the Atiyah picture and then raising indices with the metric.

Posted by: Bruce on April 4, 2007 12:41 AM | Permalink | Reply to this

picture changing

Hi Bruce,

I’d ned to think about the precise computation that you are referring to, but on general grounds I am a little surprised that you say there is a problem here, since I’d thought that we can always think of the “Atiyah picture computation” as being the “Segal picture computation” for a manifold with empty input, and all boundaries regarded as output.

Sewing/funcoriality should ensure that this is always allowed.

There are a couple of explicit computations of this sort in hep-th/0605255, some of which do use impty inputs and hence can, I think, be thought of as being “Atiyah picture”.

Could you maybe comment on these examples? Do you see a problem there, too?

Posted by: urs on April 4, 2007 9:28 AM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

There is a key statement on page 3 of Pickrell’s paper:
“In this paper we will only consider d = 1, 2.”

It was Pickrell who proved that the Mickelsson-Faddeev algebra has no unitary reps on a separable Hilbert space, in 1989. This result has a nice physical interpretation. Recall that the MF extension is a gauge anomaly, proportional to the third Casimir. Such anomalies are believed to be inconsistent, and this is confirmed by the standard model (‘t Hooft’s matching condition). However, if the MF algebra had a nice unitary rep, the Hilbert space on which it acts would define a consistent quantum theory with this kind of gauge anomaly, contradicting physical intuition.

Posted by: Thomas Larsson on April 3, 2007 6:33 AM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

You first define Atiyah data using very natural looking formulas, and then you pass to the Segal picture by using the “metric” to raise indices. Its kind of a bit artificial, and you get the feeling you’re just putting a “man-made spin” on something which the underlying maths itself was never calling out for.

But that “metric” is the inner product that makes the vector space of states into a Hilbert space. It’s part of the theory from the beginning, not something tacked on later.

Posted by: A.J. on April 3, 2007 6:34 PM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

True… but let’s see. I guess my question is “Can you give a nice formula for the operators Z(M):Z(M in)Z(M out)Z(M) : Z(\partial M_{in}) \rightarrow Z(\partial M_{out})?”

I used to think that the formula went in the standard way (eg. from Segal’s notes on topological field theories, Lecture 1). That is, each manifold MM (of dimension nn or n1n-1) has a space of fields C MC_M which lives on it. The quantum field theory

(1)Z:nCobVect Z : nCob \rightarrow Vect

assigns to a closed (n1)(n-1)-manifold Σ\Sigma the space of functions on the space of fields over Σ\Sigma,

(2)Z(Σ)=L 2(C Σ) Z(\Sigma) = L^2 (C_\Sigma)

and it assigns to a cobordism M:Σ 1Σ 2M : \Sigma_1 \rightarrow \Sigma_2 the linear operator Z(M):Z(Σ 1)Z(Σ 2)Z(M) : Z(\Sigma_1) \rightarrow Z(\Sigma_2) obtained by integrating over the interpolating fields. That is, if ψ i^Z(Σ i)\hat{\psi_i} \in Z(\Sigma_i) are wave functions sharply peaked around the fields ψ iC Σi\psi_i \in C_\Sigma_i respectively, then the matrix elements of Z(M)Z(M) are given by

(3)ψ 1^|Z(M)|ψ 0^= ψC M:ψ| Σ i=ψ ie iS[ψ]Dψ. \langle \hat{\psi_1} | Z(M) | \hat{\psi_0} \rangle = \int_{\psi \in C_M : \psi|_{\Sigma_i} = \psi_i} e^{iS[\psi]} D \psi.

(See for instance page 1 and 2 of Segal’s notes.) That’s an elegant formula for the linear operator Z(M)Z(M)! But strangely enough I haven’t yet seen an example where it actually works out in this way. When one implements it naively, or even not-so-naively, it actually is false .

Lets look at the finite group model, made rigorous by Freed and Quinn. Look at their formula in Theorem 2.13. Its written in the Atiyah picture! In other words, the elegant groupoid language they develop leads to a nice formula not for the linear operator Z(M):Z(Σ 1)Z(Σ 2)Z(M) : Z(\Sigma_1) \rightarrow Z(\Sigma_2) but rather for the vector Z(M)Z(M)Z(M) \in Z(\partial M).

Also in the recent paper by Pickrell for P(ϕ) 2P(\phi)_2 QFT’s - there again its more natural to write down formulas in the Atiyah picture.

I know it seems really strange, but… try it! Try implementing the fundamental formula above in the finite group model. You’ll need to interpret the integrals and the measure correctly… that’s no problem. We’re all grown-ups here, we know how to elegantly integrate over groupoids, and how to juggle with spans of groupoids and the like pic. Take the untwisted model where the action is zero. Its weird, but I’ve tried a few ways of writing down the formula and they always fail. I mean you can write down a formula which works for the cylinder, but then it doesn’t work for the disc (death of a circle)… or vice-versa. Annoying!

Perhaps one needs some technology from Jeffrey Morton’s toolbox, or perhaps even from the tale of groupoidification.

Posted by: Bruce Bartlett on April 4, 2007 12:23 AM | Permalink | Reply to this

Re: Oberwolfach CFT, Arrival Night

Hi Bruce,

I’ll need a few days to find time to actually write down some Dijkgraaf-Witten evolution operators. But I’ll post again after I’ve done so.

I do see your point from a different angle.

Mathematically, the progression

Vector, vector space, 2-vector space, …

looks considerably more natural than

linear transformation, vector space, bimodule,…

But I think Urs may be right, in that the sewing constraints are strong enough that we should also get the metric for free, and consequently lack the right to distinguish between H and H^*. After all, not just any 2-vector space will do; we basically must have one coming from a bimodule.

Posted by: A.J. on April 4, 2007 8:31 AM | Permalink | Reply to this
Read the post Oberwolfach CFT, Monday Evening
Weblog: The n-Category Café
Excerpt: Some random notes concerning, bundles, functorial quantum field theory and algebraic QFT.
Tracked: April 3, 2007 12:55 AM
Read the post Local Nets from 2-Transport
Weblog: The n-Category Café
Excerpt: How to obtain a local net of observables from an extended functorial QFT.
Tracked: December 9, 2007 6:04 PM

Post a New Comment