Classical String Theory and Categorified Symplectic Geometry

June 2, 2008

Classical String Theory and Categorified Symplectic Geometry

Posted by John Baez

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As categorification sweeps the land, it hits some areas sooner than others. While it’s had a big impact on fancy forms of mathematical physics like ‘topological quantum field theory’, it hasn’t yet encroached quite so visibly on more basic subjects, like classical mechanics.

However, this is typical of mathematical ideas: they’re often discovered in fancy contexts, but when it becomes clear how simple they are, their realm of application spreads. I believe categorification can be applied to classical mechanics… and then it leads to higher-dimensional field theories, including classical string theory!

Chris Rogers, Alex Hoffnung and I are writing a paper on one aspect of this topic:

Chris Roger, Alex Hoffnung and John Baez, Categorified symplectic geometry and the classical string, draft version. For a more up-to-date version try this.

Abstract: A Lie 2-algebra is a ‘categorified’ version of a Lie algebra: that is, a category equipped with structures similar to those of a Lie algebra, but where the usual laws hold only up to isomorphism. It is well known that given a manifold equipped with a symplectic 2-form, the Poisson bracket gives rise to a Lie algebra of observables. Multisymplectic geometry generalizes the classical mechanics of point particles to $n$ -dimensional field theories, decribing such a theory in terms of a ‘phase space’ that is a manifold equipped with a closed nondegenerate $(n+1)$ -form. Here, given a manifold with a closed nondegenerate $3$ -form, we construct a Lie 2-algebra of observables. We then describe how this Lie 2-algebra can be used to describe dynamics in classical bosonic string theory.

In fact, Chris just gave a series of 5 lectures on the subject, which you can see here…

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Chris Rogers did physical chemistry before switching to math and coming to UCR, so he can think and calculate like a physicist — which comes in handy! His lectures make a nice introduction to our paper, since he was explaining the ideas to some of my other graduate students, which meant that he needed to explain things from scratch, not assuming vast prior knowledge. So, if you want to get started on Hamiltonian mechanics, symplectic and multisymplectic geometry, classical string theory and Lie 2-algebras, don’t be scared — here’s a place to start!

Lectures by Chris Rogers (notes by Alex Hoffnung):

It’s amusing to note that the key idea behind categorified classical mechanics — boosting the symplectic 2-form to a multisymplectic 3-form — goes back to the work of DeDonder and Weyl in the 1930s. But only much later was it realized that 2-forms are to line bundles as 3-form are to gerbes! This makes the role of categorification more explicit. By showing that a manifold with a multisymplectic 3-form gives a Lie 2-algebra of observables, we’re making it so darn explicit that it can no longer be ignored!

Fans of Lie 2-algebras will enjoy that we actually get both a ‘semistrict’ Lie 2-algebra — where the Jacobi identity holds up to isomorphism but the bracket is skew-symmetric on the nose — and a ‘hemistrict’ one as defined by Roytenberg — where the Jacobi identity holds on the nose but the skew-symmetry holds only up to isomorphism.

But, they’re isomorphic!

I was very surprised when Chris and Alex discovered this.

Posted at June 2, 2008 8:41 PM UTC

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91 Comments & 2 Trackbacks

Re: Classical String Theory and Categorified Symplectic Geometry

Is there a version of geometric quantization in this setting yet?

Posted by: A.J. on June 2, 2008 10:22 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Good question. In fact, that’s question 3 in the Conclusions! We have some things to say about it, but I believe the really quick answer is “not yet”.

If I’m wrong, someone had better tell me quick!

Posted by: John Baez on June 2, 2008 10:40 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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One will want to talk about the “space of sections” of the gerbe that is rationally classified by the 3-form. That is the “space” of vector bundles twisted by that gerbe. They form a “category of states”.

The best known example is: the charged membrane propagating on $B G$ coupled to the Chern-Simons 2-gerbe. Transgressed to loop space of $B G$ , i.e. $G$ this leaves a gerbe on $G$ . The “space of states” is sections of this, which by Freed-Hopkins-Teleman is the category of reps of the loop group of $G$ . Which indeed is what CS theory should assign to the circle.

Posted by: Urs Schreiber on June 2, 2008 11:18 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Great. I was already waiting for that to appear.

I need to run, am on the road at the moment, but one quick comment on this:

But, they [the semistrict and the hemistrict Lie 2-algebras] are isomorphic!

I was very surprised

I haven’t seen the details, but: as far as I know roughly such kind of isomorphism was part of the motivation for Dmitry to come up with the hemistrict definition in the first place. It was well known that the Courant Lie algebroid admitted two different kinds of brackets, one skew but failing Jacobi, the other non-skew but satisfying a suitable identity. But they are equivalent.

The Courant Lie algebroid over the point is just the string Lie 2-algebra. Which is entirely controllled by the skew Jacobiator $\langle \cdot, [\cdot,\cdot]\rangle$ . Which in turn is entirely controlled by the symmetric skew-symmetrizator $\langle \cdot,\cdot \rangle$ .

I’d need to remind myself of some details of Dmitry’s work, but I think this is what is going on. So it seems that the passage semistrict $\leftrightarrow$ hemistrict mimics the relation between Lie algebra cocycles and Lie algebra invariant polynomials which are in transgression with each other.

To some extent one can see this relation already while staying entirely withing semistrict Lie $n$ -algebras, by passing to inner automorphism $(n+1)$ -algebras (dual to the Weil algebra): that also introduces a passage from anti-commuting to commuting and supports an analogous Lie $n$ -algebraic manifestation of Lie algebra cohomology and invariant polynomials.

That latter observation was the basis of our work on higher Chern-Simons connections obstructing lifts to higher String-connections, as you know.

I am really looking forward to reading your article with Chris and Alex. Printing now. Should have some spare time this evening.

Oh, and just two hours ago I had met Mike Stay at Google. They say there is no such thing as a free lunch. But at Google there is! When Mike is done with categorifying GoogleDocs he promises I might even be able to compile my LaTeX files on the Grid.

It’s seems hard not to exchange thoughts with one of your grad students these days, one way or another…

Posted by: Urs Schreiber on June 2, 2008 11:03 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

Urs wrote:
It was well known that the Courant Lie algebroid admitted two different kinds of brackets, one skew but failing Jacobi, the other non-skew but satisfying a suitable identity. But they are equivalent.

But the work in going to an sh version is a
lot easier in the one skew but failing Jacobi case - at least with our present knowledge/machinery

Posted by: jim stasheff on June 3, 2008 12:18 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Urs wrote:

It was well known that the Courant Lie algebroid admitted two different kinds of brackets, one skew but failing Jacobi, the other non-skew but satisfying a suitable identity. But they are equivalent.

Yes… I was expecting our semistrict and hemistrict Lie 2-algebras to be equivalent in the technical sense, but I was shocked when they were isomorphic.

This is no doubt an overly subtle distinction — in fact an ‘evil’ distinction in the technical sense of that term. But, I was nonetheless shocked.

Anyway, your point is a good one: I should go back and ponder this Courant Lie algebroid stuff and see what the big picture is here. Thanks for the hint about ‘invariant polynomials’ versus ‘Lie algebra cocycles’.

Oh, and just two hours ago I had met Mike Stay at Google.

Cool! I forget if you ever met before. I can visualize him welcoming you:

They say there is no such thing as a free lunch. But at Google there is!

Yes! Of course people world-wide paid for that lunch of yours… simply by clicking on links to ads. Modern civilization is weird.

It seems hard not to exchange thoughts with one of your grad students these days, one way or another…

That’s good to hear! My plan is to start a revolution by getting lots of smart young people interested in $n$ -categories. We just need a network with one person in each area of mathematics, physics and computer science.

But you count for about 10 normal smart young people. This will let me retire a decade earlier.

Posted by: John Baez on June 3, 2008 12:52 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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but I was shocked when they were isomorphic.

Ah, I see. Sure.

Thanks for the hint about ‘invariant polynomials’ versus ‘Lie algebra cocycles’.

You should make this a TWF slogan so that everybody knows it:

Lie cocycles are Jacobiators and their coherences

Lie invariant polynomials are alternizators and their coherences

(Well, the last sentence is a should-sentence :-)

I forget if you ever met before.

No, i hadn’t. It was great fun. We discussed the mysterious relation between computation and physics, your work with him and how he is applying it to make GoogleDocs run LaTeX in a secure way. Really, thought of correctly that requires 2-categories.

Modern civilization is weird.

And the relevance of advertisements in modern capitalism is astonishing. And the astonishing thing about Google is not only that everybody runs around with a T-shirt saying “I am feeling lucky”, but that they actually manage to make advertisements a pleasant (or at least not unpleasant) thing.

My plan is to start a revolution by getting lots of smart young people interested in $n$ -categories.

It’s happening. A year ago or so I would not have believed the amount of $\infty$ -category theory we have been throwing around leisurely at HIM last week I was there. At some point Freed in his lecture almost apologized for talking about an ordinary 1-groupoid. :-)

But you count for about 10 normal smart young people.

Can I have this in a letter of reference? ;-) I’ll need one or two in a few days.

Posted by: Urs Schreiber on June 3, 2008 4:16 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Thanks, Urs. I had a lot of fun, too.

Posted by: Mike Stay on June 4, 2008 12:36 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

Can you comment on the loop spaces of a symplectic manifold, and their 1-plectic, 0-plectic, and -1-plectic structures? (Or however far out it is reasonable to go.)

I imagine Floer homology should come in there.

Posted by: Allen K. on June 3, 2008 4:04 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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It ought to be true that an $n$ -plectic structure on $X$ transgresses to an $(n-k)$ -plectic structure on $L^k X$ .

If that doesn’t come out the definition is bad.

The formula on p. 19 is that transgression for $n=2$

$\tilde \omega = \int_{S^1} ev^* \omega$

where $ev : S^1 \times L X \to X$ is evaluation.

Posted by: Urs Schreiber on June 3, 2008 4:45 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Before Jim has to do it, let me do it for him: everybody notice that back in the old days people said suspension for what i just called transgression. And vice versa.

Posted by: Urs Schreiber on June 3, 2008 4:48 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

I was hoping that part of the answer would include how these notions simplify/trivialize when one gets down to very small plecticity.

Posted by: Allen K. on June 4, 2008 12:49 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Allen wrote:

I was hoping that part of the answer would include how these notions simplify/trivialize when one gets down to very small plecticity.

Yes — I liked your original question very much, but haven’t had 10 minutes to think about it yet.

Since an $n$ -plectic manifold is one equipped with a nondegenerate closed $(n+1)$ -form, 1-plectic manifolds are ordinary symplectic manifolds, and below that things become a bit weird.

A 0-plectic manifold $X$ is one equipped with a nondegenerate closed 1-form $\omega$ . In this case the nondegeneracy condition means that

$\omega(v) = 0 \Rightarrow v = 0$

for every vector field $v$ . This can only happen if $X$ has dimension $\le 1$ . In the 1-dimensional case, $\omega$ must then be a volume form.

This is very restrictive, so unless I made some stupid mistake the loop space of a 1-plectic manifold is not 0-plectic… which makes me very worried about my general claim that the loop space of an $n$ -plectic manifold is $(n-1)$ -plectic.

In fact I now think my mental argument for that general claim (which I stuck in the paper at the last minute) is wrong. The transgression of a closed form is closed, but I’m afraid the transgression of a nondegenerate form may be degenerate!

So, I need to think about this more. Thanks: your question was very useful, even if its effect was to remove misinformation instead of reveal new information.

Posted by: John Baez on June 4, 2008 3:32 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

IIRC, in Witten’s argument for Atiyah-Singer via Duistermaat-Heckman applied to loop space (written up by Atiyah in “Circular symmetry and stationary phase”), the 2-form on the loop space of a Riemannian manifold is only presymplectic. And of course the curvature of a connection on a circle bundle isn’t necessarily symplectic. So I’m going to need more convincing that nondegeneracy is so fundamental here.

(Here’s a paper on-line about the Witten stuff, in part.)

One benefit of nondegeneracy in the 1-plectic case is Darboux’ theorem. I imagine I should be looking in your references to other people’s work on n-plectic manifolds, but out of laziness I’ll ask here: is there a Darboux theorem for these higher versions? And if not, should local triviality be added to the assumptions?

Posted by: Allen K. on June 4, 2008 4:17 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Allen wrote:

I’m going to need more convincing that nondegeneracy is so fundamental here.

Well, the nondegeneracy of the $n$ -plectic structure is certainly unnecessary for some things — both the things you mentioned and the very simplest thing, namely getting a $\U(1)$ $n$ -bundle from an integral closed $(n+1)$ -form.

It does however play a big role in getting a vector field $v_F$ from an observable $F$ (details worked out for all $n$ ) and thus defining a Lie $n$ -algebra of observables with bracket $\{F,G\} = v_F(G)$ (details worked out for $n = 1,2$ ). And that’s what our paper is about.

One benefit of nondegeneracy in the 1-plectic case is Darboux’ theorem. I imagine I should be looking in your references to other people’s work on n-plectic manifolds, but out of laziness I’ll ask here: is there a Darboux theorem for these higher versions?

No, there’s not — it fails already for 2-plectic manifolds. The paper we cite by Gotay, Isenberg Marsden and Montgomery complains about this fact.

And if not, should local triviality be added to the assumptions?

They suggest doing something like that. To go beyond a certain point in the theory, it may be needed. I haven’t hit that point yet.

In short, it’s probably interesting to study all sorts of structures in parallel: closed $(n+1)$ -forms, integral ones, nondegenerate ones, and maybe ones of ‘standard local form’.

Posted by: John Baez on June 4, 2008 6:55 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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so unless I made some stupid mistake the loop space of a 1-plectic manifold is not 0-plectic

It does not seem that this is something to be worried about. While the definition of 0-plectic itself seems to make sense, you can’t get a Lie 0-algebra out of it anyway. The equation $\iota_v \omega = d F$ is pointless for $\omega$ a 1-form anyway.

What seems more interesting to me is whether the loop space of a 2-plectic manifold is 1-plectic.

And this seems to be true – for the space of unparameterized loops.

Consider first parameterized loops. Let $v$ be a vector on loop space and assume that $\int \omega(\gamma'(\sigma),v(\gamma(\sigma)), w(\gamma(\sigma))) \, d\sigma$ vanishes for all vectors $w$ on loop space. We want to find out what this implies for $v$ .

So look at sequences of such $w$ whose members are supported on ever smaller subsets of the circle. It should follow that the above integral can vanish for all $w$ if and only if the integrand

$\omega(\gamma'(\sigma),v(\gamma(\sigma)), w(\gamma(\sigma)))$

already vanishes for all $\gamma$ for any fixed $\sigma$ . But since where it is supported $w$ is still arbitrary and $\gamma'$ takes on all possible values at a given $\sigma$ as we vary $\gamma$ , this seems to imply that either $v(\gamma(\sigma))$ is proportional to $\gamma'(\sigma)$ for all $\gamma$ (at the given $\sigma$ ) or $\iota_{v(\gamma(\sigma))}\omega = 0 \,.$

Since $\omega$ itself was non-degenerate, this finally means that $v$ has to be proportional to the generator of reparameterizations on loop space. But that generator precisely vanishes as we pass to un-parameterized loops.

Posted by: Urs Schreiber on June 7, 2008 2:01 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

unparameterized loops meaning equivalence classes of loops with respect to oriented diffeos (or homeos)?

but not all thin homotopies?

Posted by: jim stasheff on June 7, 2008 4:33 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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unparameterized loops meaning equivalence classes of loops with respect to oriented diffeos (or homeos)?

A parameterized based loop is a map $\gamma : [0,1] \to X$ such that $\gamma(0) = \gamma(1)$ . If we think of the circle $S^1$ as a smooth space with basepoint, that’s the same as maps $S^1 \to X \,.$

For my above argument to make sense, it appears to be sufficient, already, to pass to the quotient where loops are identified that differ only by the choice of basepoint, i.e. which differ by precomposition with a rigid rotation of the circle.

What I had in mind when writing the comment, though, was deviding out pre-composition with all orientation-preserving diffemorphisms of the circle.

But it occurs to me now that diving out much less will do the trick, too.

Posted by: Urs Schreiber on June 9, 2008 4:24 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Thanks, Urs and Jim!

I’ll ponder this and try to turn it into a theorem.

Posted by: John Baez on June 8, 2008 7:47 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

What kind of loop space are you talking about after example 2.3? I’m guessing based not free (but I don’t think I should have to guess).

Posted by: Allen K. on June 3, 2008 4:08 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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but I don’t think I should have to guess

But it shouldn’t matter, should it?

The only potential issue that I see is that the transgressed plectic form is still nondegenerate. That might depend on what tangents exactly one allows on loops (I am unsure about this at the moment) but doesn’t seem to depend on whether or not loops are based. It seems.

But maybe I am missing something.

Posted by: Urs Schreiber on June 3, 2008 5:32 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Just finished reading your article. Imagine me, while reading, sitting on some dune over a cliff looking over the pacific ocean, silence except for a fresh wind under the blue sky.

Well, I guess you have that every second day if you want…

Anyway, very nice article. This is gonna be another classic. Here are some comments:

On p. 2:

when describing the history of how line 2-bundles and 3-bundles were realized to be present in string theory one must mention Dan Freed. It was Freed-Witten who first realized that the “B-field” is a gerbe with connection. It was Freed-Diaconescu-Moore whose first realized that the “C-field” is a 2-gerbe with connection.

I would suggest to cite the seminal

Dan Freed, Dirac Charge Quantization and Generalized Differential Cohomology

which discusses this and all its twisted generalized versions and whatnot. And all that starting from a truly foundational new look at just ordinary electromagentism.

I am predicting that in 50 years when people look back at the achievements of late 20th century mathematical physics, this article will stand out. It is the true successor of Dirac’s 1932 article.

Of course Freed there never mentions categorification or gerbes of $n$ -bundles. He phrases it in terms of differential characters. But that’s just one of the 50 ways to talk about $(n-1)$ -gerbes with connection.

p. 3: maybe mention that for a given integral closed 2-form there are in general line bundles with non-isomorphic connections realizing them.

p. 4, second paragraph “there are in general various way-s-“

p. 5: while it is implied by the list, maybe state also what the bracket of an object with a morphism is

p. 7: third line: “The reason is given an…”

p. 9 before the last displayed equation: “to a 2-term chain complexes vector spaces”

Posted by: Urs Schreiber on June 3, 2008 4:37 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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I am having weird problems with my hardware. That makes my comments come in a little scattered.

But here is another editorial-like comment on the article:

the use of the term “string theory” in the article is sub-obtimal. What you really do is look at the quantization of the Klein-gordon field in two dimensions. That can be read as the dynamics of a string. But string theory is the second quantization of that, in whichever form, which you don’t want to discuss.

Saying “string theory” when discussing quantization of a 2-plectic manifold is completely analogous to saying “QED” when discussing quantization of the 1-plectic manifold describing the phase space of an electron.

so in particular, you shouldn’t say, as you do at the beginning of section 4, that “string theory is a theory of maps $\Sigma \to X$ ”. No. The $\Sigma$ -model that string theory is the second quantization of is a theory of such maps.

Similarly, under a “solution of classical bosonic string theory” people don’t understand the equation you give, but a solution to an effective gravitational theory in 26 dimensions.

I think instead of “String theory” you want to say “the quantum string” or the like.

Posted by: Urs Schreiber on June 3, 2008 5:26 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Urs wrote:

the use of the term “string theory” in the article is sub-obtimal. What you really do is look at the quantization of the Klein-Gordon field in two dimensions.

Actually no quantization: just classical stuff!

That can be read as the dynamics of a string. But string theory is the second quantization of that, in whichever form, which you don’t want to discuss.

Right. Everything we’re doing here could work equally well for any $1+1$ -dimensional classical field theory where the action depends only on the field and its first derivatives. Our focus on the classical string is mainly just an expository tactic for explaining several analogies in a coherent way — in particular:

$A$ field : line bundle :: $B$ field : gerbe

and

$A$ field : 1-plectic structure :: $B$ field : 2-plectic structure

and

1-particle : 1-plectic structure :: 2-particle : 2-plectic structure

I think instead of “String theory” you want to say “the quantum string” or the like.

But we’re not doing anything quantum, except a little preliminary motivation from geometric quantization. That’s why we said ‘classical bosonic string theory’. Is ‘the classical string’ better?

In their work on multisymplectic geometry, Gotay, Isenberg, Marsden and Montgomery say ‘the bosonic string’. But they are experts in classical mechanics, not string theorists. I’m not a string theorist either (obviously). But, I agree that it’s good to talk in a way that doesn’t make string theorists think we’re silly.

Posted by: John Baez on June 3, 2008 7:03 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Right, sorry, “quantum” was a red herring.

The problem is that the study of that 1+1 d theory is not yet string theory, classical or not. That may sound like hair-splitting, but I think the two statements I mentioned (saying that string theory is a theory of maps from Sigma to somewhere and addressing the worldsheet eqm as a “solution to string theory”) should definitely be avoided. You could argue that the standard usage of the word “string theory” is bad, which may be true. But still.

Posted by: Urs Schreiber on June 3, 2008 7:27 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Urs wrote:

You could argue that the standard usage of the word “string theory” is bad, which may be true. But still.

Just tell me what we should say. Before you suggested “the quantum string”, which would be fine — except we’re not doing anything quantum. So what then? “The classical string”?

(My long-winded comment was just an explanation of why I don’t feel like mainly saying “the (1+1)d massless Klein–Gordon equation”.)

Posted by: John Baez on June 3, 2008 8:10 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Just tell me what we should say. Before you suggested “the quantum string”, which would be fine — except we’re not doing anything quantum. So what then? “The classical string”?

Sorry, I would have replied earlier but couldn’t due to hardware problems and being busy all day.

I would say:

title of section 4: “An application to the bosonic string” or “An application to the classical bosonic string” if you want to emphasize that you are not quantizing it.

First sentence of section 4: “The bosonic string is a theory of maps…” (no “clasical” needed here, strictly speaking)

Last sentence of the first paragraph: “A solution of the classical bosonic string is…” or “A classical trajectory of the bosonic string is…”

Posted by: Urs Schreiber on June 4, 2008 3:04 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

I like the first sentence of the Introduction:

It is becoming clear that string theory can be viewed as a `categorification’ of particle physics, in which familiar algebraic and geometrical structures based in set theory are replaced by their category-theoretic analogues.

If I’m not mistaken, this goal is one of the things that got Urs into learning categorification in the first place!

I also like what you said here:

However, this is typical of mathematical ideas: they’re often discovered in fancy contexts, but when it becomes clear how simple they are, their realm of application spreads. I believe categorification can be applied to classical mechanics…

I hope you don’t mind a slight digression that I think is still in the spirit of this post…

A former colleague of mine (before he was a colleague) once sent Urs an email asking about categorification and finance. Urs forwarded the mail to me to see what I thought. It’s a small world and I ended up working with him. He is a super smart guy and was actually good friends with Ross Street. Three years later, it might be time to revisit the idea.

One of the bedrocks of mathematical finance is the Black-Scholes equation. This equation helps determine/analyze the fair price of stock options. It involves stochastic calculus.

The Black-Scholes equation can be mapped to the Schrodinger equation. I have a writeup somewhere (or maybe on some forum somewhere) showing the details, but anyone here can easily work it out. The analogy I want to point out is that the Black-Scholes equation can be thought of as modeling the dynamics of “point prices” just as the Schrodinger equation models the dynamics of “point particles”.

There are two primary financial instruments that populate any traditional portfolio: stocks and bonds. Stocks are described by a “point price” and hence stock options are governed by the Black-Scholes equation. Bonds are more complicated because there is no 0-dimensional “point price” for bonds. Bonds depends on a 1-dimensional “price curve”. There are models to describe the dynamics of 1-d “price curves”, but nothing has had quite the impact that the original Black-Scholes model did.

If I could clone myself and if I were smarter, I would attempt to “categorify” the Black-Scholes equation to model the dynamics of the extended 1-d “price curves”.

It might sound silly, but just as string theory seems to relate to a categorification of point particle dynamics, I suspect one could develop a bond option pricing theory based on a categorification of the Black-Scholes equation (for “point prices”).

If anyone cares to work this out, I’d be glad to help with any of the relevant finance background. It should be quite simple for anyone here, but could potentially be quite significant in mathematical finance. I can’t claim to be the first to think of this idea because this is probably precisely what my old colleague had in mind, but I think it is a good example of how the “realm of application” of categories can spread.

Posted by: Eric on June 3, 2008 5:56 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

I’m glad you like those sentences, Eric. I work hard on ‘em.

Eric wrote:

A former colleague of mine (before he was a colleague) once sent Urs an email asking about categorification and finance. Urs forwarded the mail to me to see what I thought. It’s a small world and I ended up working with him. He is a super smart guy and was actually good friends with Ross Street.

I think I may know who you’re talking about, although I’m terrible with names so his name isn’t coming back to me. The idea, however, sounds familiar.

If I could clone myself and if I were smarter, I would attempt to “categorify” the Black-Scholes equation to model the dynamics of the extended 1-d “price curves”.

If you could clone yourself, you would be smarter — since nobody can do that yet. You could make money on human cloning and say goodbye to finance.

Seriously, I think it would be easier to start by guessing a stochastic differential equation for the time evolution of a price curve, and later see what that had to do with categorification. After all, there are already stochastic partial differential equations that describe the random wiggling of strings. Maybe one of these will help, since the Black–Scholes equation is just the Brownian motion of a point particle (after a change of variables).

Our paper shows (perhaps not as clearly as it should) that any 2d classical field theory is related to categorification. The same thing seems to be true for quantum field theory. So, I wouldn’t be surprised if any 2d stochastic field theory was also related to categorification.

But, I think it might be easier to start with some ideas in economics, derive the right stochastic PDE, and worry about categories later.

By the way, I know a refugee from mathematical physicist, a student of Raoul Bott who now works in finance — his name is Eric Weinstein. He gave a very nice talk at the Perimeter Institute about how arbitrage is related to gauge theory. Executive summary: if you can carry money around a loop and have it come back bigger, you’ve got a viable business model.

It’s tempting to generalize his ideas to higher gauge theory!

But, if I were going to work on mathematical finance, I would take a more crass attitude. I’d try to write down a valuable differential equation, publish it, and collect my Nobel prize as soon as possible… avoiding Black’s mistake.

Posted by: John Baez on June 3, 2008 7:42 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

By the way, I know a refugee from mathematical physicist, a student of Raoul Bott who now works in finance — his name is Eric Weinstein. He gave a very nice talk at the Perimeter Institute about how arbitrage is related to gauge theory. Executive summary: if you can carry money around a loop and have it come back bigger, you’ve got a viable business model.

Interesting. I wonder if he reads any finance forums? I wrote this on May 18, 2002.

Arbitrage and Holonomies

Hi,

I’ve got a vague glimmering of lights sparkling somewhere in the back of my head and I’m hoping someone might point me to the path of true enlightenment…

From my understanding, arbitrage amounts to simultaneously buying and shorting an asset at some point A in the “market manifold”, carrying your “portfolio” through two distinct paths P1 and P2 in the market and selling the asset you’re holding at some point B in the market as illustrated below

[Beautiful figure depicting a situation very similar to the AB effect lost in cyberspace]

If there were arbitrage, you could do this and make a profit.

But this sounds a lot like holonomy, i.e. parallel transporting a tangent vector around a closed curve. If there is no curvature in the manifold, then you get the same tangent vector as you started with at the end of the journey. If there is curvature, then you will get a rotated vector when you return.

Am I crazy, or does this have some relation to arbitrage? Could arbitrage somehow be related to curvature? Where can I read up more about this?

Thanks for any words of enlightenment.

Eric

I had never heard anyone relating arbitrage and holonomies before this (although Illinski had related gauge theories to finance in a slightly different context).

Ironically, later posts in the thread refer to some guy named Baez :D

Posted by: Eric on June 3, 2008 8:31 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Speaking of gauge theories and finance, look who’s dabbling in finance/economics:

Time and symmetry in models of economic markets

Lee Smolin

These notes discuss several topics in neoclassical economics and alternatives, with an aim of reviewing fundamental issues in modeling economic markets. I start with a brief, non-rigorous summary of the basic Arrow-Debreu model of general equilibrium, as well as its extensions to include time and contingency. I then argue that symmetries due to similarly endowed individuals and similar products are generically broken by the constraints of scarcity, leading to the existence of multiple equilibria. This is followed by an evaluation of the strengths and weaknesses of the model generally. Several of the weaknesses are concerned with the treatments of time and contingency. To address these we discuss a class of agent based models. Another set of issues has to do with the fundamental meaning of prices and the related question of what the observables of a non-equilibrium, dynamic model of an economic market should be. We argue that these issues are addressed by formulating economics in the language of a gauge theory, as proposed originally by Malaney and Weinstein. We review some of their work and provide a sketch of how gauge invariance can be incorporated into the formulation of agent based models.

Posted by: Eric on March 10, 2009 6:41 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

cf also
The thermodynamic approach to market
by Victor Segeev
jim

Posted by: jim stasheff on March 11, 2009 2:24 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

But, I think it might be easier to start with some ideas in economics, derive the right stochastic PDE, and worry about categories later.

Right. Maybe I misunderstood the basic concept of what you were saying. The economic ideas are already there in the Black-Schole formulation for “point prices” and I was hoping that getting the right “price curve” formulation would amount to simply* categorifying the Black-Scholes concepts.

*I don’t know how simple this would be.

Posted by: Eric on June 3, 2008 8:58 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Eric wrote:

I wonder if he reads any finance forums? I wrote this on May 18, 2002….

It’s possible he read that, or it’s possible he read that paper by Illinski you mentioned, Physics of finance. Or, it’s also quite possible that he reinvented this idea on his own! I think this idea was fated to come into existence as soon as physicists started getting jobs as quants.

The economic ideas are already there in the Black-Schole formulation for “point prices” and I was hoping that getting the right “price curve” formulation would amount to simply* categorifying the Black-Scholes concepts.

*I don’t know how simple this would be.

I don’t know to ‘simply’ use categorification to do something like this. It sounds hard. It sounds a lot easier to understand the relevant ideas, guess a nice stochastic differential equation describing the time evolution of price curves, and then maybe later try to see if it was a categorified version of the Black–Scholes equation.

The Wizard in me likes to do magic tricks where I put a rabbit in a hat and pull out again, saying “Presto! Now I’ve categorified it!” It’s supposed to look easy. But a lot of work goes on behind the scenes.

Posted by: John Baez on June 3, 2008 8:17 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

I think this idea was fated to come into existence as soon as physicists started getting jobs as quants.

Yeah, I totally agree. I would really like to see his stuff. Now that I know he made the switch I’ll keep my eye open and hope to bump into him some time. Thanks for pointing out the connection.

Posted by: Eric on June 3, 2008 11:50 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Readers might also be interested in

Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates

Belal E. Baaquie

Posted by: Mike Stay on June 4, 2008 2:36 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Eric wrote:

Now that I know he [Eric Weinstein] made the switch I’ll keep my eye open and hope to bump into him some time.

I hardly ever see him, but we’re pretty good friends in a way. You see, a long time ago we were both in a kind of informal physics discussion club at MIT, together with my friends Steve Sawin and Scott Axelrod. So, if you ever see him or email him, say hi.

I just noticed an abstract of a talk he likes to give, on ‘Neoclassical economics and gauge theory’.

I also see he’s going to the conference on Science in the 21st Century at the Perimeter Institute.

It all fits together, because the last time I saw him, I was at the Perimeter Institute, and he came and gave his talk on economics and gauge theory. Then we hung out and talked with Lee Smolin. Like Lee, Eric is interested in nontraditional methods of funding physics research. I can easily imagine him wanting to speak about that at this conference.

Posted by: John Baez on June 4, 2008 4:42 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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I’d like to learn more about this. Please write me at metaweta@gmail.com or post more details below. Thanks!

Posted by: Mike Stay on June 4, 2008 3:15 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

Hi Mike,

It would be great if I could get you interested in the stuff! I’ll do the best I can :)

To minimize the clutter in this thread though, I created two articles:

Categorified Option Pricing Theory

and

Black-Scholes and Schrodinger

Leaving comments with LaTeX is pretty simple. Just write $latex F = ma$ and it should render (I hope!).

Posted by: Eric on June 4, 2008 7:00 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

While I’m categorifying symplectic geometry, this grad student of mine is categorifying GoogleDocs and the stock market! I can guess who’ll end up rich.

Posted by: John Baez on June 4, 2008 8:44 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

Hi Mike,

If you are interested in learning about this, I just found a treasure trove of fantastic papers (although they weren’t exactly hidden!).

In a comment, Blake Stacey pointed out a very interesting looking a book

Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates
By B. E. Baaquie
Published 2004
Cambridge University Press

The author has a ton of papers on the arxiv as well as his web page.

In particular, this is one very neat and concise and conveys some basic ideas for the “point particle”:

Quantum Mechanics and Option Pricing

Then for a more recent article from which you can trace references that conveys the idea of yield curves as 2-d quantum field theories, have a look at this

Price of coupon bond options in a quantum field theory of forward interest rates

I have a strong hunch that this is exactly what I was looking for and can possibly be interpreted as a categorification of Black-Scholes.

Posted by: Eric on June 5, 2008 4:22 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Thanks for all the links!

In a comment, I suggested that book myself. :) Unfortunately, I haven’t had time to read it. :(

So if stocks are 0-dimensional and bonds are 1-dimensional, are there financial instruments of even higher dimension?

Posted by: Mike Stay on June 5, 2008 11:20 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

D’oh! Right. I thought it looked familiar :)

If you haven’t noticed yet, I have a very thick skull and it usually takes being told something 10-20 times before it begins to sink in.

As far as higher dimensional objects are concerned, never underestimate the ability of these quants to come up with complex financial instruments that no one understands. Ever heard of a subprime CDO? :)

I’ll try to think of a simple example that might be thought of as a higher dimensional object, but nothing obvious pops out at me. I’m still fairly new to finance myself. I left MIT Lincoln Lab at the end of 2004. Two of the subsequent years were not related to mathematical finance at all and were more about traditional (non-quantitative) investing. Only recently have I begun dusting off my old papers.

Posted by: Eric on June 5, 2008 11:37 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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It is becoming clear that string theory can be viewed as a `categorification’ of particle physics, in which familiar algebraic and geometrical structures based in set theory are replaced by their category-theoretic analogues.

If I’m not mistaken, this goal is one of the things that got Urs into learning categorification in the first place!

That’s why in the abstract of arXiv:hep-th/0509163 is says:

This stringification is nothing but categorification.

:-)

And the quick way to see it is the one we have menioned here: an $n$ -structure on $X$ transgresses to an $(n-k)$ -structure on $Maps(\Sigma,X)$ , for $\Sigma$ $k$ -dimensional.

Conversely, whenever you see an ordinary 1-structure on $Maps(\Sigma,X)$ , chances are good that it can be “localized” to a $(k+1)$ -structure on $X$ .

That’s what happens with String-structures: originally these were conceived as lifts of 1-bundles on $L X$ . Later it was realized that this corresponds to a lift of 2-bundles down on $X$ .

The same thing is now going on here: of course people knew how to describe the symplectic geometry of $n$ -dimensional field theory before, but on $Maps(\Sigma,X)$ . Now Alex, Chris and John point out how this may come from $(k+1)$ -plectic geometry on $X$ . This retains more information.

Posted by: Urs Schreiber on June 4, 2008 8:08 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

Thanks for confirming! I have a couple years of your papers and blog articles to catch up on :)

In the paper you sighted, you say

This already suggests that there is nothing more natural than replacing M with LM, the free loop space over M, d with the exterior derivative on LM, and so on. In other words this amounts to switching from the spectral triple for the configuration space M of a particle to that of the configuration space LM of a closed string.

Something like this is what I’m hoping to do with categorifying Black-Scholes. In fact, this is kind of what I thought waving the wand of categorification meant.

To borrow your line, “In other words this amounts to switching from the spectral triple for the configuration space M of a [stock price] to that of the configuration space LM of a [bond yield curve].”

Maybe a first step would be to write Black-Scholes as an arrow theory?

Posted by: Eric on June 4, 2008 3:55 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

By the way, if the words “Black-Scholes” and the relations to mathematical finance do not elicit much excitement, the problem I’m trying to solve can be equivalently described as a categorification of the heat equation. At least as a first pass.

Posted by: Eric on June 4, 2008 6:52 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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this amounts to switching from the spectral triple for the configuration space $M$ of a particle to that of the configuration space $L M$ of a closed string.

By the way, as I have mentioned a couple of times on this blog, since I wrote this a bit of progress has occurred on this aspect of “2d QFT as 2-spectral triple”, mostly in the form of Yan Soibelman having unpublished work on it.

Posted by: Urs Schreiber on June 4, 2008 9:16 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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“This is different than the usual tensor product of chain complexes” p.10

I know there are transatlantic differences, but surely ‘different from’ here. If only I had access to this.

Ah, I do. Here are our uses:

$\array{ & From & to & than \\ UK writing & 87.6\% & 10.8\% & 1.5\% \\ UK speech & 68.8\% & 27.3\% & 3.9\% \\ US writing & 92.7\% & 0.3\% & 7.0\% \\ US speech & 69.3\% & 0.6\% & 30.1\% }$

About a sample of recordings of language in situations of greater and lesser formality, Iyeiri et al. say

…the proportion of different than to the total of relevant examples is the smallest in the White House files, whereas the ratio is the largest in the files of the national meetings on reading tests, where both men and women use different than more frequently than different from. Furthermore, there is a clear tendency for women to use different than less frequently than men except in the files of the reading tests, where the gender distinction is very slight, although we do admit that the absence of different than in the White House female files may be ascribable to lack of evidence. Supposing that formal settings discourage the use of different than, we could surmise that the settings of the White House press conferences are the most uptight while the settings of the reading committee meetings are the most relaxed. Furthermore, it is also a reasonable conjecture deduced from the above graph that women are slower than men to feel laid back. Men are inclined to use different than reasonably often in the settings of the faculty meetings and the national meetings on mathematics tests, whereas even in the same settings women are careful about their use of language. It is only in the setting of the meetings on reading tests that women start using different than at the same level as men.

Clearly you were too relaxed (and too male) while writing this paper.

Posted by: David Corfield on June 3, 2008 2:24 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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David quoted a study showing:

…the proportion of different than to the total of relevant examples is the smallest in the White House files, whereas the ratio is the largest in the files of the national meetings on reading tests, where both men and women use different than more frequently than different from.

See? This proves that in the US, people tend to use different from in meetings dominated by an illiterate idiot, while experts in reading prefer different than.

It’s all a question of how you interpret the data…

More seriously: introspecting, I think I might tend to say “different from” if I’m in the midst of explaining how A differs from B in some particular aspect, e.g.: “this yoghurt is different from that one: it’s more runny”.

On the other hand, I might tend to say “different than” if I’m simply trying to note that $A \ne B$ , e.g.: “William Bennett, the conservative pundit, is different than Bill Bennett, the British comedian”. Here I’m not trying to explain how they’re different — e.g. I’m not thinking “…. because Bill Bennett is three inches taller”. I’m simply asserting that they’re not the same guy.

However, all this could be an illusion. I’ll have to keep tabs on myself to see what I say.

But anyway: so writing “different than” in this context really looks uneducated to the British eye? If so, I’ll change it, since to me the difference is negligible.

Thanks!

Posted by: John Baez on June 3, 2008 6:17 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

I expect the easiest way for you to imagine how it sounds to me is to read it with ‘different to’ instead.

I’m sure the whole thing is horribly complicated and not worth worrying about.

Posted by: David Corfield on June 3, 2008 8:27 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

Thanks - I’m glad someone else also finds it grating on the ear or, in this case, on the eye. Only exception: this difference might be more different than that one.

Even worse is the tendency for as to be totally supplanted by like
even in the construction
a:b::c:d
read as
a is to b AS c is to d

Posted by: jim stasheff on June 4, 2008 2:43 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

John Baez said:

“ I believe categorification can be applied to classical mechanics… and then it leads to higher-dimensional field theories, including classical string theory!”

As far as I remember reading this blog, and searching anything quantum on 4D. I tried looking for it, and the best I could find was this.

I would like to study quantum on this dimension, using categories, but it seems too misterious, too weird and no where I found something that could help. Any tips on how should I proceed?

Thanks!

Posted by: Daniel de França MTd2 on June 3, 2008 2:32 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

I really need some guidance… I will try to beg you guys better later.

Posted by: Daniel de França MTd2 on June 3, 2008 8:17 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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It’s hard to guide you, since you’re not saying what you already know — or what you want to know.

Also, to be brutally frank, your inexpert use of English makes your writing very strange, e.g. “I would like to study quantum on this dimension, using categories”. I mention this only because everyone else on this blog will be too polite to do so: instead, they just won’t reply to your question!

Anyway, I get the impression you’re trying to study something about 4-dimensional quantum field theory using categories — but something that isn’t too ‘mysterious’ or ‘weird’.

That’s a vague request, but okay… have you considered the Doplicher–Roberts theorem, which reconstructs the gauge group for a 4d quantum field theory starting from its algebras of local observables?

It’s a nice result, and understanding it is a good way to learn some category theory. The abstract core of the result was proved here:

S. Doplicher and J. Roberts. A new duality theory for compact groups. Inventiones Mathematicae, 98:157–218, 1989.

but you’ll only understand the point if you know some algebraic quantum field theory (which is worth learning anyway):

Rudolf Haag, Local Quantum Physics: Fields, Particles, Algebras, Springer-Verlag, 1992.

Urs Schreiber provides a link to a more streamlined proof by Michael Müger and a summary of the abstract result. But again, this will seem mysterious and weird if you don’t know the physics context.

Anyway, that’s something that comes to mind — but it could be way too hard for you, or way too easy.

Posted by: John Baez on June 3, 2008 8:42 PM | Permalink | Reply to this

Doplicher-Roberts

That’s a vague request, but okay… have you considered the Doplicher–Roberts theorem, which reconstructs the gauge group for a 4d quantum field theory starting from its algebras of local observables?

The gauge group? Maybe the global symmetry group, but surely not the gauge group.

Posted by: Jacques Distler on June 3, 2008 9:58 PM | Permalink | PGP Sig | Reply to this

Re: Doplicher-Roberts

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Jacques wrote:

Maybe the global symmetry group, but surely not the gauge group.

Right, I was being sloppy.

Posted by: John Baez on June 4, 2008 12:00 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

“I mention this only because everyone else on this blog will be too polite to do so: instead, they just won’t reply to your question”

I’m sorry John! I merely copied from my Openoffice Word and pasted here. Anyway, even there I couldnt see very well what I was writing, because the autocorrection was set to portuguese. All the text was scrambled, and couldn’t see heads or tails…

You asked me about the status of my knowledge… I can give you some clues, something that aproximately covers my most uptodate knowledge:

-Nakahara, the book on topology, except for the last chapter (intro. to bosonic string).
-Vertex Algebra for Beginners, Kac, except for the last chapter, which I just read very superficialy (applications for what was done in the rest of the book).
-Modular Functions and Dirchlet Series, Apostol, up to chapter 1-5,6. Basicaly, the content of the book’s title.
-QFT, I can’t come up with a book right now, but I can comprehend renormalization, regularization, and a little bit of how to get rid of ghosts in QCD. I still don’t know the BRST formalism.

I am completely lost when it comes to category theory. I took this as a textbook:

-Category Theory, Awodey. I studied up to chapter 5. So, I didnt yet study formaly functors, and diddn’t even touch neither Yoneda Lemma nor Adjoints.

And that’s it.

BTW, John, I really like weirdness and craziness. I can’t stand normal things, so I got a job completely unrelated to physics (trademark analyzer at a patent office), so that I could afford my tastes.

So, after 4 years away from physics, last year, I was idling on wikipedia and found about exotic smoothness in 4 manifolds here, and this shocked me O_O O_o: “For any positive integer n other than 4, there are no exotic smooth structures on Rn; in other words, if n ≠ 4 then any smooth manifold homeomorphic to Rn is diffeomorphic to Rn.” That happened in the same time Garrett Lisi’s article was posted on arxiv. I tried to figure out what was e8, and found this by coincidence. So, I rushed to get this book, “Exoctic Smoothness and Physics”, Asselmeyer and Brans, only to find out that it was related to all sorts of weird stuff that I ALWAYS wanted to knowand MORE!!! :O For example, I was always curious to know what Seiberg-Witten invariant was all about, so, I found somewhere to start reading. Well, to be fair, I still don’t know anything about it…

Also, because of Garrett’s discussion, I also found this blog and this. I was overwheled by the level of the discussion on this forum, and thought that my struck of luck couldn’t be over. Also kept reading because I found arrows cute, just like those from Cvitanović’s book are also cute. But I really got interested in category theory when you posted the Rosetta stone article. When I saw that, I thought that I really could cover and manipulate much more easily concepts in several different mathematica fields.

Recently, I bought “The Wild World of 4-Manifolds”, from Scorpan, but I didn’t read it, although I hope I can someday read it. I want to find a crazy path for me… And right now, I’d like to have a hint on how to use categories with exotic 4-dimensional smooth manifolds, and come up with something cool.

Posted by: Daniel de França MTd2 on June 4, 2008 3:45 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

Hi People,

I’m sorry for adding noise to this board.

See you someday. :)

Posted by: Daniel de França MTd2 on June 4, 2008 11:42 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Hi Daniel,

John is right that you might want to spend a minute more polishing your comment before you post, to increase the chance that readers will reply to you, but it seems to me you are serious about what you write and are asking good questions. So I’ll be glad to reply as far as I can.

When I hear the keywords “4d manifolds” and “categories” I think: Seiberg-Witten theory.

That is a 4-d QFT which knows about topological invariants of 4d manifolds. As every QFT, in its functorial description this comes from a functor on a cobordism category. Try to google around a bit for “Seiberg-Witten” and see how far you get. Probably you’ll run into more questions. Feel free to come back with them.

BTW, I don’t think it has anything to say about exotic smooth structures on 4-manifolds. And generally I don’t know much about such exotic structures. I don’t think they have really made it into the physics side of life yet. But who knows.

Posted by: Urs Schreiber on June 4, 2008 3:37 PM | Permalink | Reply to this

Donaldson Theory

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BTW, I don’t think it has anything to say about exotic smooth structures on 4-manifolds.

Au contraire! That’s the whole point. In the Math literature, Seiberg-Witten Theory is known as “Donaldson Theory”. And the whole beauty (and subtlety) of Donaldson theory is that it is a “TQFT” that is actually sensitive to the smooth structure of the 4-manifold (and not just to its topology).

Posted by: Jacques Distler on June 4, 2008 4:25 PM | Permalink | PGP Sig | Reply to this

Re: Donaldson Theory

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Au contraire!

Oh, thanks!

This means that without knowing I gave a more helpful answer than I thought I did. :-)

Posted by: Urs Schreiber on June 5, 2008 3:09 AM | Permalink | Reply to this

Re: Donaldson Theory

I updated my post, please, would mind telling me any new sugestion?

Posted by: Daniel de França MTd2 on June 5, 2008 4:15 AM | Permalink | Reply to this

Re: Donaldson Theory

In the Math literature, Seiberg-Witten Theory is known as “Donaldson Theory”.

Not so; Donaldson theory predates Seiberg-Witten, and was much more unpleasant and technical to work with. (Indeed, it has somewhat fallen into disuse.) Both of them, of course, are differential and not topological invariants. There are hundreds of papers in the math literature quite properly calling Seiberg-Witten theory by its rightful name.

Posted by: Allen K. on June 5, 2008 9:21 AM | Permalink | Reply to this

Re: Donaldson Theory

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What we have here, is an unfortunate clash between what the physicists call various things, and the names the mathematicians use.

So … a little history.

Donaldson’s original formulation could, roughly, be characterized as: study intersection theory on instanton moduli space (the space of anti-self-dual Yang Mills connections on $X$ ).

Witten showed that this could be reformulated as a 4D TQFT: a twisted version of $N=2$ Super Yang Mills.

In 1994, Seiberg and Witten presented a “solution” to $N=2$ SYM (in the linked-to paper, to the pure $SU(2)$ theory, but this was easily generalized).

This is what physicists call “Seiberg-Witten theory”.

At least, when $b_2^+(X)\gt 1$ , this leads to a dual formulation of the $SU(2)$ Yang Mills instanton problem in terms of an abelian gauge theory: the theory of the “magnetic” gauge field coupled to the monopole which becomes light at a certain strong-coupling point on the $u$ -plane.

When $b_2^+(X)= 1$ , Seiberg and Witten’s solution cannot be localized at a point on the $u$ -plane. Instead, one must integrate over the $u$ -plane.

The formulation in terms of abelian gauge theory (when SW theory localizes in the $u$ -plane) is, indeed, much simpler to work with. But it’s not some “different” theory. It’s the same theory, formulated in dual variables.

Which reminds me of one of the funniest comments I ever heard emerge from a Park City Summer School. During the triumphant Park City session, after the SW solution appeared, someone was heard to remark, ” Yippee! Now we don’t have to learn gauge theory any more.”

Posted by: Jacques Distler on June 5, 2008 10:27 AM | Permalink | PGP Sig | Reply to this

Re: Donaldson Theory

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I should have added one more reference to my “historical” review, which would probably clarify some of the main points, for the uninitiated.

The $N=2$ Super Yang Mills Theory has a moduli space of vacua (in the case of pure $SU(2)$ , it is one complex-dimensional, and I’ve been calling it “the $u$ -plane”).

There is a deformation of the theory, which break $N=2$ to $N=1$ , producing a mass gap and isolated vacua. Prior to coming up with the Seiberg-Witten solution, Witten showed that this deformation is compatible with the topological twisting and, provided $b_2^+(X)\gt 1$ , that this provides a nice heuristic derivation of the form of the Donaldson Invariants.

Posted by: Jacques Distler on June 5, 2008 4:18 PM | Permalink | PGP Sig | Reply to this

Re: Donaldson Theory

In the Math literature, Seiberg-Witten Theory is known as “Donaldson Theory”.
…
In 1994, Seiberg and Witten presented a “solution” to N=2 SYM (in the linked-to paper, to the pure SU(2) theory, but this was easily generalized). This is what physicists call “Seiberg-Witten theory”.

Now I’m really confused. Presumably since Donaldson did his work before 1994, there should be some theory known as “Donaldson theory”? And if “in the math literature, Seiberg-Witten theory is known as Donaldson theory”, what name would you give to the thing mathematicians are talking about when they (frequently) refer to Seiberg-Witten theory?
I get the impression that you have a clear opinion on what should be called what (possibly, depending on whether one is publishing in a math or physics journal!) but I haven’t figured out how it works.

(I’m not trying to be obtuse – it just comes naturally!)

Posted by: Allen K. on June 10, 2008 1:06 AM | Permalink | Reply to this

Re: Donaldson Theory

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Now I’m really confused. Presumably since Donaldson did his work before 1994, there should be some theory known as “Donaldson theory”?

The realization that the Donaldson Invariants are the observables of a certain 4D TQFT (twisted $N=2$ SYM) followed after Donaldson’s original work, in 1988. I (and most physicists) would call that particular TQFT “Donaldson Theory”.

The solution to that theory came later, in 1994. I’d call that solution, “Seiberg-Witten Theory.”

And if “in the math literature, Seiberg-Witten theory is known as Donaldson theory”, what name would you give to the thing mathematicians are talking about when they (frequently) refer to Seiberg-Witten theory?

They’re just dual descriptions of the same 4D TQFT. In fact, since the deformation mentioned above localizes the theory at the point where the “magnetic” description is weakly-coupled, it’s quite natural that the latter provides a simpler computation of the same observables than that provided by the original “electric” description.

If you want to reserve the name “Seiberg-Witten Theory” for just the computations done in this weakly-coupled magnetic description, then what name would you ascribe to the vexing case of $b_2^+(X)=1$ , where there is no localization, and one must integrate the Seiberg-Witten solution over the whole $u$ -plane?

Posted by: Jacques Distler on June 10, 2008 5:49 AM | Permalink | PGP Sig | Reply to this

Re: Donaldson Theory

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Personally I’d be happy to call most of the work in this book ‘Donaldson theory’:

S. K. Donaldson and P. B. Kronheimer, The Geometry of Four–Manifolds, Oxford Univ. Press, Oxford, 1990.

I’d say Donaldson theory is not a physical theory, but rather what mathematicians call a ‘theory’: a body of interconnected theorems, like ‘Galois theory’.

Witten was certainly involved in coming up with some of the ideas here — but so were ADHM (Atiyah, Hitchin, Drinfel’d and Manin) and many other people. Donaldson clearly did enough to win a Fields prize.

I guess his most shocking theorem, dating to 1983, was this: “If the intersection form of a smooth, closed, simply connected 4-manifold is positive or negative definite then it is diagonalizable over the integers.” Together with the work of Michael Freedman, who classified the allowed intersection forms of closed simply-connected topological manifolds, this winds up implying the existence of non-smoothable 4-manifolds, and also (more subtly) exotic $\mathbb{R}^4$ ’s.

Posted by: John Baez on June 10, 2008 1:28 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Hi Urs,

Thank you for you answer. :)

” I don’t think they have really made it into the physics side of life yet.”

That’s why I told you about the book from Asselmeyer and Brans “Exotic Smooth and Physics”. The idea is “simple” as 1+2:

1.Only fourth dimensional have uncontable many infinite non equivalent exotic structures

2.General relativity is defined on a 4-manifold

1+2: Use these to come up with well defined irregularities, maybe particles and the cosmological constant, dark matter etc, to naturaly arise in 4-d space.

So, why do I insist on this exotic stuff? Because the existance of such structure are found because of casson handle. A casson handle is a kind of thing that works as a kind of h-cobordism in 4 dimension. But this 4 dimensional entity is formed by tying together an infinite number number of 2-handles, think of a 4 dimensional fractal constructed with 2 by filling the space with progressively smaller closed-looop ferns.

So, my idea is relating each of this 2-handles with a string, to come up with a QFT, more clearly, an exotic QFT. But I need some guidence on how to relate all this infinite struture to categories (you said “stringfication is categorification”). Maybe I need somehow to make calculus using fractals and k-skeletons?

Posted by: Daniel de França MTd2 on June 4, 2008 4:28 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Here are some articles, that constitutes can give an idea of what that book has:

Exotic Smoothness and Physics

Localized Exotic Smoothness

Exotic Smoothness on Spacetime

Cosmological Anomalies and Exotic Smoothness Structures

Differential Structures - the Geometrization of Quantum Mechanics

Posted by: Daniel de França MTd2 on June 4, 2008 4:53 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

I got at home, and rewrote the text… I hope it is clearer now:

*******************************

Hi Urs,

Thanks for answering. :)

” I don’t think they have really made it into the physics side of life yet.”

This is the reason why I cited the book from Asselmeyer and Brans, “Exotic Smooth and Physics”. It includes some of the key subjects I want to study. The motivation comes from some “simple” ideas, just like 1+2:

1.Only fourth dimensional manifolds have uncontable many infinite non equivalent exotic kinds.

2.General relativity is defined on a 4-manifold

1+2: Fields would arise naturaly from the exotic “crystal defects” of some manifolds, just because they are 4 dimensional, which is generaly accepted as a “realistic” number of dimesion.

These exotic manifolds appear when you try to fix the failure of h-cobordism theorem in 4 dimensions, that is, when you define the cobordant manifold as a casson handle
This object is formed by tying together an infinite number number of 2-handles. A possible mental picture that of that is a 4 dimensional fern fractal,
in which you make the computer monitor a 4 dimension screen and each dot in the screen correspond to a 2-handle.

So, my idea is to relate each of these 2-handles to a string, and by extension, to a QFT. Using the analogy above, we would naturaly get a fractal 4d QFT. What is cool it is that this casson handle can be nicely connected to an usual and well behaved 4 manifold, so that it can be just a small substet of this universe, like a particle or a black hole is. Also, these casson handles do not “obey” Seiberg-Witten Theory, I think.

To accomplish this I need to use categories, so I’d like some sugestions how to relate all these infinite packed strutures to categories (Urs said “stringfication is categorification”).

So, I would like to know if you guys have any sugestion or if I just should shut up and canculate everything someday.

Posted by: Daniel de França MTd2 on June 5, 2008 3:02 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Since $TOP = PL$ in dimension 4, the existence of exotic smooth structures on $\mathbb{R}^4$ can be translated into this amazing fact about triangulations:

There are lots of more or less obvious ways to chop $\mathbb{R}^4$ into infinitely many 4-dimensional simplexes. And, all the ways we know are related by (possibly an infinite sequence of) steps called Pachner moves, which play a basic role in topological quantum field theory.

But, there are also ‘exotic’ ways to glue together infinitely many 4-simplexes and get a space homeomorphic to $\mathbb{R}^4$ : that is, ways that can’t be turned into any of the ‘obvious’ ways using Pachner moves.

Nobody has any explicit description of any of these exotic ways! But, we know they must exist, because you can get them by triangulating exotic $\mathbb{R}^4$ ’s.

To make matters still more frustrating, we know there are uncountably many exotic $\mathbb{R}^4$ ’s.

This sort of puzzle has led Igor Frenkel, Louis Crane and David Yetter to spend a lot of time seeking a purely combinatorial formulation of Donaldson theory or Seiberg–Witten theory — that is, some invariant of triangulated 4-manifolds, computed in a purely combinatorial way, that can detect exotic smooth structures. You could say this is the holy grail of topological quantum field theory.

Indeed, the very first paper on categorification was about this subject, and I wrote about it in the week38, back in 1994.

David Yetter recently thought he had succeeded in finding this holy grail. But alas — he made a mistake.

So, my advice would be: don’t worry about categories much at first! Spend some time learning about 4-manifolds, the original formulation of Donaldson theory in terms of the self-dual Yang–Mills equations, and its later formulation of the Seiberg–Witten equation. If you learn this stuff, you’ll get lots of ideas for research projects.

I’d start with these introductory texts. For 4-manifolds, the book you mentioned is probably the best:

Alexandru Scorpan, The Wild World of 4-Manifolds.

What’s great is that it’s fun to read and packed with math that’s important even for people who don’t wind up studying 4-manifolds — lots of big ideas!

I wish I owned that book. Instead I own a different introductory text, which uses diagrammatic methods to describe these manifolds. These diagrammatic methods are closely related to things you’ve seen in my Rosetta Stone paper:

Robert E. Gompf and András I. Stipsicz, 4-Manifolds and Kirby Calculus

For Donaldson theory, this book is great:

Daniel Freed and Karen Uhlenbeck, Instantons and Four-Manifolds.

For Seiberg–Witten theory, I’m not sure I love any of the introductory textbooks… I don’t know them well enough to have a strong opinion. But, it’s probably good to look at these:

Liviu I. Nicolaescu, Notes on Seiberg–Witten Theory
John Morgan, The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds.

Posted by: John Baez on June 5, 2008 6:38 AM | Permalink | Reply to this

Sharpei

Thank you for the tips John. I forgot to mention that I also bought Gompf’s book together with Scorpan.

I also have the electronic version of the other books you cited on Seiberg-Witten, but aesthetically judging, it is better to start with Scorpan chapters on this subject, right? It would keep me inside a style I am used to… And they have more than a hundred pages about it, without counting the cross references, I guess.

I also have an argument in which I find that any black hole is nothing but a hollow wrinkled compact mass â€œfloatingâ€ above a wrinkled horizon, making it an object diffeomorphically isomorphic to a exotic 4-manifold. Proving this or not, is a self-motivation to work with this weird stuff…

As for only getting familiar with 4-manifolds first, I guess it wouldn’t work for me. I must know categories, since I will surely use it in the future… After all, I wouldn’t like to get stuck later without intuition, since I am doing it all independent.

Posted by: Daniel de França MTd2 on June 5, 2008 4:26 PM | Permalink | Reply to this

Re: Sharpei

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Hi Daniel,

you wrote:

I find that any black hole is nothing but a hollow wrinkled compact mass floating above a wrinkled horizon

Just in case you might appreciate that information: this statement sounds pretty crackpottish. You want to be very sure that you go around saying things like this only after you have completely mastered the standard description of black holes – if then you still feel like saying this. I predict that after you learn the standard theory of black holes, for instance from a good set of lecture notes, you will not want to say this anymore :-).

It’s good to be interested in exciting questions, and it is good to be daring in which thoughts to think. But care has to be exercised.

A good strategy is the following: if in your studies you come to a point where you are struck by a visionary insight which you feel should revolutionize the field and which was missed by all the experts before you, and if you feel you need to share this with the rest of the world, then a good strategy is to start this by phrasing questions.

For instance you might ask:

“Is there any chance that a black hole is a hollow wrinkled object?”

And I would reply:

“No, for one because one of the deep facts about black holes is that they are everything but wrinkled. One of the big theorems about black holes (in 4d) is that they are entirely characterized by their mass, charge and angular momentum. That means a black hole is suprisingly alike an elementary particle.”

“The only thing hollow about a black hole is possibly its horizon: for holes in 4d the black hole horizon has the topology of a 2-dimensional sphere. There is a certain sense in which the interior of this sphere is physically irrelevant. So if by “hollow” one means that a black hole in spacetime is a bit like a 2-ball cut out of spacetime, this has some truth to it.”

This is something I would reply if somebody asked me if a black hole might be a hollow wrinkled object. But if that somebody just states he has an argument that a black hole is a hollow wrinkled object, most people will deduce that this somebody is behaving like a crackpot.

Please take this the way it is meant: as a friendly advice.

Posted by: Urs Schreiber on June 6, 2008 4:35 AM | Permalink | Reply to this

Re: Sharpei

I know it sounds crackpotish, I am sorry. But notice that I am not saying that I believed that, and no matter what, I would brove that true. I said it was a self motivational exercise, to learn some maths so that I could merely try to see if that was true.

Any way, sorry for being abrupt… What I stated above, is not related to the stability of the black hole solutions, but its possible observed form.

Once, I was thinking how come a black hole comes to exist in our universe if an observer could never see a particle enter it. I searched for an answer once in a library, a long time ago. I found someone (in a book from Ellis about gravitaion, I will check later) said that when you come closer to a black hole, you would actualy see the star which formed it in the last moments of existence. So, there wouldn’t be actualy a black hole, but that the black hole would be a limiting solution.

Later, I read the black hole could form because a particle close to the horizon would “tunnel” into it. But I was not convinced of that because of the Hawkin radiation.

From the point of view of a particle, a black hole would tend to shrink infinitely fast, thus lose mass infinitely fast.

So, somedays ago a found an article on arxiv that tried to calculate tha hawking radiation from the perspective of an infalling observer. he conclusion somewhat had some relation to what I was thinking, and so, I sent it. Assuming their conclusion as true, I made this reasoning. I will copy the email I sent them: (no answer, so, one more thanks to you to avoid crackpotish language :) )

************************************

Hi Mr. Eric,

I read your article, and I agree that the termal radiation is infinite when an infalling observer crosses the horizon. Here is a very naive handwaving argument.There must be an agreement of how many particles are in the physical system, if 2 observers do observe the same physical system, despite the reference frame. So, if an observer far away from the horizon sees a small temperature (low particle emition), one can expect that, as in this referencial system the clock of the infalling observer slows down, you can expect that from the infalling point of view the total number of particles counted per clock tick by the infalling guy tend to diverge as the the total is conserved, right?

But, I was thinking, the black hole as a finite mass, so if the irradiated energy goes to infinity, so its mass can tend to 0 faster and faster. The observer would never enter the black hole, he would just see it vanishing, shining. So, the hawking radiation, in the end, would be a kind of cosmic censorship against entering a black hole. Generalizing this for any particles, a real world black hole would be a hollow creature, where all particles would be really close to the horizon.

Considering one more step ahead, we should avoid that the density of the particles near the black hole should be bigger than the black hole itself, so a real black hole would have a really wrinkled surface, since the curvature would be smaller, allowing a higher density of mass (smaller black holes have higher densities).

So, I’d like your opinion. Are real black holes hollow and wrinkled?

Thanks for taking your time!

***********************

Posted by: Daniel de França MTd2 on June 6, 2008 5:49 AM | Permalink | Reply to this

Re: Sharpei

Following the advice from Urs, I will ask a couple questions to everyone, with some clarifications from what I said before.

“is not related to the stability of the black hole solutions, but its possible observed form.”

What I want to know: Is a black hole a reasonable solution for something that can actualy exist in our Universe? How can a big star becomes a black hole, if nothing can be seen entering it? In the end, we would have no black hole at all, because it would take an infinite to any particle to cross it, from an external observer.

I was not addressing the question of a black hole stability. I was actualy aware that it can be just described by its mass, charge and spin, and also that its horizon had a 2d topology.

But, I trying to imagine how is the actual state of a star in which surpasses TOV limit . Even though there is no mechanism to keep the matter from contracting even more, I’ve shown my doubts concerning that we should never see a black hole, and I tried to imagine what we could see instead.

So, in fact, I made language abuse by calling that thing that surpassed the TOV limit a black hole, but that is not actually a black hole from GR.

Excuse me…

Posted by: Daniel de França MTd2 on June 6, 2008 2:36 PM | Permalink | Reply to this

Re: Sharpei

How can a big star becomes a black hole, if nothing can be seen entering it? In the end, we would have no black hole at all, because it would take an infinite to any particle to cross it, from an external observer.

Do you have access to a copy of Gravitation by Misner, Thorne and Wheeler? If you do, try reading section 33.1; if you don’t, I’ll paraphrase their argument for you.

Posted by: Greg Egan on June 6, 2008 3:10 PM | Permalink | Reply to this

Re: Sharpei

No, I don’t have it…

Posted by: Daniel de França MTd2 on June 6, 2008 3:35 PM | Permalink | Reply to this

Re: Sharpei

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Suppose there is a star that exceeds the TOV limit, and you are an observer somewhere outside the star as it begins to collapse.

If you examine the spacetime solution that contains the collapsing star, it will contain an event horizon, and there will be a null surface known as “the surface of last influence”, which consists of all ingoing light rays that cross the event horizon at the same moment as some part of the surface of the star.

Once you have crossed the surface of last influence (i.e. once you wait long enough that a beam of light you sent towards the star would not reach it before it reached the event horizon), even though you will still observe exponentially-decaying light coming from the surface of the star, forever, there is nothing you can do to prevent the surface of the star falling through the horizon. If you send a little robot spaceship down to try to grab a piece of the star and bring it back, then (despite the fact that it looks to you as if the star is still hovering there, waiting to be sampled), you are guaranteed not to succeed, because your spaceship would need to travel faster than light (to cross backwards through the surface of last influence) in order to reach any part of the star before it crossed the horizon.

Posted by: Greg Egan on June 6, 2008 4:05 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

John wrote:

> Since TOP=PL

Did you mean, “Since Diff=PL”?

Posted by: Dan Christensen on June 13, 2008 5:04 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Dan wrote:

Did you mean, “Since Diff=PL”?

Yes.

But I just realized I don’t know how good the Pachner moves are at going between different triangulations of the same noncompact piecewise-linear manifold. You could easily need to use infinitely many moves! Will it always work? Is it even clear what we mean, exactly, by getting from one triangulation to another via an infinite sequence of Pachner moves?

Anyway: there are these ‘weird’ triangulations of $\mathbb{R}^n$ that correspond to exotic smooth structures, but nobody has an explicit description of them.

Posted by: John Baez on June 14, 2008 6:51 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

I hope necroposting is not frowned upon around here but I just came across John’s comment about the lack of explicit description of exotic smooth structures and had a quick question. I believe there are sourceless gravitational wave solutions to Einstein’s field equations which are not diffeomorphic to each other because they represent physically distinct situations. Do these solutions not correspond to exotic smooth structures?

Posted by: J.R. van Meter on July 2, 2010 6:33 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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I’d never heard of ‘necroposting’. We like to have conversations that go on forever here, because people raise tough questions that can take years to answer. Necroposting has been defined as ‘posting in a forum thread that is too old to matter any more’, but a lot of our threads will take centuries to become too old to matter.

The usual gravitational wave solutions in general relativity don’t correspond to ‘exotic smooth structures’; they are correspond to different metrics $g, g'$ on the same smooth manifold $M$ .

More precisely: there’s no diffeomorphism $f: M \to M$ mapping one solution $g$ to the other, $g'$ . But it’s the same underlying smooth manifold in both solutions.

Posted by: John Baez on July 2, 2010 7:13 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

Thanks. I think I’m still missing something elementary though. A metric is typically expressed in a particular coordinate system, which I associate with an atlas, which in turn I associate with a smooth structure. So it seems odd to me that non-diffeomorphic metrics don’t imply non-diffeomorphic smooth structures.

In the context of general relativity, I’m accustomed to thinking of a spacetime manifold as being completely specified by a given topology and metric. But it seems a complete specification must also include the smooth structure, which is somewhat independent of the other two properties.

Posted by: J.R. van Meter on July 3, 2010 12:29 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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J. R. wrote:

So it seems odd to me that non-diffeomorphic metrics don’t imply non-diffeomorphic smooth structures.

It shouldn’t seem odd. There’s a lot more information in a (smooth structure + metric) than in just a smooth structure. So it’s a lot harder for two (smooth structure + metric)s to be diffeomorphic than two smooth structures.

In other words, whenever two (smooth structure + metric)s are related by a diffeomorphism, so are the smooth structures — this is automatic. But not conversely.

In the context of general relativity, I’m accustomed to thinking of a spacetime manifold as being completely specified by a given topology and metric. But it seems a complete specification must also include the smooth structure, which is somewhat independent of the other two properties.

People don’t usually think this way. They usually stack these structures on one at a time. First they specify some set that’s supposed to describe spacetime. Then they specify the topology on this set. Then, given this, they specify the smooth structure. Then, given this, they specify the Lorentzian metric.

There are good reasons for doing this! It’s hard to make sense of the Lorentzian metric if you don’t know the smooth structure, and it’s hard to make sense of the smooth structure if you don’t know the topology.

In our conversation so far, I’d been implicitly holding the topology fixed, since you were only talking about the higher layers of structure: the smooth structure and the metric.

Posted by: John Baez on July 3, 2010 1:57 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

There is a very nice introduction to exotic smooth structures and their interpretation in GR:

Torsten Asselmeyer-Maluga, Carl Brans: “Exotic Smoothness and Physics”, see ZMATH.

Posted by: Tim van Beek on July 6, 2010 1:23 PM | Permalink | Reply to this

exotic smooth structure in physics

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$n$ Lab: exotic smooth structure

Posted by: Urs Schreiber on July 6, 2010 5:24 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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The link to a summary of Michael Müger’s proof of Doplicher-Roberts is this.

The proof appears as the appendix of

Hans Halvorson, Michael Mueger, Algebraic Quantum Field Theory

which is a description of AQFT very much revolving around this result. Halvorson describes lots of math in detail, but doesn’t discuss genuine physics a lot. I can recommend it as one of the best texts on AQFT.

In my article I point out the following simple fact which I haven’t seen mentioned in the AQFT literature:

since algebras are just on-object Vect-enriched categories, a local net of algebras naturally lives in a 2-category. By general abstract nonsense the endomorphism 1-category of that net in that 2-category is monoidal. There is a certain braided monoidal subcategory sitting inside, which one can interpret as the category of superselection sectors. Then by the astract Doplicher-Roberts nonsense we know this is equivalent to the category of reps of some group, which we then identify with the global gauge group.

Posted by: Urs Schreiber on June 4, 2008 3:20 PM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

You claimed, once upon a time, that string dynamics could be reinterpreted as 2-brane statics. Shouldn’t then what you’re doing in this paper on classical string theory bear some relation to work on minimal surfaces?

Posted by: David Corfield on June 5, 2008 9:19 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

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Hmm, that’s interesting! We start Section 4 by recalling that a string minimizes area, but in Minkowski spacetime rather than Euclidean space. Then we march on. But I believe everything we do could also be done for minimal surfaces in Euclidean space. The only difference is some minus signs sprinkled here and there. For example, strings are described by a variant of the wave equation:

$\frac{\partial^2 \phi}{\partial t^2} - \frac{\partial^2 \phi}{\partial x^2} = 0$

while minimal surfaces are described by a variant of the Laplace equation:

$\frac{\partial^2 \phi}{\partial t^2} + \frac{\partial^2 \phi}{\partial x^2} = 0$

So, I think there’s a nondegenerate closed 3-form lurking in the theory of minimal surfaces!

Maybe someone who does minimal surfaces can figure out something cool to do with this fact. I would certainly enjoy seeing a paper with ‘soap bubble’ and ‘gerbe’ in the title.

Posted by: John Baez on June 7, 2008 1:17 AM | Permalink | Reply to this

Re: Classical String Theory and Categorified Symplectic Geometry

The only paper on the ArXiv I could find (within 20 seconds of searching) which contains both ‘soap bubble’ and ‘gerbe’ is Amitabha Lahiri’s Parallel transport on non-Abelian flux tubes. Funnily enough it uses Lie 2-groups. But the use of ‘soap bubble’ is just to designate what lies between two configurations of a flux tube.

Naturally enough, people here know about this paper.

Posted by: David Corfield on June 7, 2008 10:45 AM | Permalink | Reply to this

Read the post A Groupoid Approach to Quantization
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Excerpt: On Eli Hawkins' groupoid version of geometric quantization.
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Read the post Categorified Symplectic Geometry and the Classical String
Weblog: The n-Category Café
Excerpt: In this new version of our paper, we systematically explain how n-dimensional field theories give n-plectic manifolds. We also say how a B field affects the 2-plectic structure for a string.
Tracked: August 2, 2008 2:38 PM

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