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March 12, 2005

M ∩ Φ

Over at the String Coffee Table, Eric Forgy asks:

Hanging out at Harvard

Do you ever think that maybe all this abstract mathematics is not how nature really operates? I think that physicists come to a turning point early in their careers where they need to decide on a philosophy. Are you going to try to develop theories that might describe some kind of phenomological aspect of nature, or are you going to really try to understand the true nature of the universe at the most fundamental level.

Are you pursuing this because you truly believe that nature operates according to the rules of gerbes and n-categories? It’s kind of a silly philosophical question, I suppose, but what is it that drives you?

Back in the early 1980s, vector bundles, the Atiyah-Singer Index Theorem, and basic ideas of homotopy theory were considered pretty cutting-edge1 in high-energy theory. If you used them in a seminar, you generally had to make a few apologies to your audience before proceeding. There was, shall we say, a certain resistance to incorporating highbrow mathematics. When I started applying certain techniques of algebraic geometry (in particular, coherent sheaves) to problems in string compactification, I quickly became known as “Mr. Sheaf” among my graduate student colleagues.

It’s fair to say that two decades of string theory have broken down that resistance. All sorts of highbrow math have shown up (in, sometimes, surprising ways) in string theory. Indeed, many suspect that whole new mathematical frameworks may need to be invented, in order for us to formulate string theory properly.

Subtle questions require subtle mathematical techniques to study them. K-theory — to pick one example — is essential to the formulation of the Index Theorem. But, for the applications that were of interest in the early 1980s, you really didn’t need that extra baggage. Ordinary cohomological formulæ were sufficient. Only when you start dealing with more subtle matters, like mod-2 index theorems, does the K-theoretic formulation really come into its own. The converse is also true. Subtle physics often suggests interesting new mathematics. The past two decades provide numerous examples of new mathematical developments inspired by string theory constructs.

I’ve no idea whether n-categories will eventually play an important role in string theory. (Gerbes have already made an appearance, but not in the guise the Urs seems to want for them.) So far, there doesn’t seem to be any compelling reason to think that they do.

Urs is sniffing around various branches of string theory, looking for hints that they might be reformulated in a 2-categorical form. That’s a fair endeavour. It’s not clear he will succeed. But it’s not (I think) a philosophical statement about how he “truly believe[s] that nature operates.”

If there’s anything we ought to avoid, it’s approaching Mother Nature with such philosophical preconceptions.

1 Some people, apparently, still view these as cutting-edge stuff; to them, I suggest a digital watch.

Posted by distler at March 12, 2005 1:57 PM

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Re: M ∩ Φ

Recently I had an interesting conversation with Aaron on what exactly nonabelian gerbes (and hence nonabelian 2-bundles) might apply to in string theory. Following Aschieri&Jurčo and also vague statements by Witten I was arguing that they should describe aspects of stacks of M 5-branes. The other possibility is that they are related to elliptic cohomology somehow.

The argument by Aschieri&Jurčo is that the coupling of the supergravity 3-form to the membrane certainly should be described by an abelian (Chern-Simons-)2-gerbe (what else could it be??) and that then (analogous of how the nonablian (-1)-gerbe=bundle coupling of the string boundary derives from the coupling of its bulk to the KR field described by an abelian 1-gerbe) the boundary of the membrane must couple to a nonabelian 1-gerbe. (Is there anything wrong with this argument?)

Later I saw that Hisham Sati is arguing that both aspects are actually part of the same underlying phenomenon. In the closing remarks in section 6.3 of hep-th/0404013 he writes:

There is also a purely mathematical side of these phenomena: In [54], a model of elliptic cohomology was proposed based on \mathcal{E}-equivariant stringy bundles over an elliptic curve \mathcal{E}. A stringy bundle is a variant of what [45,31] call an elliptic object, i.e. a ‘conformal field theory indexed by \mathcal{E}’. The authors of [54] could not figure out what was the role of the elliptic curve \mathcal{E} in the ‘spacetime’ part of the theory. But the present context suggests that this may be perhaps interpreted as an interesection between an MM2-brane and an MM5-brane, and that the roles of the MM-branes should be further investigated to enhance geometric interpretation of elliptic cohomology.

On p. 5 of hep-th/0501245 Sati makes a related comment.

Of course I would like to make these connections between string physics and 2-bundles more explicit. So currently I am speculating that maybe the explicit description of the 6D worldvolume theories of the 5-branes by means of dimensional deconstruction might be of help. The idea is that modelling dimensions by quivers naturally gives rise to 2-algebras whose modules are 2-bundles with a relation to the elliptic cohomology studied by Baas,Dundas&Rognes.

Given all this as well as what I have now learned of derived categories and their relation to D-branes I find that my original hunch expressed in ‘strings as categorified points’ is maybe rather naïve but not that far off.

In any case, it’s fun. :-)

Posted by: Urs Schreiber on March 13, 2005 12:36 PM | Permalink | Reply to this

Speculative extrapolations…

So, if you were to guess, do you think physics will ever finish it’s job? My bet is that it’s going to keep on going as it has been: we will always be finding new phenomenon that need to be united in an ever more complex mathematical framework. This is based on the reasonable assumptions that the universe really is a mathematical structure and that all mathematical structures exist. You’d then always find that your current theories are only approximations to more complex theories, for statistical reasons - there are many more complex theories than simple ones. Well, we’ll see! If things plateau out over time then I think this idea is falsified, and alternatively the longer that basic science continues to advance the more confidence we can have in it.

Posted by: Travis Garrett on March 13, 2005 1:46 PM | Permalink | Reply to this

Re: M ∩ Φ

Well, it is true that for a foreigner it is not clear the way coming from F=ma to higher mathematics. But once you notice the action principle, it is not so rare. And one you come to representation theory, you can go via Morita Takesaki even to higher mathematics, and so on.

But with categories is also a different question: if we are interested on it due to its fundational character, and if so, why?

Posted by: Alejandro Rivero on March 13, 2005 3:26 PM | Permalink | Reply to this

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