Hello to all,

This question will probably be closed down as being off-topic faster than one can say "string theory", but here it goes: I've noticed that the problems I'm working on -the structure of derived categories of some interesting surfaces- have a very important physical meaning. Unfortunately, I have in no way any idea as to why. Is there anyone out there who could give a mathematical explanation -or a link to a paper- as to why physicists would be interested in such highly abstract gizmos like derived categories, mutation, orbifolding, tilting...(the list goes on and on)

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    arxiv.org/abs/hep-th/0608056 – Gjergji Zaimi Jun 11 '10 at 15:24
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    I think this is an interesting question. But since you are really asking for a list of applications of these things, you should probably community wiki the question. – GMRA Jun 11 '10 at 15:29
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    louis is asking for an explanation, not a list of applications. – j.c. Jun 11 '10 at 15:30
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    community wiki done – louis de Thanhoffer de Völcsey Jun 11 '10 at 15:37
  • @jc. Sorry it was badly worded. What I meant to say was that he is looking for an explanation for a list of things. That means a list of eplanations, hence community wiki. – GMRA Jun 11 '10 at 15:52

The short but ahistorical answer is that topological string theories turn out to be examples of $(\infty,1)$-categories. The mathematical formulation of this statement is in Lurie's classification of topological field theories http://www.math.harvard.edu/~lurie/papers/cobordism.pdf (building on work of Atiyah, Segal, Getzler, Costello, Baez-Dolan, Kontsevich and probably a bunch more I'm forgetting.)

The content of this statement is that when you write down the axioms for a topological string theory, the collection of "boundary conditions" or "D-branes" look like the collection of objects in an $(\infty,1)$ category.

Of course, you can ask why the derived category of coherent sheaves. Historically, the answer to that is that it is very easy to write down a boundary condition for a holomorphic vector bundle in the topological B-model. It's not a huge leap from there to coherent sheaves, and if you start mumbling words like tachyon condensation, you can get to the derived category with a fair bit of hand waving.

That's from the physics side of things. On the math side, Kontsevich got there first, possibly by noting that the space of closed string states in the B-model ($H^\bullet(\wedge^\bullet TX)$) is exactly the Hochschild coohomology of the derived category of coherent sheaves. He then followed up by associating the (still not yet defined?) Fukaya category with the A-model and conjecturing that mirror symmetry is an equivalence of the two (with some Hodge structure goodies thrown in). Subsequently, it looks like you have to add in some things called coisotropic branes to cover all your bases, but the basic idea is right.

Kontsevich formulated all this in terms of $A_\infty$ categories which in the Lurie language turn into $(\infty,1)$ categories which are just TQFTs in disguise. So, Kontsevich's homological mirror symmetry is then the statement that two TQFTs are the same, just like mirror symmetry in string theory.

From the physics side of things, this was all a bit of a mess, but we now understand that the derived category really arises via Block's construction of the derived category (I'm being intentionally vague as to which version of the derived category) as arising from integrable super-connections of graded smooth vector bundles http://www.math.upenn.edu/~blockj/papers/BottVolume.pdf. You can see this explicitly in the physics from a few sources, particularly Kapustin, Rozansky and Saulina, and Herbst, Hori and Page, but I'm rather fond of my own contribution http://arxiv.org/abs/0808.0168.

I'm reminded of Manin's comment, that physics contains just about everything mathematical, but in no particular order. I believe that "relevant to physics" needs strongly scary quotes. Mathematicians find some difficuly in recognising the "what" in physics - the conventional mathematics behind less familiar notation, for example. But that's a dictionary-type issue that can in principle be solved. The "why" in physics is much less likely to translate into mathematicians' terms: ideas such as thermodynamics that are central to certain kinds of physical thinking are not really mathematical possessions, or where they are, may be rather misleading. In other words if you accept facile answers to questions such as you pose, they may be worth rather little.

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    "Ideas such as thermodynamics that are central to certain kinds of physical thinking are not really mathematical possessions, or where they are, may be rather misleading." Why do you say that? – Vectornaut Jun 11 '10 at 22:20
  • I was thinking, for example of ergodic theory, as it appears to different kinds of views. The molecules in the air in my room are not all going to rush into one corner (physics), but in mathematical terms they might, and indeed are expected to for some tiny proportion of "all time". In units of a billion years, it's probably not something for me to worry about. – Charles Matthews Jun 12 '10 at 7:49
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    Maybe I'm being a bit unfair, I've generally found that such sentiment belies a profound misunderstanding of mathematics -- e.g. the idea that mathematicians are incapable of considering the notion of approximation. – Hurkyl Mar 12 '12 at 21:15

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