The n-Category Café

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!

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 $BG$ coupled to the Chern-Simons 2-gerbe. Transgressed to loop space of $BG$, 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.

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

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.

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 $n=2$

$$\tilde{\omega }={\int }_{S^1}{\mathrm{ev}}^{*}\omega$$

where $$\mathrm{ev}:S^1\times LX\to X$$ is evaluation.

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.

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.

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.

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

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 $\mathrm{Maps}(\Sigma ,X)$, for $\Sigma$ $k$-dimensional.

Conversely, whenever you see an ordinary 1-structure on $\mathrm{Maps}(\Sigma ,X)$, 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 $LX$. 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 $\mathrm{Maps}(\Sigma ,X)$. Now Alex, Chris and John point out how this may come from $(k+1)$-plectic geometry on $X$. This retains more information.

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!

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

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