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January 28, 2007

Another Interview

Posted by David Corfield

It’s worth taking a look at an interview Mikio Sato gave to Emmanuel Andronikof in 1990, published in February’s Notices of the American Mathematical Society. Sato is famous for algebraic analysis, D-modules, and the like, about which I know next to nothing. Perhaps if Urs continues to post on Geometric Langlands we’ll hear something about D-modules, as they appear to very relevant. You won’t learn much mathematics from the interview, but it gives a fascinating account of an indirect path to becoming one of the world’s leading mathematicians.

Concerning future directions, this passage caught my eye:

While methods of mathematical physics in quantum field theory have profited various branches of mathematics (topology, braid theory, number theory, geometry), the converse is not necessarily true. Today [remember this is 1990 - DC], mathematical physicists mostly use number theory or algebraic geometry. Mathematical physics is receptive only to higher developed areas of mathematics, some of which are exploited in superstring theory, though not to its full extent. Mathematics has not succeeded in providing a more effective way of computation than perturbation expansions. Of course, there are some primitive methods of computation, like the Monte-Carlo method. All these are kind of brute force computations, not refined mathematics, surely not refined enough for the problems physics is now confronted with, like determining the mass of particles or quarks. All these things are discussed on a very abstract level, not on a quantitative level. So I think that mathematical analysis should be developed much further to match the reality of physics.

Also see Pierre Schapira’s description of Sato’s work in the same edition of the Notices.

Posted at January 28, 2007 8:33 AM UTC

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D-modules

Perhaps if Urs continues to post on Geometric Langlands

I should really take the time, at some point, and actually study Frenkel’s lectures myself. I started typing some notes from our last seminar on Langlands, but I get the feeling there is little point in posting these notes here unless and until I have read Frenkel myself.

I mean, read it seriously, i.e. spent some time with it.

Always a difficult decision, since there are so many other things that demand their time.

On the other hand, the relation of Langlands to CFT that Frenkel emphasizes might be just what I need to thiunk about anyway.

Very roughly, Frenkel notes the following:

In ordinary CFT, the correlator on a given surface is something that depends on the complex structure of the surface. It turns out that, more precisely, pre-correlators, called conformal blocks, form a (projective, flat) vector bundle on the moduli space of conformal structures.

The point is that the space of sections of this vector bundle is a D-module on that moduli space.

As Frenkel puts it, we can therefore think of conformal blocks as “factories” that produce lots of D-modules for us.

Now, for geometric Langlands we want D-modules not on a moduli space of conformal structures of some surface, but on the moduli space of GG-bundles on that surface.

So he introduces a certain twist to the construction of conformal blocks: instead of demanding that they are functionals on reps of an affine Lie algebra that are invariant under the action of that Lie algebra, he twists that action by a GG-bundle on the surface and demands invariance under the resulting twisted action.

The resulting space of invariant functionals now does depend on the chosen GG-bundle, too. Accordingly, the resulting space of such functionals becomes a vector bundle also on the moduli space of GG-bundles on our surface Σ\Sigma. Its sections then form a D-module on Bun G(Σ)\mathrm{Bun}_G(\Sigma).

This way, we get a “factory” of D-modules on Bun G\mathrm{Bun}_G from something like twisted conformal blocks.

Well, that’s at least what I understood so far. It’s all explained in Frenkel’s lecture in more detail, and I still haven’t really read that in detail.

Posted by: urs on January 28, 2007 6:14 PM | Permalink | Reply to this

Re: D-modules

The fundamental construction seems to me to be this localization functor. As I remember it, it goes something like this: Let VV be a representation of gg, the Lie algebra of GG for some group GG (which will eventually be a loop group). We can take VO GV \otimes O_G which is a trivial bundle over GG. This is obviously a O GO_G module. In fact, since the Lie algebra of GG can be identified with the tangent space to GG at a point, and VV is a representation of the Lie algebra, this is in fact a D-module. Now, let HH be a subgroup of GG. We want to construct a D-module on H\GH\backslash G. For a representation VV, VO H\GV \otimes O_{H\backslash G} is no longer has a nice action of differential operators. However, we can take a tangent vector and lift it to an element in gg and act that way. To make things uniquely defined, we take a quotient. In particular, gO H\Gg \otimes O_{H\backslash G} is a Lie algebroid. Call the anchor map aa. Then, the module VO H\G/Ker(a)VO H\G V\otimes O_{H\backslash G} / Ker(a) \cdot V \otimes O_{H\backslash G} is a D-module on H\GH\backslash G. If we work equivariantly with a further subgroup K, we can go from (g,K)(g,K) modules (ie, the action of the lie algebra of KK exponentiates to an action of KK) to D-modules on H\G/KH\backslash G/K.

Finally, it turns out that any GG-bundle (different GG now) on a curve can be trivialized on the complement of a point. Thus, Bun GBun_G is isomorphic to G out\LG/LG +G_{out}\backslash LG/LG_+ where the definition of the various groups is left as an exercise for the reader.

The interesting question, of course, is how does this show up in the physics? In particular, in the physics, the A-branes are more fundamental than the D-modules. Extracting which parts are needed for the physics story and which aren’t is a fun thing to think about.

Posted by: Aaron Bergman on January 28, 2007 9:15 PM | Permalink | Reply to this

Re: D-modules

It might be worth pointing out the relation
of localization to “localization”.
If a commutative ring R maps to functions
on a space X, we have natural (adjoint) functors between R-modules and modules
over functions on X, or (quasicoherent) sheaves on X for algebraic geometrs.

Namely, given an R-module M we may tensor
it with functions O_X on X, as modules over R, obtaining a new module for O_X.
Conversely given an O_X module it (its global sections) carries an action of R.
When X=Spec R this is the usual equivalence between R-modules and sheaves on X, which to an open set in X - whose functiosn are a localization of O_X - assigns
a corresponding localization of the module.

The story with D-modules is the same.
Namely if a Lie group G acts on a space
X, its Lie algebra acts by vector fields
\g —> TX, and its enveloping
algebra acts by differential operators
U\g —-> D_X.

Therefore we again have (adjoint)
functors from modules for \g (or equivalently U\g) to
D_X-modules and back — tensor product over U\g and global sections.
Beilinson-Bernstein proved that if X=G/B
is a flag variety then this is an equivalence, like for affine schemes and O-modules.

Beilinson-Drinfeld then said we can apply
this to moduli of bundles on a curve, and
the result is the construction of (G-twisted) conformal blocks. Here
the Lie algebra is the loop or Kac-Moody
algebra, and X= space of G-bundles + local
trivialization at a point.
(To get down from X to Bun_G one needs the
D-module upstairs to be equivariant
for changes of coordinates, so one needs
the representation to be integrable to
the positive half of the loop group).

Posted by: David Ben-Zvi on January 29, 2007 2:46 AM | Permalink | Reply to this

Re: D-modules

(G-twisted) conformal blocks

What would be the 2dQFT where these GG-twisted conformal blocks appear? Is that known?

Maybe interestingly, it seems that this would have to be a 2-dimensional theory part of whose field content is a map from the worldsheet into BGB G.

So it vaguely looks like a hybrid in between CFT and Chern-Simons.

Aaron seems to hint at some relation to A-branes, but I don’t see it yet.

What I know is that there is something like a decategorification of Chern-Simons, where one considers strings on BGB G.

Posted by: urs on January 29, 2007 7:10 PM | Permalink | Reply to this

Re: D-modules

The theory involved is not really a CFT,
but just a chiral algebra, from which
one constructs a kind of modular functor.
The chiral algebra involved is the affine Kac-Moody vertex algebra at level k a complex number (for k a positive integer, this theory has an RCFT as a quotient,
which gives WZW at level k). Beilinson-Drinfeld use this theory for k=-n (for SL_n), where it is not related
to RCFT and is not terribly physical from what I understand (eg has zero partition function — though we still get interesting D-modules, capturing all the differential equations the partition function ought to satisfy!)

Since this chiral algebra carries a loop group symmetry it can be coupled to principal G-bundles, ie to a map to BG.
I think of this as extending the modular
functor to be a functor of curves+bundles,
rather than just of curves. But this
is not a subtle statement: whenever one identifies a piece of the local
symmetries of a field theory (like conformal, or gauge)
then one can “gauge” it by those symmetries.

I’ll let Aaron explain the identification
of A-branes on a holomorphic cotangent bundle T^*M to D-modules on M :-) — this
is discussed in Kapustin-Witten,
and recent work of Nadler-Zaslow and Nadler gives it some strong support,
but it is still far from established (if true). Likely A-branes on T^*M
are “semiclassical D-modules”…

Posted by: David Ben-Zvi on January 29, 2007 8:34 PM | Permalink | Reply to this

Re: D-modules

Since this chiral algebra carries a loop group symmetry it can be coupled to principal G-bundles, ie to a map to BGB G.

While I understand that one can consider this coupling, I have no good idea what kind of QFT that would correspond to.

But this

is not a subtle statement: whenever one identifies a piece of the local symmetries of a field theory (like conformal, or gauge) then one can “gauge” it by those symmetries.

Okay. So should I imagine some flavor of gauged WZW model playing a role here?

Posted by: urs on January 29, 2007 8:57 PM | Permalink | Reply to this

Re: D-modules

While not really helping, I’d just like to point out that what makes it difficult for me to identify a relevant QFT is already the statement:

Beilinson-Drinfeld use this theory for k=-n (for SL_n)

since it (as mentioned) implies that it does not correspond to a RCFT. Some people believe that such theories (WZW models at ‘critical level’) describe tensionless strings, and I also believe there may be applications in statistical mechanics. Still I have not seen something that would help me understand what kind of QFT this is.

Posted by: Jens Fjelstad on January 29, 2007 9:46 PM | Permalink | Reply to this

QFTs for twisted chiral algebras

Hi Jens,

the case where the conformal blocks are twisted by a GG-bundle on the world sheet looks very interesting to me.

Almost smells like we are dealing with a mixture of a CFT and a homotopy quantum field theory, where in addition to the CFT degrees of freedom we have a map of the worldsheet to BGB G in the game.

You did think about homotopy QFT once? Every encountered anything like this?

Posted by: urs on January 30, 2007 2:47 PM | Permalink | Reply to this

Re: QFTs for twisted chiral algebras

Well I agree that it smells like something involving homotopy quantum field theory, but I have not encountered anything like this before.

It would still be a good thing to complete the description of “Equivariant RCFT” from HQFT though, I’ve sadly neglected that during the past year and a half.

Posted by: Jens Fjelstad on January 31, 2007 11:57 AM | Permalink | Reply to this

Re: Another Interview

Mathematics has not succeeded in providing a more effective way of computation than perturbation expansions.

It seems that some of the work by Connes and Kreimer is at least aimed at providing a deeper mathematical understanding of perturbative QFT.

Posted by: urs on January 28, 2007 6:19 PM | Permalink | Reply to this

Re: Another Interview

I think the keywords are succeeded (in the past tense) and computation. If Connes-Kreimer really do improve computations, why don’t all phenomenologists use it?

CFT and Yang-Baxter really have succeeded in simplifying computations in 2D statmech, compared to various perturbative techniques (epsilon, 1/N, high-temp, low-temp expansions, etc.). OTOH, perturbation theory isn’t limited to 2D.

Posted by: Thomas Larsson on January 29, 2007 9:47 AM | Permalink | Reply to this

Re: Another Interview

What I found interesting was that Sato mentioned that differential algebra was a source for the theory of D-modules. They’re obviously related conceptually, so it always puzzled me that they seemed to exist in completely different worlds.

Posted by: Walt on January 28, 2007 7:06 PM | Permalink | Reply to this

Re: Another Interview

differential algebra was a source for the theory of D-modules.

Part of it seems to be that the world is still awaiting theorems as powerful as GAGA. There is some work that looks at both, in particular D-modules cohérents et holonomes is a nice introduction. [Some of the chapters are in english, others in french].

The algebraic theory is indeed well understood, and extremely computational. See for example Non-commutative Elimination in Ore Algebras Proves Multivariate Identities. (Algebraic D-modules are “the same as” Weyl Algebras, which are a special case of Ore Algebras). They also show up in another unexpected place - see Computer algebra libraries for combinatorial structures. Yes, there is a link with species there!

Some of this discussion has recently recently appeared in another blog, Lambda the Ultimate, where we got started by discussing the derivative of data structures.

Posted by: JacquesC on January 30, 2007 1:21 AM | Permalink | Reply to this

Re: Another Interview

I’m glad to see that people are pursuing the connection. I always thought it was a shame that differential algebra was such an obscure field.

I think that from the right point of view the connection between differential algebra and species is obvious. Maybe I’ll write up what I mean at some date.

Posted by: Walt on January 30, 2007 3:07 AM | Permalink | Reply to this

Re: Another Interview

Differential algebra is obscure within mathematics, but is alive and vibrant within the computer science sub-community interested in closed-form solutions of differential equations, and more generally within the Computer Algebra community.

In one direction, the link between differential algebra and species is indeed straightforward. The direction that is less simple is when you try to solve differential operators in terms of species (and give meaning to that). D-modules of species anyone?

Posted by: JacquesC on January 30, 2007 3:44 AM | Permalink | Reply to this

Re: Another Interview

I actually learned about differential algebra through computer algebra.

Posted by: Walt on January 31, 2007 5:16 PM | Permalink | Reply to this

Re: Another Interview

Mathematics has not succeeded in providing a more effective way of computation than perturbation expansions.

This reminds me of what my theoretical physics professor back in South Africa used to say : ‘The mathematicians haven’t been doing their jobs, so now we have to roll up our sleeves and do it for them!’

Posted by: Bruce Bartlett on January 28, 2007 7:20 PM | Permalink | Reply to this

Re: Another Interview

Mikio Sato supposedly said:

Today [remember this is 1990 - DC], mathematical physicists mostly use number theory or algebraic geometry.

That’s a strange sentence. What do you think he meant by it? Surely mathematical physicists ‘mostly’ use group theory, differential geometry, partial differential equations and the like.

Posted by: John Baez on January 30, 2007 3:00 AM | Permalink | Reply to this

Re: Another Interview

I assumed that was a typo, and it was missing a “don’t”.

Posted by: Walt on January 31, 2007 5:19 PM | Permalink | Reply to this

Re: Another Interview

Oh, duh! You’re probably right. The typo, if it is one, is not David’s; it appears in the original article.

Posted by: John Baez on February 1, 2007 6:08 AM | Permalink | Reply to this

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