### 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*.

## D-modules

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 $G$-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 $G$-bundle on the surface and demands invariance under the resulting twisted action.

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

This way, we get a “factory” of D-modules on $\mathrm{Bun}_G$ 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.