Saturday, March 28, 2009

Re-announcement of quantum geometric Langlands conjecture

Today, I presented a refinement of my previous presentation at the Strings 2005 conference, Toronto.
My work of this presentation was on the quantization / localization of the chiral (Hecke) - factorization (Galois) equivalence of right /left $\mathcal{D}$-modules, which is a modern version of the quantization of mirror symmetry.

I also tried to explain the recent development of the open string theory coming from the Chan-Paton factor of the gauge theory attached to the orbifold singularity points.

The details of my presentation slides are avaible at my homepage.
By the way, Tokyo has the cherry blossoms at this weekend, as is shown at the right figure.

Sunday, March 08, 2009

Algebraic Analysis and Deformation Quantization

There is a workshop announcement on the "algebraic analytical geometry" at the following website;

Take care.

Thursday, March 05, 2009

Birth and Origin of Quantum Geometric Langlands Program

To the best of my knowledge, the conjecutres on quantum geometric Langlands and Mirror symmetry for D-modules (not in the form of quantum D-module of Givental, but in the form of Beilinson-Drinfeld chiral algebras) were first proposed by me (Makoto Sakurai) at the Strings 2005 conference in Toronto (Fields Institute), and my slides of presentations on the web were delivered to Dennis Gaitsgory next month at the Seattle 2005 summer instiute held by AMS. Ivan Mirkovic and David Ben-Zvi know my presentations, becuase I e-mailed to them at the end of the Seattle 2005 conference.

"Duality between open Gromov-Witten invariants and Beilinson-Drinfeld chiral algebras" (July 2005, Fields institute, Strings 2005)(available ppt, pdf)

Of course, I did not write the quantum groups formalism, and I did not mention about the irrational quantum deformation parameter formalism, which was originally claimed at the Kashiwara conference by Dennis Gaitsgory. Rather, I am now working on supersymmetric Poisson sigma model, as a stratification (globalization) of quantum groups (non-commutative schemes), which is a generalization of the quantum flag varieties (affine Kac-Moody algebras as the fiber) and this has the Euler obstruction classes as summation (Riemann-Roch theorem) of the 1st and 2nd cohomology class of the gerbe cohomology of chiral differential operators.

Of course, the quantum deformation by 1 parameter or 2 parameters are not yet sufficiently done in my work, although I am working on the quantum cohomology and elliptic genus (algebraic hom of cobordism due to Thom and character (supertrace of conformal blocks) of super conformal field theory, elaborating Eguchi-Sugawara-Taormina's representation of non-rational conformal field theory). I will just talk about the relation between the algebraic cobordism ("open-closed duality of cylinder amplitude") and the double loop group ("double Kac Moody" algebras) at the end of this month at the Mathematical Society of Japan.

Also, Braverman said that Dennis should mention about the work by Feigin-Stoyanovsky (while at RIMS = arXiv:math/0610974 ?) where I could not find out exacly what was going on.

Since I do not want to argue about who was the first (like Newton and Leibnitz on calculus...), but we must be more careful on which point of our works owes the idea to who.

Tuesday, March 03, 2009

Kyoto & Clay on strings & gauge theory

At the morning session, Bezrukavnikov was on noncommutative resolution of singularities. After the lecture, I asked him whether we can construct noncommutative projective schemes after Artin, Stafford, and ven den Bergh, in higher dimensional cases.

We went to a Japanese noodle restaurant for lunch, where we ate Japanese "healthy" noodles with vegetable or fish. In Japan, we have snows everywhere (in Kyoto it was first snowflakes, but soon became stronger). In Boston and New York, it was also snowing.

At the afternoon session (14:30-), Nakajima's quiver gauge theory started from the historic Donaldson theory of instantons where we have two compactifications of translations and bubbles. We dealt with Uhlenbeck's compactification. He explained that the order to take affinization and Langlands dual group (co-weight, co-root) is important, which was different from that of Gaitsgory. We will work on infinite dimensional Grassmannian, but we have a good approximation by finite dimensional geometry; so, don't be afraid. We worked on the intersection cohomology, usual tensor product (not fusion product), and level-rank duality by Igor Frenkel (due to instanton number and the order of the cyclic group acting on the space (orbifold singularities)), which will be talked in more detail.

Gaitsgory's (about one third of) talk was overlapped with Nakajima's introducion to geometric Satake correspondence. He introduced

1) something like a Tannakian category (rigid tensor category, which can be identified with a representation of algebraic group)


2) the notion of "compactly generated" category


3) many limits... (direct, inverse 2-limit, ...)


4) non-degenerated Killing bilinear form


5) deformation 1-parameter $\kappa$ for the deformation of quantum groups

To make a good algebraic theory, he made $\kappa$ an irrational parameter at the Kashiwara conference, but there are some progresses after that. He claimed the last theorem / conjecture that the Whittaker sheaf (or rather, Whittaker category) $Whit_{\kappa} (G)$ is equivalent to $KL_{{\kappa}^{-1}}$ (${}^L G$). (not in Riemann-Hilbert correspondence)

In the last note, he used some representation of Heisenberg algebra and semi-infinite cohomology depending on "$n (\kappa)$", which should be a nilradical part (to construct the W-algebras) and I asked that point, because the audience was very silent after the lecture. He said "Yes, it is a nilpotent, Laurent." I am awaiting for the global quantum geometric Langlnads correspondence of tomorrow's talk. (I know local quantum geometric Langlands and affine Kac-Moody algebras, but I am not satisfied with the localization functor.)

See you tomorrow!

Clay public lectures for quantization, Langlands, and Riemann hypothesis

Today was a public lecture day, so the talks were not aimed at specialists.

The first speaker was Roman (MIT) who suggested, at the positive characteristic cases, the quantum nature becomes easier to study. But, for the time reason, he did not explain its relation to the recent developements of Bridgeland's stability conditions in detail.

The next speaker was Dennis Gaitsgory (Harvard), whom I first met at the Settle'05 summer institute on algebraic geometry. I could have talked to him at the reception time.

The final speaker (James Carlson) was from Clay institute who claims the Riemann hypothesis for the non-trivial zero point of Riemann zeta function. An audience asked what if somebody would find a counter example? Then the speaker said there are some conditions for the $1,000,000 award, but as was in the Hodge conjecture, which was negatively stated by Voisin, the problem statement will be improved. So maybe if someone who might find a zero point of the zeta function outside the critical line (and of course the trivial zero-point in the real axis), they can obtain the money.

The reception was at the Holiday-Inn Kyoto. The toast ("Kanpai") speech was done by Prof. Kashiwara. We could have good conversations with participants from far abroad, although there were many Korean student groups who were invited to the conference. In the middle of the dinner time, we took some group photos in front of the golden wall ("Byobu").

Monday, March 02, 2009

Clay Symposium with Kyoto Univerisity

From tomorrow, we will have a series of lectures by Bezrukavnikov (MIT), Gaitsgory (Harvard), and Nakajima (Kyoto). The main theme is on the quantum algebras and "quantum" geometric Langlands program, (and a lot of number theory...).

For the moment, I am preparing (namely, reading the original papers of the speakers and myself, and some recent research books.) for the discussion in my hotel, where the network connection is very unstable. So I am using au WIN cable (cellular phone connection) to write this blog. I will ask my friends to let me use better internet environment from tomorrow.

The picture today is the city center of Kyoto-Fu (local government).

I hope we will enjoy this workshop.