Wednesday, October 15, 2008

Informal Seminar with Maxim Kontsevich in Keio

I went to Keio University, Yagami Campus, to hear the informal seminar with Maxim Kontsevich from 13:30 to about 17:40. There were three talks: Toric degenerations, A_\infty category, and pseudo-groups. Maxim made some comments on the talks of speakers.

Then we took a dinner of Japanese sake bar (Izakaya), so that we brightened the conversation on some culture (not always on mathematics).

Today was the end of Kontsevich's visit in Tokyo, and he is going to Kyoto University.

Tuesday, October 14, 2008

Kontsevich's 5 hour seminar at the IPMU on wall-crossing

Maxim Kontsevoch explained his recent joint work "wall-crossing" with Y.Soibleman at the IPMU (University of Tokyo, Kashiwa campus) from 13:30 o'clock to after 19:00, including a half hour break and the dinner at one of the cafeterias (also he had a half-hour interview with Kyoji Saito for some journal).

His first half of the talk was

1. Physics background \mathcal{N} = 2, d = 4

super Yang-Mills
supergravity (black hole)

2. Mathematics background

Stability condition in 3-dim Calabi-Yau categories
Donaldson-Thomas invariants (and their generalization)
Hyperkaehler metrics
? Donaldson invariant, and Borcherds Kac-Moody algebras

3. Stability data (Z, (a_{\gamma})_{\gamma \in \Gamma} (infinite set)), where \Gamma \cong \mathbb{Z}^d is a lattice, and \mathcal{G} = \oplus_{\gamma \in \Gamma} \mathcal{G}_{\gamma} / \mathbb{Q}

Then he defined quadratic form Q on \Gamma_{\mathbb{R}} := \Gamma \otimes \mathbb{R}. Thereafter he fixed Q, and he reformulated the stability data (Z, (A_V)), where V is a strict sector ("chamber" of Donaldson theory) in \mathbb{R}^2 \cong \mathbb{C}.

He claimed that, under some assumptions, even if move Z which a little bit preserves "the" condition, A_{\Triangle} should not change. (He was working on a topology structure.)

Now he defined a continuous path in stab (\mathcal{G}) with a parameter in [0,1] for a continuous family Z_t in order to study 2 examples. The 1st example was simpler than the 2nd example: \mathcal{G} = \mathcal{gl}_n and \Gamma = \mathbb{Z}^{n-1}. In this case the walls look like a geometric vector of 2-dim Euclidean space.

4. After the tea break (15:30-), M. Kontsevich examined the 2nd example: \Gamma skew symmetric, with < , >: \Gamma \otimes \Gamma \to \mathbb{Z}, and \Gamma = \otimes_{\gamma \in \Gamma} \mathbb{Q} e_{\gamma} (e is a basis of the Fourier modes). Then he introduced a conjecture on the Donaldson-Thomas invariant \Omega (\gamma) \in \mathbb{Q} (\gamma \neq 0).

Here came the non-commutative geometry and motives; X will be a 3-dim compact Calabi-Yau equipped with a holomorphic (nowhere vanishing (3,0)-form) \Omega^{3,0}; rather than just a K3 surface. Assume that the Calabi-Yau 3-folds are endowed with a Kaehler form [\omega^{1,1}] \in H^2 (X, \mathbb{Z}). We therefore have a \Gamma = H_3 (X, \mathbb{Z}), < , > := intersection Z(\Gamma) := \int_{\gamma} \Omega^{3, 0} \in \mathbb{C}, and \Omega (\gamma) := "number" of special submanifolds of X (whose definition for the "calibrated submanifold" L \in X is such that \omega^{1,1}|_L = 0 and vol (L) = |Z (\gamma)|, but I was not certain about how to define a computable definition of this "number".) Well, we will deform Z <=> change complex str on \Omega^{3,0} by tensoring $\mathbb{C}$ with "B-field" or complexified Kaehler form.

Hereafter he explained the categorical aspects of A_{\infty}-categories / field k (similar to the Fukaya category). The object is once defined as a set of object(s) \mathcal{E}, \mathcal{F}, and the morphism is \mathbb{Z}-graded Hom^\ast (\mathcal{E}, \mathcal{F}) such that \Sum_i dim Hom^i (\mathcal{E}, \mathcal{F}) < \infty, and for all n \ge 1; \Sigma_0, ..., \Sigma_n; m_n of the following;

Hom(\mathcal{E}_0, \mathcal{E}_1) \otimes ... \otimes Hom(\mathcal{E}_{n-1}, \mathcal{E}_n) \to \Hom(\mathcal{E}, \mathbb{E}_n) of degree Z:

\Sum \pm m_i (\alpha_1 \otimes \alpha, m_1 (\alpha...) ... \alpha_n) = 0,

which is something like a unital DGA of homotopy algebras of some "mutation" like enhanced triangulated category with multiplucation structure. Then he changed the object of the category from "set" to a constructible set for any triangulated category. He compared semi-stable objects a_{\gamma} with Lie algebra A (V) of the left and the right. This comparison was done in a following way; Stab(\mathcal{C}) \to \Gamma^{\dual} \otimes \mathbb{C} is locally homeomorphism. And 2 special cases:

1. the base field k = \mathbb{F}_q
2. Hall algebras H (\mathbb{C}) (after To\"en's formula which was more complicated) = \oplus_{\gamma \in \Gamma} \H (\mathcal{C}) \gamma

The equivalence from associative algebras to Lie algebras was;

\mathcal{A} (V) := \Sum_{\mathcal{E} \in \mathcal{A} (V) / isom.} \frac{[\mathcal{E}]}{# \Aut (\mathcal{E})}. This Hall algebra in the char k = 0 will be, for example, a 3-dim Calabi-Yau. In this Calabi-Yau case, we will deal with ind-constructible A_{\infty} algebras / k. The target space is a scheme of finite type, and the object is a constructible maps. The Hall algebras will be "motivic" in the sense of motivic integration in terms of symbol $\mathbb{L}$. He used the motivic integration by means of \mathcal{U} := Ext^1 (\mathcal{E}, \mathcal{E}), with a use of formal (non-convergent) power series on \mathcal{U}. He introduced the "(super)potential" (Morse function) W (\alpha) (\alpha \in \mathbb{C} infinitely close to 0) to study the Milnor fiber (also Denef-Loeser) and vanishing cycles. The talk came to the computation of the "weight" of \mathcal{E} (with a 5-term relation for quantum torus).

Thursday, October 09, 2008

Prof. Kontsevich on BPS counting and wall-crossing

Maxim Kontsevich (IHES) was on "BPS counting as hyperkaehler metric".

He started from axioms, its background physics and mathematics such as Donaldson-Thomas, quiver cluster algebras, collapsing hyperkaehler geometry, geometric Langlands and classical integrable systems (Hitchin system and Seiberg-Witten theory).

Today was the end of the joint meeting of Japan and France.

Wednesday, October 08, 2008

Prof. Kazuya Kato and Voisin at the IHES 50th anniversary and the Mathematical Society of Japan joint workshop,"Perspectives in mathematical sciences"

We are holding a joint meeting of Universities of Tokyo, Keio, and IHES (France) on the mathematical science in Japan.

Today I heard the talks of Kato and Voisin. Kato was on the moduli space and complex / p-adic mixed Hodge structure analogies. Voisin was on the geometric structure on the cohomology algebras of compact Kaehler manifolds. (Kontsevich will talk tomorrow morning at Keio, Mita Campus.)

The talk of Tohru Eguchi (and conversation between Prof. Kontsevich and Prof. Eguchi) was cancelled because of the Nobel prize in physics. Note that Eguchi was a student of Emer. Prof. Yoichiro Nambu in Chicago University more than 30 years ago. As I used to be a Ph.D. student of Eguchi, I am one of the grandchildren of Nambu.

My work on chiral algebra depends on the Higgs mechanism, which is a kind of symmetry breaking of gauge theory, is somehow a generalization of Nambu-Goldstone bosons (massless particles due to broken "global" symmetry, such as pi mesons inside atoms) into the "local" gauge symmetry (acquiring masses) of "the" standard model of elementary particle physics.

Although the Higgs boson(s) of the standard model is (or "are") now in search of LHC (Large Hadron Collider) in Europe (Geneve, Swiss), the symmetry mechanism in the quantum / statistical field theory is everywhere in theoretical physics (including condensed matter physics) or even in modern mathematics including algebro-geometry. (This standard model is still not established experimentally, because we do not know about the Higgs boson masses or their supersymmetric version (e.g. MSSM), or string theory versions.)

Tuesday, October 07, 2008

Nobel Laureate Yoichiro Nambu

Dear all;

Prof. Nambu will receive the Nobel prize in physics 2008. Nambu was the supervisor of my previous supervisor Tohru Eguchi.

We must celebrate this occasion.

Sunday, October 05, 2008

5th Takagi Lectures

This weekend we held the 5th Takagi Lectures at the University of Tokyo, Komaba Campus.

At the conference, I talked to several well-known people on several well-known conjectures on mathematics and mathematical physics.

The staffs offered speeches on the occasion of 50th anniversary of IHES, France. They also encouraged younger researchers by inspiration from honored professors from abroad.

Finally, the wine party today was nice, so that we appreciated our conversations.