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)
superstring
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).
, 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.
