## Saturday, May 23, 2009

### IPMU: Focus Week on New Invariants and Wall Crossing

In this week, I was staying near the Kashiwa campus of the University of Tokyo, which is a little far (> 1 hour) from my regular office at the Komaba campus. The purpose of my visit was to look for the recent developments of my old work on topological string theory (open / closed Gromov-Witten) and topological field theory (generalization of Donaldson type invariants of gauge theories). Although my recent research is more on the mathematics side (because I am now a postdoc of mathematics), I hope there will be a good communication with my old friends.

Well, we were afraid of two things; one is that of new influenza (type A; H1N1) panic from pandemic. Thanks to the efforts of the organizers and staffs, the panic could be avoided, (surprisingly, they prepared medical kits (such as disinfection liquid and surgical masks) as well as a thermography camera). The other is the possible miscommunication between proper mathematicians and sincere physicists. The misunderstandings due to the culture of either physics or mathematics were also overcame by the active questions from the participants from far abroad.

OK, let me summarize some of the academic topics of the conference. The 1st day talks were delivered by Murayama (opening address), Szendroi (refinement of virtual Poincare polynomials by the virtual localization (fixed point) theorem for DT (Donaldson-Thomas) sheaf invariants), Jim Bryan (orbifold & crepant resolution conjecture), Toda (strong rationality conjecture on DT with some examples), and Krefl (work with Walcher, orientifold by "O-plane" by involution on the worldsheet).

The 2nd day was mostly the 2-hour talks by Nakajima, Neitzke, and Verlide. (It was because we were awaiting for the banquet at the evening). Nakajima was on the renewed viewpoint on the old and new on the $t$-structure by Beilinson-Bernstein-Deligne (famous French paper on perverse sheaf) and $Z$: central charge of $\mathcal{C}$: heart (Harder-Narasimhan).
Neitzke was on the 1) review on the Seiberg-Witten data and Kontsevich-Soibelman wall-crossing formula, 2) construction on hyperkahler manifolds (of the moduli space of above-mentioned supersymmetric gauge theory; naive first approximation and quantum corrections by instantons), and 3) examples of moduli space of rank 2 ramified stable Higgs bundles over curve (generalization of Hitchin system). Verlinde was more on vertex algebras (of Borcherds) for $\mathcal{N} = 4$ dyons (electric & magnetic charges) by conformal field theory embedded to K3 x T^2. At the double point ($a = 0$ of a * y^2 + b * y + c = 0), they call a "wall" for Weyl reflection, and Weyl "chamber".

At the 3rd day, everyone became tired. Soibelman was on the joint work with Maxim Kontsevich (, and works in progress). After addressing the tools for construction of motivic integration (motivic functions of Denef-Loeser, their stack version by Joyce, and ind-constructible familes). He also mentioned some Hall algebras, which was reminiscent of the 5-hour talk by Kontserich at IPMU. Mikhalkin was on the tropical geometry over non-field. I am not sure whether its "manifold" is a variety or a scheme of locally ringed space. Cheng was a contributed talk on the Borcherds-Kac-Moody algebra for orbifolds and moduli for degenerated metric inner form, and hyperbolic geometry of Poincare sphere with an infinite set-sum of generically (except the Leech lattice) finte number of chambers. Some recent works on wall-crossing dyons were with Lotte Hollands. Nagao was on a big conjecture on the analogy for "open" non-commutative Donaldson-Thomas invariants by elaborating the work of Szendroi.

The 4th day was Maulik, Denef, Ohkawa, and Yamazaki. Maulik was on the generalization of the work of Bryan-Leung for the higher genus curve of poralized K3 surfaces. Denef was on the "local Calabi-Yau" from moduli space of Seiberg-Witten (\mathcal{N} = 2) gauge theory and its prepotential. Ohkawa was a computation of difference of Betti number under the change of "theta-stability" by flips (wall-crossing). Yamazaki was on the collaboration with Ooguri, on the path-algebra of quiver diagrams for (co-) amoebas.

The 5th day was by Fukaya, Hanany, Dimofte, and Konishi. Fukaya was not on Donaldson-Thomas, but on the open Gromov-Witten invariants / Floer homology of Lagrangian submanifold (A-brane with the Maslov (topological) index condition) of symplectic manifold $X = (X, \omega)$. He considered the convergence condition of quantum cohomology utilizing the Novikov ring and Cho-Oh, and the famous preprint by Fukaya-Oh-Ohno-Ohta. The first 1 hour was on the results on toric (Fano) cases, and the latter 1 hour was on the conjecture on Calabi-Yau cases. It was something like the bubbles of Donaldson theory for the Uhlenbeck compactification of Chern-Simons theory. Hanany was on meson / baryon counting of chiral operators, by illuminating algebraic surfaces / orbifolds with at most 2 Kahler moduli, with the help of quiver diagrams, subset of crystals, and toric diagrams. Dimofte was on the refined / motivic wall-crossing, where it was not certain to consider the rigidness or existence of motives (dilogarithm after Beilinson-Deligne). Yukiko Konishi was on the decreasing / increasing filtration structure of mixed Hodge structure for Kaehler variety with singularities ("local Calabi-Yau", namely the total space of canonical bundle of smooth nef surface, e.g. $\mathbb{P}^2$), and its application to the symplectic "open" manifolds for the Yukawa coupling.

Some of the participants will stay in Tokyo for a few more days, and I believe that all appreciated this occasion of meeting.