Workshop on Geometry and Physics of the Landau-Ginzburg model
May 31 - June 4, 2010
Abstracts


Serguei Barannikov
A-infinity gl(N)-equivariant matrix integrals and intersections on the moduli spaces
My talk will consist of two parts. First part("genus zero"): the negative cyclic homology subspace moving inside periodic cyclic homology defines for the noncommutative varieties the analogue of the variations of Hodge structure, which I've described more than 10 years ago. For the deformations of the derived categories of coherent sheaves of Calabi-Yau hypersurfaces the periods of these noncommutative Hodge structures give the generating function of the totality of the genus zero Gromov-Witten invariants of the mirror manifolds. Second part ("arbitrary genus"): I'll describe the higher dimensional analogs of the matrix Airy integral, which I've introduced in my 2006 paper on noncommutative Batalin-Vilkovisky formalism (hal-00102085). My matrix integrals are constructed from Calabi-Yau A-infinity algebras. The asympthotic expansions of my matrix integrals are related with cohomology of the compactified moduli spaces.

Boris Dubrovin
Pairs of LG potentials and an infinite-dimensional Frobenius stucture
We will introduce a structure of an infinite-dimensional Frobenius manifold on the space of pairs of functions analytic inside/outside the unit disk with a simple pole at infinity/zero. The connection of this construction with the theory of 2D Toda hierarchy will also be explained. The talk is based on a joint work with G.Carlet and L.P.Mertens.

Carel Faber
Tautological and non-tautological classes on the moduli space of curves
I will first report on the tautological ring of the moduli space of smooth curves of genus 24, where the standard relations don't seem to give a Gorenstein quotient. Then I will report on the cohomology of the moduli spaces of pointed curves of genus three, where non-tautological classes abound and where the first examples of classes not coming from Siegel modular forms have been detected.

Huijun Fan
Geometry of section bundle system (SBS)
We define section bundle systems (SBS), a new geometrical object whose structure is closely related to topological field theory, mirror symmetry and CY/LG correspondence. As an exmple, the SBS given by a holomorphic function will be discussed. Based on the spectrum theory of Schrodinger operators, we can construct the tt*-bundle structure (and so a harmonic Frobenius manifold structure due to Hertling, Sabbah as well as a harmonic Higgs bundle structure due to Simpson). This provides a brigde between the singularity theory and the nonlinear sigma-model of the related Kähler manifolds. The definitions and properties discussed here are based on the work of physicists, Witten, Cecotti, Vafa and others.

Hiroshi Iritani
Matrix factorization and enumeration of spin curves
The Landau-Ginzburg/Calabi-Yau correspondence has been discussed both in the context of D-branes and also in the context of (topological) closed string theory. For D-branes, Orlov showed the equivalence of the derived category of a Calabi-Yau hypersurface and the category of matrix factorizations of the corresponding Landau-Ginzburg model. For closed string theory, Chiodo and Ruan recently showed that the Gromov-Witten theory of a quintic hypersurface analytically continues to the FJRW theory (counting of W-spin curves) of the Landau-Ginzburg model at genus 0. In this talk I will describe joint work with Alessandro Chiodo and Yongbin Ruan about a relationship between the above two correspondences for a weighted projecitve Calabi-Yau hypersurface. In particular, we will see that Orlov's equivalence determines Chiodo-Ruan's analytic continuation.

Claus Hertling
A generalization of Hodge structures and oscillating integrals.
The physicists Cecotti and Vafa considered in 1991 a generalization of Hodge structures which is related to Simpson's harmonic bundles. Mathematicians took this up under different names (TERP structure, integrable twistor structure, non-commutative Hodge structure). It consists of a holomorphic vector bundle on P^1C with a flat connection on C* with poles of order 2 at 0 and infinity, with a flat real structure and a certain flat pairing on the bundle on C*. It arises via oscillating integrals in the case of Landau-Ginzburg models. Several results on Hodge structures have generalizations in this setting: the notions nilpotent orbit and mixed Hodge structure and their correspondence, the curvature of classifying spaces. In the case of tame functions, the structures are pure and polarized. In the case of germs of functions and their unfoldings, there are nilpotent orbits and mixed structures. Here Hodge structures and Stokes structures come together.

Kentaro Hori
Gauged Landau-Ginzburg Models
I discuss 2d (2,2) supersymmetric gauge theories with simple gauge groups. They are close cousins of Landau-Ginzburg orbifolds but carry new features of interest. CY/GLG correspondence will also be discussed.

Tyler Jarvis
Mirror symmetry and integrable hierarchies for the D4 singularity.
Almost twenty years ago, Witten made a conjecture for the simple (ADE) singularities singularities, relating intersection theory on certain moduli spaces associated to each singularity and certain integrable hierarchies arising from the singularity. That conjecture was proved in the case of An singularities by Faber, Shadrin, and Zvonkine in 2006. Ruan, Fan, and I proved the conjecture for the D and E singularities last year, except for the case of D4, which, surprisingly, was much harder to prove than the others. In this talk I will provide a survey the problem and its background, and then describe how we, together with my student Evan Merrell, complete the proof in the case of D4.

Albrecht Klemm
Counting Donaldson-Thomas invariants with modular forms
In this talk we discuss how Donaldson Thomas invariants on Calabi-Yau threefolds are encoded in terms of modular forms. Emphasis is laid on the interplay between modularity and non-holomorphicity. In particular the holomorphic anomaly lead to the construction of almost holomophic forms for counting of D6-D2-D0 BPS states, which are related to DT and Gromov-Witten invariants. We report on recent observation that link the problem of counting D4-D2-D0 BPS states to Mock modular forms.

Yongbin Ruan
Landau-Ginzburg/Calabi-Yau correspondence
A far reaching correspondence from physics suggests that the Gromov-Witten theory of Calabi-Yau hypersurfaces of weighted projective spaces (more generally a toric variety) can be computed by means of the singularity theory of its defining polynomial. In this talk, I will present some of the works (in collaboration with Alessandro Chiodo) towards establishing this correspondence mathematically as well as some surprises and speculations.

Claude Sabbah
Examples of non-commutative Hodge structures
Non-commutative Hodge structures occur in various ways in Mirror symmetry. They produce the tt* geometry on Frobenius manifolds (Cecotti-Vafa, Dubrovin and more recently Hertling, Iritani). I will explain the simplest non-trivial structures of this kind in terms of Stokes matrices.

Kyoji Saito
Towards Primitive Forms of types A∞/2 and D∞/2
This is a report of a work in progress. We consider two entire functions, named fA and fD, in two variables, which have only two critical values 0 and 1. The associated maps: C2 -> C define local trivial fibrations on C\{0,1}, where the general fiber is an infinite genus (non-algebraic) curve. The fibers over 0 and 1 carries infinitely many simple critical points and associated vanishing cycles in the middle homology group of the generic fiber form a bipartite decomposition of the quivers of type A∞/2 and D∞/2, respectively. In the talk, we try to describe primitive forms and associated period maps for those lattices of vanishing cycles, respectively.

Sergey Shadrin
A polynomial bracket for Dubrovin-Zhang hierarchies
We define an integrable system of Hamiltonian PDEs associated to an arbitrary tau-function in the semi-simple orbit of the Givental group action on genus expansions of Frobenius manifolds. We prove that the right hand sides of the equations, the Hamiltonians, and the bracket are weighted-homogeneous polynomials in the derivatives of the dependent variables with respect to the space variable. In the particular case of a conformal (homogeneous) Frobenius structure, our integrable system coincides with the Dubrovin-Zhang hierarchy that is canonically associated to this Frobenius structure.
Our approach allows to reprove the polynomiality of the equations and the Hamiltonians of the Dubrovin-Zhang hierarchies and to prove the polynomiality of one of the Poisson structures (that was also conjectured by Dubrovin and Zhang).
It is a joint work with A. Buryak and H. Posthuma.

Eric Sharpe
An overview of progress on quantum sheaf cohomology
In this talk we will outline progress towards understanding `quantum sheaf cohomology,' an analogue of quantum cohomology adapted from heterotic strings, which involves a complex manifold together with a holomorphic vector bundle. Whereas ordinary quantum cohomology involves deforming classical cohomology rings, quantum sheaf cohomology is a deformation of a classical sheaf cohomology ring. We will also outline the A/2 and B/2 `holomorphic field theories' from which quantum sheaf cohomology is derived, and also discuss related aspects of Landau-Ginzburg models (both ordinary and heterotic) over nontrivial spaces.
PDF

Yefeng Schen
LG/CY correspondence for quotients of elliptic singularities
I will report on work in progress with Marc Krawitz, on LG/CY correspondence for quotients of elliptic singularities.

Atsushi Takahashi
Strange duality of weighted homogeneous polynomials
We consider a mirror symmetry between invertible weighted homogeneous polynomials in three variables. We define Dolgachev and Gabrielov numbers for them and show that we get a duality between these polynomials generalizing Arnold's strange duality between the 14 exceptional unimodal singularities. This is a joint work with Wolfgang Ebeling.

Arkady Vaintrob
Matrix factorizations and cohomological field theories
This is a report on a joint work with Alexander Polishchuk. Starting with a quasihomogeneous isolated singularity W and its diagonal group of symmetries G, we construct a cohomological field theory whose state space H is the equivariant Milnor algebra of W. This theory is an algebraic counterpart of the Fan-Jarvis-Ruan version of the A-model for Landau-Ginzburg orbifolds. We use as the main tool categories of matrix factorizations which appeared previously as B-type D-branes for the Landau-Ginzburg model.

Johannes Walcher
Residues and normal functions
I will review real mirror symmetry from the point of view of Landau-Ginzburg models, emphasizing the role played by the Griffiths' infinitesimal invariant of normal functions.