NOTE: Colloquium by Prof. Eduard Looijenga was CANCELED.
Titles and Abstracts
Nathan Priddis (U. Michigan)
Title: A Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic
Abstract: I will briefly introduce FJRW-theory, which should be understood as the LG A-model for a singularity. Then I will describe the connection of this theory with the Gromov-Witten theory for the mirror quintic in genus zero. This correspondence involves mirror symmetry--in particular it involves analytic continuation on the B-model together with a certain symplectic transformation.
Satoshi Sugiyama (U. Tokyo)
Title: On the Fukaya-Seidel categories of surface Lefschetz fibrations.
Abstract: Seidel defined the Fukaya-Seidel category for exact Lefschetz fibration, that is a Lefschetz fibration equipped with a exact symplectic structure compatible with the fibratoin, and showed that the derived category of the Fukaya-Seidel category is an invariant of the exact Lefschetz fibration.
Let us fix a (not necessarily exact) surface Lefschetz fibration over a disc. If the fibration has a structure of exact Lefschetz fibration, then we can define the Fukaya-Seidel category of that. We proved that the derived category of the Fukaya-Seidel category is independent of choice of the structure. And we establish a necessary and sufficient condition for having the structure of exact Lefschetz fibration.
Atsushi Takahashi (Osaka U)
Title: From Calabi-Yau dg categories to Frobenius manifolds via primitive forms: a work in progress
Abstract: It is one of the most important problems in mirror symmetry to construct functorially Frobenius manifolds from Calabi-Yau dg categories. This is because one expects that the homological mirror symmetry should imply the classical one, the isomorphism of Frobenius manifolds between the one from Gromov-Witten theory and the one from the deformation theory. In this talk, we will give an approach to this problem based on the theory of primitive forms. Under a formality assumption and a technical assumption, we shall construct formal primitive forms for Calabi-Yau dg categories, which enable us to have formal Frobenius manifolds.
| Monday 10 |
Tuesday 11 (Holiday) |
Wednesday 12 | Thursday 13 | Friday 14 | |
| 09:00-0930 | Drinks & snacks | Breakfast | |||
| 09:30-10:45 |
Lecture I-1 |
Lecture II-2 |
Fan |
Lecture III-3 |
Iritani |
| 10:45-11:15 | Break |
10:45-10:50 Photo session
10:50-11:15 Break
|
Break | ||
| 11:15-12:30 |
Lecture III-1 |
Lecture I-2 |
Short communications II |
Lecture II-3 |
Shen |
| 12:30-13:45 | Lunch | ||||
| 13:45-15:00 |
Lecture II-1 |
Lecture III-2 |
No talk |
Lecture I-3 |
Milanov |
| 15:00-15:30 | IPMU teatime | Break | IPMU teatime | ||
| 15:30-16:45 | Barannikov | Takahashi |
Looijenga CANCELED (Colloquium) |
Zhang | Pomerleano |
| 16:45-17:00 | Break | ||||
| 17:00-18:15 |
Short communications I |
Hori-Mauricio | No talk | Losev | No talk |
| 18:30-20:00 | Reception | ||||
| February 10 (Monday) | |
| 09:00-09:30 | Coffee, tea and snacks |
| 09:30-10:45 |
Kenji Fukaya (SCGP), Yong-Geun Oh (IBS), Hiroshi Ohta (Nagoya U), Kaoru Ono (KURIMS) Lecture I-1: Frobenius manifold structure and Lagrangian Floer theory for toric manifolds |
| 10:45-11:15 | Break |
| 11:15-12:30 |
Changzheng Li (Kavli IPMU), Si Li (Boston U), Kyoji Saito (Kavli IPMU) Lecture III-1: LG-model via Kodaira-Spencer gauge theory. |
| 12:30-13:45 | Lunch |
| 13:45-15:00 |
Huijun Fan (Peking U), Tyler Jarvis (Brigham Young U), Yongbin Ruan (Michigan U) Lecture II-1: Introduction to FJRW theory and a mathematical approach to the Gauged Linear Sigma Model. |
| 15:00-15:30 | IPMU teatime |
| 15:30-16:45 |
Serguei Barannikov (Jussieu) On the noncommutative Batalin-Vilkovisky formalism and EA matrix integrals |
| 16:45-17:00 | Break |
| 17:00-18:15 |
Short communications I Yuuki Shiraishi (Osaka U.): On Weyl group and Artin group associated to orbifold projective lines. [PDF] Nathan Priddis (U. Michigan): A Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic Mohammad Reza Rahmati (CIMAT): Hodge Theory of Isolated Hypersurface Singularities |
| 18:30-20:00 | Reception |
| February 11 (Tuesday) | |
| 09:00-09:30 | Breakfast |
| 09:30-10:45 |
Huijun Fan (Peking U), Tyler Jarvis (Brigham Young U), Yongbin Ruan (Michigan U) Lecture II-2: Introduction to FJRW theory and a mathematical approach to the Gauged Linear Sigma Model. |
| 10:45-10:50 | Photo session (in front of the main entrance door) |
| 10:50-11:15 | Break |
| 11:15-12:30 |
Kenji Fukaya (SCGP), Yong-Geun Oh (IBS), Hiroshi Ohta (Nagoya U), Kaoru Ono (KURIMS) Lecture I-2: Frobenius manifold structure and Lagrangian Floer theory for toric manifolds |
| 12:30-13:45 | Lunch |
| 13:45-15:00 |
Changzheng Li (Kavli IPMU), Si Li (Boston U), Kyoji Saito (Kavli IPMU) Lecture III-2: LG-model via Kodaira-Spencer gauge theory. |
| 15:00-15:30 | IPMU teatime |
| 15:30-16:45 |
Atsushi Takahashi (Osaka U) From Calabi-Yau dg categories to Frobenius manifolds via primitive forms: a work in progress [PDF] |
| 16:45-17:00 | Break |
| 17:00-18:15 |
Kentaro Hori (Kavli IPMU), Mauricio Romo (Kavli IPMU) The parameter delta |
| February 12 (Wednesday) | |
| 09:00-09:30 | Breakfast |
| 09:30-10:45 |
Huijun Fan (Peking U) Analytic construction of quantum invariant of singularity [PDF] |
| 10:45-11:15 | Break |
| 11:15-12:30 |
Short communications II Boris Bychkov (HSE): On the number of coverings of the sphere ramified over given points Michel van Garrel (KIAS): Integrality of relative BPS state counts of toric Del Pezzo surfaces Satoshi Sugiyama (U. Tokyo): On the Fukaya-Seidel categories of surface Lefschetz fibrations. [PDF] Alexey Bondal (Kavli IPMU), Ilya Zhdanovskiy (MITP): Critical points of a functional and orthogonal pairs of Cartan subalgebras |
| 12:30-13:45 | Lunch |
| 13:45-15:00 | No talk |
| 15:00-15:30 | IPMU teatime |
| 15:30-17:00 |
Eduard Looijenga (U Tsinghua) CANCELED IPMU Colloquium: Arrangements complements, Frobenius structures and constant holomorphic curvature metrics. |
| February 13 (Thursday) | |
| 09:00-09:30 | Breakfast |
| 09:30-10:45 |
Changzheng Li (Kavli IPMU), Si Li (Boston U), Kyoji Saito (Kavli IPMU) Lecture III-3: LG-model via Kodaira-Spencer gauge theory. |
| 10:45-11:15 | Break |
| 11:15-12:30 |
Huijun Fan (Peking U), Tyler Jarvis (Brigham Young U), Yongbin Ruan (Michigan U) Lecture II-3: Introduction to FJRW theory and a mathematical approach to the Gauged Linear Sigma Model. |
| 12:30-13:45 | Lunch |
| 13:45-15:00 |
Kenji Fukaya (SCGP), Yong-Geun Oh (IBS), Hiroshi Ohta (Nagoya U), Kaoru Ono (KURIMS) Lecture I-3: Frobenius manifold structure and Lagrangian Floer theory for toric manifolds |
| 15:00-15:30 | IPMU teatime |
| 15:30-16:45 |
Youjin Zhang (Tsinghua U) On the genus two free energies for semisimple Frobenius manifolds [PDF] |
| 16:45-17:00 | Break |
| 17:00-18:15 |
Andrey Losev (HSE) K.Saito theory of primitive form, generalized harmonic theory and mirror symmetry |
| February 14 (Friday) | |
| 09:00-09:30 | Breakfast |
| 09:30-10:45 |
Hiroshi Iritani (Kyoto U) Gamma Conjecture for Fano manifolds |
| 10:45-11:15 | Break |
| 11:15-12:30 |
Yefeng Shen (Kavli IPMU) Mirror symmetry for exceptional unimodular singularities |
| 12:30-13:45 | Lunch |
| 13:45-15:00 |
Todor Milanov (Kavli IPMU) The phase form in singularity theory |
| 15:00-15:30 | IPMU teatime |
| 15:30-16:45 |
Daniel Pomerleano (Kavli IPMU) Deformation theory of affine symplectic manifolds |
Titles and Abstracts
Lecture I.
Kenji Fukaya (SCFG), Yong-Geun Oh (IBS), Hiroshi Ohta (Nagoya U), Kaoru Ono (KURIMS)
Title: Frobenius manifold structure and Lagrangian Floer theory for toric manifolds
Abstract: We will discuss Frobenius manifold structure arising from Lagrangian Floer theory for smooth compact toric manifolds. This is based on our joint work by Fukaya, Oh, Ohta, Ono, and partially with Abouzaid.
Kenji Fukaya (SCFG), Yong-Geun Oh (IBS), Hiroshi Ohta (Nagoya U), Kaoru Ono (KURIMS)
Title: Frobenius manifold structure and Lagrangian Floer theory for toric manifolds
Abstract: We will discuss Frobenius manifold structure arising from Lagrangian Floer theory for smooth compact toric manifolds. This is based on our joint work by Fukaya, Oh, Ohta, Ono, and partially with Abouzaid.
Lecture II.
Huijun Fan (Peking U), Tyler Jarvis (Brigham Young U), Yongbin Ruan (Michigan U)
Title: Introduction to FJRW theory and a mathematical approach to the Gauged Linear Sigma Model.
Abstract: I will give an introduction to a Landau-Ginzburg A-model theory that is sometimes called "FJRW theory." This will include some discussion of Landau-Ginzburg mirror symmetry as well as the Landau-Ginzburg/Calabi-Yau correspondence. Once this background is established, I will discuss a generalization, joint with Hiujun Fan and Yongbin Ruan, which gives a mathematical approach to the Gauged Linear Sigma Model (GLSM). This mathematical GLSM is expected to provide a framework that could help explain the LG/CY correspondence and some aspects of mirror symmetry.
Huijun Fan (Peking U), Tyler Jarvis (Brigham Young U), Yongbin Ruan (Michigan U)
Title: Introduction to FJRW theory and a mathematical approach to the Gauged Linear Sigma Model.
Abstract: I will give an introduction to a Landau-Ginzburg A-model theory that is sometimes called "FJRW theory." This will include some discussion of Landau-Ginzburg mirror symmetry as well as the Landau-Ginzburg/Calabi-Yau correspondence. Once this background is established, I will discuss a generalization, joint with Hiujun Fan and Yongbin Ruan, which gives a mathematical approach to the Gauged Linear Sigma Model (GLSM). This mathematical GLSM is expected to provide a framework that could help explain the LG/CY correspondence and some aspects of mirror symmetry.
Lecture III.
Changzheng Li (Kavli IPMU), Si Li (Boston U), Kyoji Saito (Kavli IPMU)
Title: LG-model via Kodaira-Spencer gauge theory.
Abstract: In these lectures, we will explain the extension of Kodaira-Spencer gauge theory (BCOV theory) to Landau-Ginzburg models, and present a perturbative approach to Saito's theory of primitive forms.
Changzheng Li (Kavli IPMU), Si Li (Boston U), Kyoji Saito (Kavli IPMU)
Title: LG-model via Kodaira-Spencer gauge theory.
Abstract: In these lectures, we will explain the extension of Kodaira-Spencer gauge theory (BCOV theory) to Landau-Ginzburg models, and present a perturbative approach to Saito's theory of primitive forms.
Serguei Barannikov (Jussieu)
Title: On the noncommutative Batalin-Vilkovisky formalism and EA matrix integrals
Alexey Bondal (Kavli IPMU), Ilya Zhdanovskiy (MITP)
Title: Critical points of a functional and orthogonal pairs of Cartan subalgebras
Abstract: We show how the problem on describing orthogonal pairs of Cartan subalgebras in sl(n,C) is related to symplectic geometry and to finding critical points of some functional.
Boris Bychkov (HSE)
Title: On the number of coverings of the sphere ramified over given points
Abstract: We describe the generating function for the numbers of isomorphism classes of coverings of the two-dimensional sphere by the genus g compact oriented surface not ramified outside of a given set of m + 1 points in the target, fixed ramification type over one point, and arbitrary ramification types over the remaining m points. We present the genus expansion of this generating function and prove, that it satisfies the KP hierarchy.
Title: On the noncommutative Batalin-Vilkovisky formalism and EA matrix integrals
Alexey Bondal (Kavli IPMU), Ilya Zhdanovskiy (MITP)
Title: Critical points of a functional and orthogonal pairs of Cartan subalgebras
Abstract: We show how the problem on describing orthogonal pairs of Cartan subalgebras in sl(n,C) is related to symplectic geometry and to finding critical points of some functional.
Boris Bychkov (HSE)
Title: On the number of coverings of the sphere ramified over given points
Abstract: We describe the generating function for the numbers of isomorphism classes of coverings of the two-dimensional sphere by the genus g compact oriented surface not ramified outside of a given set of m + 1 points in the target, fixed ramification type over one point, and arbitrary ramification types over the remaining m points. We present the genus expansion of this generating function and prove, that it satisfies the KP hierarchy.
Huijun Fan (Peking U)
Title: Analytic construction of quantum invariant of singularity
Abstract: In this talk, I will explain the analytic construction of the quantum singularity invariants(FJRW) by Fan-Jarvis-Ruan. This includes how to define the Witten map and its perturbation in infinite dimensional space, the compactness, Fredholm theory of the linearized operators, gluing and the wall-crossing formula.
Title: Analytic construction of quantum invariant of singularity
Abstract: In this talk, I will explain the analytic construction of the quantum singularity invariants(FJRW) by Fan-Jarvis-Ruan. This includes how to define the Witten map and its perturbation in infinite dimensional space, the compactness, Fredholm theory of the linearized operators, gluing and the wall-crossing formula.
Kentaro Hori (Kavli IPMU) and Mauricio Romo (Kavli IPMU)
Title: The parameter delta
Abstract: I will discuss the meaning of the parameter "delta" in Kyoji Saito's original paper on primitive forms, which is denoted as "1/t" in the more recent paper by C.Li-S.Li-Saito, and also as "h bar" (in early works) or "z" (in more recent works) by Givental.
Title: The parameter delta
Abstract: I will discuss the meaning of the parameter "delta" in Kyoji Saito's original paper on primitive forms, which is denoted as "1/t" in the more recent paper by C.Li-S.Li-Saito, and also as "h bar" (in early works) or "z" (in more recent works) by Givental.
Hiroshi Iritani (Kyoto U)
Title: Gamma Conjecture for Fano manifolds
Title: Gamma Conjecture for Fano manifolds
Abstract: In this talk I will describe joint work with Sergey Galkin and Vasily Golyshev. We associate a characteristic class to a Fano manifold (or monotone symplectic manifold) as a limit of a solution (J-function) to the quantum differential equation.
We say that a Fano manifold satisfies Gamma Conjecture if the characteristic class equals the Gamma class. I will explain relationships to (homological) mirror symmetry and Dubrovin conjecture, and sketch a proof for Grassmannians.
Andrey Losev (HSE)
Title: K.Saito theory of primitive form, generalized harmonic theory and mirror symmetry
Abstract: I will show that the type B topological strings do need a version of K.Saito theory of the primitive form as an imput. I will review my old work (1998) showing that generalized harmonic theory leads to such an imput. I will address mirror symmetry along the lines of my A-I-B paper with E.Frenkel and show that it implies the effective construction of the analogue of primitive form theory for rational functions on $(C^*)^{ otimes N}$.
Title: K.Saito theory of primitive form, generalized harmonic theory and mirror symmetry
Abstract: I will show that the type B topological strings do need a version of K.Saito theory of the primitive form as an imput. I will review my old work (1998) showing that generalized harmonic theory leads to such an imput. I will address mirror symmetry along the lines of my A-I-B paper with E.Frenkel and show that it implies the effective construction of the analogue of primitive form theory for rational functions on $(C^*)^{ otimes N}$.
Todor Milanov (Kavli IPMU)
Title: The phase form in singularity theory
Abstract: In a recent joint work with B. Bakalov we have constructed a certain twisted representation of the lattice vertex algebra corresponding to the Milnor lattice of a simple singularities and use it to obtain a differential operator constraints for the so called total descendant potential. Our construction is based on the period mappings associated with a primitive form in the sense of K. Saito and it is tempting to apply the same ideas to other singularities as well. However, the key ingredient in our construction is a certain multivalued analytic function, called the phase factor, whose properties seems to depend quite heavily on the structure of the monodromy representation. The latter is quite complicated in general which is probably the main obstacle to generalizing our ideas. It turns out that the phase factor can be taken also as a recursion kernel in order to set up an Eynard--Orantin topological recursion, which gives an alternative (very efficient) way to reconstruct the total descendant potential. The main goal of my lecture is to introduce the phase factor, formulate several conjectures about it, and to explain its role in the Eyanard--Orantin recursion and in our work with B. Bakalov.
Title: The phase form in singularity theory
Abstract: In a recent joint work with B. Bakalov we have constructed a certain twisted representation of the lattice vertex algebra corresponding to the Milnor lattice of a simple singularities and use it to obtain a differential operator constraints for the so called total descendant potential. Our construction is based on the period mappings associated with a primitive form in the sense of K. Saito and it is tempting to apply the same ideas to other singularities as well. However, the key ingredient in our construction is a certain multivalued analytic function, called the phase factor, whose properties seems to depend quite heavily on the structure of the monodromy representation. The latter is quite complicated in general which is probably the main obstacle to generalizing our ideas. It turns out that the phase factor can be taken also as a recursion kernel in order to set up an Eynard--Orantin topological recursion, which gives an alternative (very efficient) way to reconstruct the total descendant potential. The main goal of my lecture is to introduce the phase factor, formulate several conjectures about it, and to explain its role in the Eyanard--Orantin recursion and in our work with B. Bakalov.
Daniel Pomerleano (Kavli IPMU)
Title: Deformation theory of affine symplectic manifolds
Abstract: I will describe some conjectures concerning the deformation theory of Fukaya categories of affine varieties and some work in progress with Sheel Ganatra where we attempt to prove them.
Title: Deformation theory of affine symplectic manifolds
Abstract: I will describe some conjectures concerning the deformation theory of Fukaya categories of affine varieties and some work in progress with Sheel Ganatra where we attempt to prove them.
Nathan Priddis (U. Michigan)
Title: A Landau-Ginzburg/Calabi-Yau correspondence for the mirror quintic
Abstract: I will briefly introduce FJRW-theory, which should be understood as the LG A-model for a singularity. Then I will describe the connection of this theory with the Gromov-Witten theory for the mirror quintic in genus zero. This correspondence involves mirror symmetry--in particular it involves analytic continuation on the B-model together with a certain symplectic transformation.
Mohammad Reza Rahmati (CIMAT)
Title: Hodge Theory of Isolated Hypersurface Singularities
Abstract: We study the asymptotic of Polarization and Riemann-Hodge bilinear relations on Mixed Hodge structure arising from isolated hypersurface singularites. For such a germ of singularity a limit MHS can be defined on the cohomology of the Milnor fibers, due to W. Schmid and J. Steenbrink.
This limit MHS can also be defined using pure analysis of singularity in a different way and one can prove that the filtration induced on the weight graded pieces of the both definitions are the same. The asymptotic Mixed Hodge Structure is polarized. There always exists an extension of the cohomology bundle over the puncture. It is called Deligne extension.
A MHSstructure can be defined on the new fiber. The question is how the polarization or Riemann-Hodge bilinear relations can be formulated on extended fiber. The polarization on the asymptotic of the fibers is a modification of residue product re-flexing the properties of Saito pairing.
Title: Hodge Theory of Isolated Hypersurface Singularities
Abstract: We study the asymptotic of Polarization and Riemann-Hodge bilinear relations on Mixed Hodge structure arising from isolated hypersurface singularites. For such a germ of singularity a limit MHS can be defined on the cohomology of the Milnor fibers, due to W. Schmid and J. Steenbrink.
This limit MHS can also be defined using pure analysis of singularity in a different way and one can prove that the filtration induced on the weight graded pieces of the both definitions are the same. The asymptotic Mixed Hodge Structure is polarized. There always exists an extension of the cohomology bundle over the puncture. It is called Deligne extension.
A MHSstructure can be defined on the new fiber. The question is how the polarization or Riemann-Hodge bilinear relations can be formulated on extended fiber. The polarization on the asymptotic of the fibers is a modification of residue product re-flexing the properties of Saito pairing.
Yefeng Shen (Kavli IPMU)
Title: Mirror symmetry for exceptional unimodular singularities
Abstract: I would like to talk about the mirror symmetry between the Saito-Givental theory of Arnold's 14 exceptional unimodular singularities on Landau-Ginzburg B-side and the Fan-Jarvis-Ruan-Witten theory of their Berglund-H"ubsch-Krawitz mirror partners on Landau-Ginzburg A-side. On the B-side, we compute the genus-zero generating function from a perturbative formula of primitive forms introduced by Li-Li-Saito recently. This computation matches the orbifold-Grothendieck-Riemann-Roch calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction techniques. This establishes the first examples of LG-LG mirror symmetry of all genera for weighted homogeneous polynomials of central charge greater than one. This is joint with Changzheng Li, Si Li and Kyoji Saito.
Title: Mirror symmetry for exceptional unimodular singularities
Abstract: I would like to talk about the mirror symmetry between the Saito-Givental theory of Arnold's 14 exceptional unimodular singularities on Landau-Ginzburg B-side and the Fan-Jarvis-Ruan-Witten theory of their Berglund-H"ubsch-Krawitz mirror partners on Landau-Ginzburg A-side. On the B-side, we compute the genus-zero generating function from a perturbative formula of primitive forms introduced by Li-Li-Saito recently. This computation matches the orbifold-Grothendieck-Riemann-Roch calculations in FJRW theory on the A-side. The coincidence of the full data at all genera is established by reconstruction techniques. This establishes the first examples of LG-LG mirror symmetry of all genera for weighted homogeneous polynomials of central charge greater than one. This is joint with Changzheng Li, Si Li and Kyoji Saito.
Yuuki Shiraishi (Osaka U.)
Title: On Weyl group and Artin group associated to orbifold projective lines.
Abstract: We associate a generalized root system in the sense of Kyoji Saito to an orbifold projective line via the derived category of finite dimensional representations of a certain bound quiver algebra. We generalize results by Saito-Takebayshi and Yamada for elliptic Weyl groups and elliptic Artin groups to the Weyl groups and the fundamental groups of the regular orbit spaces associated to the generalized root systems. Moreover we study the relation between this fundamental group and a certain subgroup of the autoequivalence group of a triangulated subcategory of the derived category of 2-Calabi-Yau completion of the bound quiver algebra. This is based on the joint work with Atsushi Takahashi and Kentaro Wada.
Title: On Weyl group and Artin group associated to orbifold projective lines.
Abstract: We associate a generalized root system in the sense of Kyoji Saito to an orbifold projective line via the derived category of finite dimensional representations of a certain bound quiver algebra. We generalize results by Saito-Takebayshi and Yamada for elliptic Weyl groups and elliptic Artin groups to the Weyl groups and the fundamental groups of the regular orbit spaces associated to the generalized root systems. Moreover we study the relation between this fundamental group and a certain subgroup of the autoequivalence group of a triangulated subcategory of the derived category of 2-Calabi-Yau completion of the bound quiver algebra. This is based on the joint work with Atsushi Takahashi and Kentaro Wada.
Satoshi Sugiyama (U. Tokyo)
Title: On the Fukaya-Seidel categories of surface Lefschetz fibrations.
Abstract: Seidel defined the Fukaya-Seidel category for exact Lefschetz fibration, that is a Lefschetz fibration equipped with a exact symplectic structure compatible with the fibratoin, and showed that the derived category of the Fukaya-Seidel category is an invariant of the exact Lefschetz fibration.
Let us fix a (not necessarily exact) surface Lefschetz fibration over a disc. If the fibration has a structure of exact Lefschetz fibration, then we can define the Fukaya-Seidel category of that. We proved that the derived category of the Fukaya-Seidel category is independent of choice of the structure. And we establish a necessary and sufficient condition for having the structure of exact Lefschetz fibration.
Atsushi Takahashi (Osaka U)
Title: From Calabi-Yau dg categories to Frobenius manifolds via primitive forms: a work in progress
Abstract: It is one of the most important problems in mirror symmetry to construct functorially Frobenius manifolds from Calabi-Yau dg categories. This is because one expects that the homological mirror symmetry should imply the classical one, the isomorphism of Frobenius manifolds between the one from Gromov-Witten theory and the one from the deformation theory. In this talk, we will give an approach to this problem based on the theory of primitive forms. Under a formality assumption and a technical assumption, we shall construct formal primitive forms for Calabi-Yau dg categories, which enable us to have formal Frobenius manifolds.
Youjin Zhang (Tsinghua U)
Title: On the genus two free energies for semisimple Frobenius manifolds
Title: On the genus two free energies for semisimple Frobenius manifolds
Abstract: For a semisimple Frobenius manifold the genus g >0 free energies are defined by solving recursively the associated loop equation, in particular one can obtain an explicit formula of the genus two free energy in terms of the canonical coordinates of the Frobenius manifold.
In this talk, we show how to represent the genus two free energy of an arbitrary semisimple Frobenius manifold as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so-called "genus two G-function". Conjecturally the genus two G-function vanishes for a series of important examples of Frobenius manifolds associated with simple singularities as well as for P1-orbifolds with positive Euler characteristics. We explain the validity of this Conjecture for the Frobenius manifolds associated to simple singularities.
