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Matthew Ballard (University of South Carolina)
Title: Some Homological Algebra for Factorization Categories
Abstract:The utility of Landau-Ginzburg models in understanding phase transitions in physics and relating derived categories of coherent sheaves on varieties in algebraic geometry has become quite apparent in recent years. In this talk, I will review some general notions for categories of B-branes associated to LG models and then discuss their natural generalizations. Afterwards, some basic facts, very analogous to those for derived categories of coherent sheaves, will be covered. This is joint work with D. Favero, D. Delius, M.U. Isik, and L. Katzarkov.
Alessandro Chiodo (Institute de Mathematiques de Jussieu)
Abstract:The utility of Landau-Ginzburg models in understanding phase transitions in physics and relating derived categories of coherent sheaves on varieties in algebraic geometry has become quite apparent in recent years. In this talk, I will review some general notions for categories of B-branes associated to LG models and then discuss their natural generalizations. Afterwards, some basic facts, very analogous to those for derived categories of coherent sheaves, will be covered. This is joint work with D. Favero, D. Delius, M.U. Isik, and L. Katzarkov.
Alessandro Chiodo (Institute de Mathematiques de Jussieu)
Title:"Is it really easier to compute? "
Abstract: This question is often asked on the recent approach to the Landau-Ginzburg model due to Fan-Jarvis-Ruan, Chang-Li-Li, Polishchuk-Vaintrob... I will show some successful examples both in cohomology and quantum cohomology. There, Jérémy Gu'er'e has recently provided a new effective method via K-theory to handle the virtual cycle and establish mirror symmetry even in some non-concave cases.
Will Donovan (University of Edinburgh)
Emily Clader (University of Michigan)
Titile: The Landau-Ginzburg/Calabi-Yau Correspondence and Wall-Crossing
Abstract: The Landau-Ginzburg/Calabi-Yau correspondence is a proposed equivalence between two enumerative theories associated to a homogeneous (or, more generally, quasihomogeneous) polynomial: the Gromov-Witten theory of the hypersurface cut out by the polynomial in projective space, and the Landau-Ginzburg theory of the polynomial when viewed as a singularity. In this talk, I will describe a perspective on the LG/CY correspondence via variation of stability conditions. This interpretation allows the correspondence to be generalized from hypersurfaces to complete intersections, and it also points toward recent results and work-in-progress on wall-crossing formulas relating the two theories.
Titile: The Landau-Ginzburg/Calabi-Yau Correspondence and Wall-Crossing
Abstract: The Landau-Ginzburg/Calabi-Yau correspondence is a proposed equivalence between two enumerative theories associated to a homogeneous (or, more generally, quasihomogeneous) polynomial: the Gromov-Witten theory of the hypersurface cut out by the polynomial in projective space, and the Landau-Ginzburg theory of the polynomial when viewed as a singularity. In this talk, I will describe a perspective on the LG/CY correspondence via variation of stability conditions. This interpretation allows the correspondence to be generalized from hypersurfaces to complete intersections, and it also points toward recent results and work-in-progress on wall-crossing formulas relating the two theories.
Will Donovan (University of Edinburgh)
Title:Mixed braid group actions and B-brane transport
Abstract: The heuristics of B-brane transport in gauged σ-linear models can be translated into mathematical predictions concerning the derived categories of varieties associated to certain phases of these models. In this talk I will discuss some toric Calabi{Yau examples which arise as deformations of small resolutions of type A surface singularities. In joint work with Ed Segal (arXiv:1310.7877) we construct mixed braid group actions on the derived categories of coherent sheaves for these varieties. We prove the braid relations by an analysis of the grade restriction rules (or `windows') which control B-brane transport in these examples.
Richard Eager (IPMU)
Richard Eager (IPMU)
Title: Supersymmetric gauge theories and the hemisphere partition function
Abstact: T.B.A.
Abstact: T.B.A.
David Favero (University of Alberta)
Title: Comparing Derived Categories for Various Mirror Constructions
Abstract:Given a Calabi-Yau complete intersection in a toric variety, mirror duals were proposed, for example, by Berglund and Hubsch, and independently by Batyrev and Borisov. These constructions do not always agree when one varies complex structure on the complete intersection, however, recent individual works of Kelly, Shoemaker, and Clarke illustrate that these various mirrors are in fact birational. I will discuss a theorem of myself and T.Kelly which shows that these various mirrors have equivalent derived categories.
Abstract:Given a Calabi-Yau complete intersection in a toric variety, mirror duals were proposed, for example, by Berglund and Hubsch, and independently by Batyrev and Borisov. These constructions do not always agree when one varies complex structure on the complete intersection, however, recent individual works of Kelly, Shoemaker, and Clarke illustrate that these various mirrors are in fact birational. I will discuss a theorem of myself and T.Kelly which shows that these various mirrors have equivalent derived categories.
Daniel Halpern-Leistner (Columbia University)
Title:The structure of instability in moduli theory
Abstract: In many examples of moduli stacks which come equipped with a notion of stable points, one tests stability by considering "iso-trivial one parameter degenerations" of a point in the stack. To such a degeneration one can often associate a real number which measures "how destabilizing" it is, and in these situations one can ask the question of whether there is a "maximal destabilizing" or "canonically destabilizing" degeneration of a given unstable point. I will discuss a framework for formulating and discussing this question which generalizes several commonly studied examples: geometric invariant theory, the moduli of bundles on a smooth curve, the moduli of Bridgeland-semistable complexes on a smooth projective variety, the moduli of K-stable varieties. The key construction assigns to any point in an algebraic stack a topological space parameterizing all possible iso-trivial degenerations of that point. When the stack is BG for a reductive G, this recovers the spherical building of G, and when the stack is X/T for a toric variety X, this recovers the support of the fan of X.
Abstract: In many examples of moduli stacks which come equipped with a notion of stable points, one tests stability by considering "iso-trivial one parameter degenerations" of a point in the stack. To such a degeneration one can often associate a real number which measures "how destabilizing" it is, and in these situations one can ask the question of whether there is a "maximal destabilizing" or "canonically destabilizing" degeneration of a given unstable point. I will discuss a framework for formulating and discussing this question which generalizes several commonly studied examples: geometric invariant theory, the moduli of bundles on a smooth curve, the moduli of Bridgeland-semistable complexes on a smooth projective variety, the moduli of K-stable varieties. The key construction assigns to any point in an algebraic stack a topological space parameterizing all possible iso-trivial degenerations of that point. When the stack is BG for a reductive G, this recovers the spherical building of G, and when the stack is X/T for a toric variety X, this recovers the support of the fan of X.
Kentaro Hori (Kavli IPMU)
Title: Gauge Theory Duality and Linear Sigma Models
Abstract: I will explain physical results on 2d gauge theories such as low energy behavior and Seiberg-like duality. When applied to linear sigma models, they yield interesting mathematical correspondences, such as Rodland (Pfaffian vs Grassmannian), Hosono-Takagi (double symmetric determinantal variety vs Reye congruence) as well as their generalizations.
Abstract: I will explain physical results on 2d gauge theories such as low energy behavior and Seiberg-like duality. When applied to linear sigma models, they yield interesting mathematical correspondences, such as Rodland (Pfaffian vs Grassmannian), Hosono-Takagi (double symmetric determinantal variety vs Reye congruence) as well as their generalizations.
Shinobu Hosono (University of Tokyo)
Title: Mirror symmetry and geometry of double quintic symmetroids
Abstract: Based on recent works with Hiromichi Takagi since 2011(which can be found in math.arXiv), I will talk about an example of Calabi-Yau threefold which has a nontrivial Fourier-Mukai partner. This example has many similar properties to the well-studied Grassmann-Pfaffian Calabi-Yau threefolds, but requires detailed studies of quadrics (or double quintic symmetroids) to describe the derived equivalence to its Fourier-Mukai partner. I will try to present interesting (birational) geometries of the double quintic symmetroids which have appeared in the proof of the derived equivalence. Also, I will comment some interesting observations on the BPS numbers.
Abstract: Based on recent works with Hiromichi Takagi since 2011(which can be found in math.arXiv), I will talk about an example of Calabi-Yau threefold which has a nontrivial Fourier-Mukai partner. This example has many similar properties to the well-studied Grassmann-Pfaffian Calabi-Yau threefolds, but requires detailed studies of quadrics (or double quintic symmetroids) to describe the derived equivalence to its Fourier-Mukai partner. I will try to present interesting (birational) geometries of the double quintic symmetroids which have appeared in the proof of the derived equivalence. Also, I will comment some interesting observations on the BPS numbers.
Hiroshi Iritani (Kyoto University)
Title:Gamma Conjectures for Fano Manifolds
Abstract:The Gamma class gives a natural integral structure in the A-model variation of Hodge structure and plays an important role in mirror symmetry. In this talk, I will explain a conjectural relationship between the Gamma classes and quantum cohomology differential equations of Fano manifolds, which we call Gamma Conjecture. This can be viewed as a square root of the index theorem and partially refines Dubrovin’s conjecture. This is a joint work with Sergey Galkin and Vasily Golyshev.
Abstract:The Gamma class gives a natural integral structure in the A-model variation of Hodge structure and plays an important role in mirror symmetry. In this talk, I will explain a conjectural relationship between the Gamma classes and quantum cohomology differential equations of Fano manifolds, which we call Gamma Conjecture. This can be viewed as a square root of the index theorem and partially refines Dubrovin’s conjecture. This is a joint work with Sergey Galkin and Vasily Golyshev.
Young-Hoon Kiem (Seoul National University)
Title:Cosection localized virtual fundamental classes
Abstract: When there is a cosection of the obstruction sheaf of a perfect obstruction theory, the virtual fundamental class can be localized to the zero locus of the cosection. I will discuss this localization phenomenon and the pushforward behavior of the cosection localized virtual fundamental classes. If time permits, I will also talk about Le Potier’s stability for quasi-maps and wall crossing.
Abstract: When there is a cosection of the obstruction sheaf of a perfect obstruction theory, the virtual fundamental class can be localized to the zero locus of the cosection. I will discuss this localization phenomenon and the pushforward behavior of the cosection localized virtual fundamental classes. If time permits, I will also talk about Le Potier’s stability for quasi-maps and wall crossing.
Johanna Knapp (ITP)
Title: Exotic Calabi-Yaus from one- and two-parameter non-abelian GLSMs
Abstract: We explain how to construct exotic Calabi-Yau threefolds from non-abelian GLSMs and discuss five new one-parameter examples found in joint work with K. Hori. A surprising result is that CYs which come from two different GLSMs and have different Hodge numbers h2,1 still appear to have the same K¨ahler moduli space. We further report on a new non-abelian two-parameter model with two geometric phases.
Abstract: We explain how to construct exotic Calabi-Yau threefolds from non-abelian GLSMs and discuss five new one-parameter examples found in joint work with K. Hori. A surprising result is that CYs which come from two different GLSMs and have different Hodge numbers h2,1 still appear to have the same K¨ahler moduli space. We further report on a new non-abelian two-parameter model with two geometric phases.
Jun Li (Stanford University)
Title: Toward an effective theory of GW invariants of quintic CY threefolds.
Abstract:We introduce the moduli of MSP fields (mixed Spin-P fields) as a mean to realize the wall crossing of two LG-theory, that will give us effective algorithm to evaluate GW invariants of quintic CY threefolds. This is a joint work with HL. Chang, WP. Li and M. Liu.
Abstract:We introduce the moduli of MSP fields (mixed Spin-P fields) as a mean to realize the wall crossing of two LG-theory, that will give us effective algorithm to evaluate GW invariants of quintic CY threefolds. This is a joint work with HL. Chang, WP. Li and M. Liu.
Jeongseok Oh (KAIST)
Title: Quasimaps to relative GIT quotients and Applications
Abstracts: T.B.A.
Nathan Priddis (Leibniz University Hannover)
Title: Quasimaps to relative GIT quotients and Applications
Abstracts: T.B.A.
Nathan Priddis (Leibniz University Hannover)
Title: Landau-Ginzburg/Calabi-Yau Correspondence and the Crepant Transformation Conjecture
Abstract:I will discuss current work with Y.P. Lee and Mark Shoemaker, in which prove the Landau {Ginzburg/Calabi{Yau correspondence in genus zero for certain hypersurfaces in weighted projective space. The LG/CY correspondence is a relationship between the Gromov-Witten theory of a Calabi{Yau and the Landau-Ginzburg model, given by FJRW theory, for the corresponding potential and group of symmetries. This correspondence is proved via the Crepant Tranformation Conjecture.
Abstract:I will discuss current work with Y.P. Lee and Mark Shoemaker, in which prove the Landau {Ginzburg/Calabi{Yau correspondence in genus zero for certain hypersurfaces in weighted projective space. The LG/CY correspondence is a relationship between the Gromov-Witten theory of a Calabi{Yau and the Landau-Ginzburg model, given by FJRW theory, for the corresponding potential and group of symmetries. This correspondence is proved via the Crepant Tranformation Conjecture.
Yongbin Ruan (University of Michigan)
Title: Landau-Ginzburg/Calabi-Yau Correspondence
Abstract:This is a survey talk on the topic of Landau-Ginzburg/Calabi-Yau Correspondence, a famous duality from physics to connect Gromov-Witten theory (CY-model) to FJRW-theory (LG-model).
Abstract:This is a survey talk on the topic of Landau-Ginzburg/Calabi-Yau Correspondence, a famous duality from physics to connect Gromov-Witten theory (CY-model) to FJRW-theory (LG-model).
Emanuel Scheidegger (University of Freiburg)
Title: Topological strings and quasimodular forms
Abstract: We discuss the relation between topological strings on Calabi-Yau threefolds and modular forms. We show that there is a special differential ring of functions which plays the analogous role of the ring of quasimodular forms in the case of elliptic curves. The existence of this ring is due to special geometry and the topological string amplitudes are polynomials in this ring. We exhibit a Lie algebra action on this ring. Finally, we will explain a conjecture that expresses the Gromov-Witten invariants at higher genus for elliptically fibered Calabi-Yau threefolds in terms of quasimodular forms. In particular, there is a recursion relation which governs these modular forms.
Abstract: We discuss the relation between topological strings on Calabi-Yau threefolds and modular forms. We show that there is a special differential ring of functions which plays the analogous role of the ring of quasimodular forms in the case of elliptic curves. The existence of this ring is due to special geometry and the topological string amplitudes are polynomials in this ring. We exhibit a Lie algebra action on this ring. Finally, we will explain a conjecture that expresses the Gromov-Witten invariants at higher genus for elliptically fibered Calabi-Yau threefolds in terms of quasimodular forms. In particular, there is a recursion relation which governs these modular forms.
Yefeng Shen (Kavli IPMU)
Title: Modularity of Gromov-Witten correlation functions for elliptic orbifold curves
Abstract: Modularity appears naturally in the study of Gromov-Witten theory of elliptic curves and their quotients. I will provide both an A-model proof and an B-model proof of the modularity of Gromov-Witten correlation functions for quotients of elliptic curves. This also relates to LG/CY correspondence and global mirror symmetry of simple elliptic singularities. This talk briefly introduces works by Todor Milanov, Yongbin Ruan, Jie Zhou and myself.
Abstract: Modularity appears naturally in the study of Gromov-Witten theory of elliptic curves and their quotients. I will provide both an A-model proof and an B-model proof of the modularity of Gromov-Witten correlation functions for quotients of elliptic curves. This also relates to LG/CY correspondence and global mirror symmetry of simple elliptic singularities. This talk briefly introduces works by Todor Milanov, Yongbin Ruan, Jie Zhou and myself.
Yukinobu Toda (Kavli IPMU)
Title: S-duality conjecture for Calabi-Yau 3-folds
Abstract: In this talk, I give the transformation formula of Donaldson-Thomas invariants counting two dimensional torsion sheaves on Calabi-Yau 3-folds under flops. The error term is described by the Dedekindeta function and the Jacobi theta function, and the result gives evidence of a 3-fold version of Vafa-Witten’s S-duality conjecture. As an application, I show that the generating series of Donaldson-Thomas invariants on the local projective plane with any positive rank is described in terms of modular forms and theta type for indefinite lattices.
Abstract: In this talk, I give the transformation formula of Donaldson-Thomas invariants counting two dimensional torsion sheaves on Calabi-Yau 3-folds under flops. The error term is described by the Dedekindeta function and the Jacobi theta function, and the result gives evidence of a 3-fold version of Vafa-Witten’s S-duality conjecture. As an application, I show that the generating series of Donaldson-Thomas invariants on the local projective plane with any positive rank is described in terms of modular forms and theta type for indefinite lattices.
Chris Woodward (Rutgers)
Title:The minimal model program and Floer theory
Abstract:Many years ago Li-Ruan mentioned a connection between birational geometry and quantum cohomology. An observation of Gonzalez and mine is that in the toric case, quantum cohomology has a splitting according to transitions in the minimal model program (mmp). I will discuss the conjecture that this is true in general. A recent result in this direction is that mmp flips produce Floer-non-trivial Lagrangian tori. These are expected to be the generators of the Fukaya category corresponding to the mmp decomposition in cases of varieties with smooth mmp runnings, and generate a splitting of the quantum cohomology according to the open-closed string map.
Abstract:Many years ago Li-Ruan mentioned a connection between birational geometry and quantum cohomology. An observation of Gonzalez and mine is that in the toric case, quantum cohomology has a splitting according to transitions in the minimal model program (mmp). I will discuss the conjecture that this is true in general. A recent result in this direction is that mmp flips produce Floer-non-trivial Lagrangian tori. These are expected to be the generators of the Fukaya category corresponding to the mmp decomposition in cases of varieties with smooth mmp runnings, and generate a splitting of the quantum cohomology according to the open-closed string map.