Simons Collaboration on Homological Mirror Symmetry Annual Meeting 2018
 Organized by

Tony Pantev, Ph.D.University of Pennsylvania
The third annual meeting of the Simons Collaboration on Homological Mirror Symmetry focused on new developments bridging symplectic topology, Hodge theory, noncommutative geometry, nonArchimedean analysis and arithmetic quantum invariants. Talks highlighted major progress in constructing and understanding Fukaya categories, establishing the associativity of quantum products and proving Homological and Hodge theoretic Mirror Symmetry in a variety of geometric setups.
The annual meeting showcased exciting novel axiomatic and symplectic field theory approaches to wrapped Fukaya categories, the construction of nonArchimedean mirrors in neighborhoods of immersed lagrangians and LandauGinzburg theories and the family Floer and noncommutative gluing in the proof of StromingerYauZaslow mirror symmetry. Talks also surveyed new results on categorical Kaehler and arithmetic structures and in noncommutative Hodge theory. The applications will feature connections with deformed HermitianYangMills theory and proofs of classical and categorified mirror conjectures in geometric representation theory, hyperKaehler geometry, and higher dimensional quantum field theory. Discussions emphasized emerging open problems and future research directions.
Collaboration website: https://schms.math.berkeley.edu/
Report
Yau reported on work from the past three years focusing on the interactions among the categorical, analytic and metric aspects of mirror symmetry. Major progress here incudes new, precise understanding of the differential geometry of SYZ fibrations of a CalabiYau manifold, its connections to Bridgeland stability, and consequences for the quantum WeilPetersen geometry on the moduli space for the Bmodel. On the categorical side, Yau described explicit computations of Floertheoretic deformations of singular torus fibers and immersed Lagrangians in symplectic manifolds and the homological construction of rigid analytic neighborhoods of mirrors. He also discussed a newly discovered class of CalabiYau mirrors arising from fractional complete intersections which exhibit unexpected enumerative and arithmetic features. The new features suggest remarkable connections between mirror symmetry and other enumerative dualities, and Yau described several results establishing these connections in dimensions 2 and 3. In the last part of his talk, Yau outlined several important open problems for the HMS/SYZ program going forward.
Kontsevich and Soibelman presented a new, streamlined definition and construction of the Fukaya category based on the concept of a singular Lagrangian skeleton. A major advance reported by Kontsevich is a solution of the big outstanding problem of defining quantum corrections for skeleta. The solution uses tools from Symplectic Field Theory and builds on deep work in contact and symplectic topology by EkholmLekili and Ganatra. It provides a purely algebraic formalism of Fukaya categories. The new definition has farreaching ramifications and leads to unexpected insights into the structure of symplectic invariants. In particular, it gives new existence predictions for Bridgeland stabilities on Fukaya categories for symplectic manifolds with trivial first Chern class and provides a concrete description of interactions between coisotropic Abranes. A fascinating outcome is a nonlinear version of the RiemannHilbert correspondence which identifies coherent sheaves on a quantized complex symplectic manifold and Fukaya objects with singular Lagrangian supports. Soibelman discussed this new theory in the special case of symplectic surfaces and made a connection to the classical RiemannHilbert correspondence for differential, qdifference and elliptic difference equations, to nonabelian Hodge theory in dimension one and to periodic monopoles.
Ganatra presented a series of new structural results for partially wrapped Fukaya categories, and their role in the recent proof of an equivalence between the partially wrapped Fukaya category of a cotangent bundle stopped at any subanalytic singular isotropic Λ and a category of “large” sheaves on the zero section microsupported along the same Λ. The proof is a formal consequence of the structural results and ties up with a powerful novel axiomatic characterization for general wrapped Fukaya categories, which is currently being developed by Ganatra, Pardon and Shende.
Iritani described a remarkable progress in understanding the birational geometry of noncommutative Hodge theoretic invariants and specifically of the Gammaintegral structures of KatzarkovKontsevichPantev. He discussed the change of quantum cohomology of toric orbifolds under ips and explained how the Gammaintegral structure of quantum cohomology are functorially related under toric blowups. These results fit in a program jointly developed by Abouzaid, Ganatra, Iritani and Sheridan, which proposes a new method for computing asymptotics of periods using tropical geometry. In this framework, Riemann zeta values appear naturally as error terms in tropicalization. The method also explains how the Gamma class should arise from the SYZ conjecture and gives a new proof of the Gamma conjecture for Batyrev pairs of mirror CalabiYau hypersurfaces.
Abouzaid, Fukaya and Tonkonog presented important technical advances in Floer theory that are essential for the algebraic construction of mirrors. Fukaya described the relations of the various constructions of Floer homology and focused on the remaining technical problems for understanding the requisite bounding cochains and deformation theory for tropical or analytic disk counting in noncompact manifolds with a normal crossing boundary. Tonkonog reported on deep results relating the ad hoc SYZ gluing constructions of LandauGinzburg mirrors of Fano varieties and new, direct constructions in Floer theory. He also discussed the period integral formula for quantum periods of Fanos and the quantum Lefschetz formula. Abouzaid gave a comprehensive survey of the newly developed localtoglobal proofs of mirror symmetry. He reviewed the progress made in the proof of mirror equivalences via families of Lagrangians, including the proof of the equivalence for local models of the degenerations studied by Gross and Siebert. He also formulated the precise challenges in the remaining steps for the proof of mirror symmetry in the case of CalabiYau families admitting toric degenerations and explained how many of these challenges can be solved by using his recent joint results with Ganatra and Sylvan as well as recent work of Groman and Varolgunes.
Videos
ShingTung Yau
Harvard University
HMS/SYZ Mirror Symmetry: Recent Progress and Going Forward
Yau will give a summary report on work that the Boston HMS/SYZ team has done toward the program over the last few years of collaboration. Progress includes:
 understanding analysis of SYZ fibration of a CalabiYau manifold, its connections to Bridgeland stability and quantum WeilPetersen geometry on the moduli space for Bmodel and the role of Floertheoretic deformations of singular SYZ fibers and immersed Lagrangians in HMS; and
 understanding an important class of CalabiYau mirrors with new techniques from Hodge theory and algebraic geometry of cyclic covers and fractional complete intersections, as well as investigating their enumerative and arithmetic ramifications in low dimensions; relations between mirror symmetry and other enumerative dualities in dimensions two and three are also established.
Yau will also discuss some of the important open problems for the HMS/SYZ program going forward.
Yan Soibelman
Kansas State University
Fukaya Categories and the RiemannHilbert Correspondence
Mirror symmetry understood vaguely as an equivalence of the category of Lagrangian Abranes and the category of holomorphic Bbranes should have a generalization in which coisotropic Abranes are allowed.
The classical RiemannHilbert correspondence and its generalizations should have an interpretation in terms of this hypothetical bigger category of Abranes.
In mathematical language, the RHcorrespondence should be a statement that describes the category of certain coherent sheaves on a quantized complex symplectic manifold in terms of the Fukaya category of this manifold, in which singular Lagrangian supports of objects are allowed.
Soibelman will discuss this proposal in the case of symplectic surfaces, which arise in the RiemannHilbert correspondence for differential, qdifference and elliptic difference equations. If time permits, he will explain the corresponding nonabelian Hodge theory in dimension one and its relation to periodic monopoles.
The talk is a part of a joint project with Maxim Kontsevich.
Maxim Kontsevich
IHES
On the Definition of the Fukaya Category
Kontsevich will present an approach to the definition of Fukaya category, which is based on the concept of a singular Lagrangian support. The problem of defining quantum corrections is resolved by tools from symplectic field theory (joint work with Y. Soibelman). This new definition allows us to formulate a conjecture on the existence of Bridgeland stability on Fukaya category for symplectic manifolds with trivial first Chern class (a generalization of the original idea of M. Douglas on Dbranes).
Dmitry Tonkonog
University of California, Berkeley
Disk Potentials and Mirror Symmetry
It is well known that the mirror of a Fano manifold is a LandauGinzburg model; the SYZ approach constructs it by appropriately gluing together the holomorphic disk potentials of Lagrangian tori. Tonkonog will explain how to approach mirror symmetry predictions using Floer theory, staying purely within the holomorphic disk point of view on the mirror LG model. He will focus on two examples: the period integral formula for quantum periods of Fanos and the quantum Lefschetz formula.
Sheel Ganatra
University of Southern California
Microlocal Morse Theory of Wrapped Fukaya Categories
Ganatra will describe a series of new structural results for (partially) wrapped Fukaya categories and their role in the recent proof of an equivalence between the partially wrapped Fukaya category of a cotangent bundle (stopped at any subanalytic singular isotropic Lambda) and a category of “large” sheaves on the zero section (microsupported along the same Lambda). The proof is largely formal after one knows aforementioned structural results, and gives evidence for an emerging axiomatic characterization (in progress) for general wrapped Fukaya categories. This is joint work with John Pardon and Vivek Shende.
Kenji Fukaya
Simons Center for Geometry and Physics
Lagrangian Floer Theory in Mirror Symmetry and Topological Field Theory
Lagrangian Floer homology was invented by A. Floer around 40 years ago. It is the main object to be studied on the symplectic side of (homological) mirror symmetry. In this talk, Fukaya will survey:
 its basic properties and expected role in mirror symmetry;
 various problems with its definition and establishing its basic properties and present the status of study;
 various variant versions of its generalizations; and
 some other expected applications, such as those to symplectic toplogy and gauge theory.
Hiroshi Iritani
Kyoto University
Toric Blowups and the Gamma Integral Structure
In this talk, Iritani will discuss the change of quantum cohomology of toric orbifolds under flips and how the Gammaintegral structure of quantum cohomology are functorially related under toric blowups.
Mohammed Abouzaid
Columbia University
Toward LocaltoGlobal Proofs of Mirror Symmetry
Abouzaid will review the progress made in the proof of mirror equivalences via families of Lagrangians, including the proof of the equivalence for local models of the degenerations studied by Gross and Siebert. He will then survey the remaining steps for the proof of mirror symmetry in the case of CalabiYau families admitting toric degenerations.