- 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, non-commutative geometry, non-Archimedean 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 non-Archimedean mirrors in neighborhoods of immersed lagrangians and Landau-Ginzburg theories and the family Floer and non-commutative gluing in the proof of Strominger-Yau-Zaslow mirror symmetry. Talks also surveyed new results on categorical Kaehler and arithmetic structures and in non-commutative Hodge theory. The applications will feature connections with deformed Hermitian-Yang-Mills theory and proofs of classical and categorified mirror conjectures in geometric representation theory, hyper-Kaehler geometry, and higher dimensional quantum field theory. Discussions emphasized emerging open problems and future research directions.
Collaboration website: https://schms.math.berkeley.edu/
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 Calabi-Yau manifold, its connections to Bridgeland stability, and consequences for the quantum Weil-Petersen geometry on the moduli space for the B-model. On the categorical side, Yau described explicit computations of Floer-theoretic 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 Calabi-Yau 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 Ekholm-Lekili and Ganatra. It provides a purely algebraic formalism of Fukaya categories. The new definition has far-reaching 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 A-branes. A fascinating outcome is a nonlinear version of the Riemann-Hilbert 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 Riemann-Hilbert correspondence for differential, q-difference and elliptic difference equations, to non-abelian 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 Gamma-integral structures of Katzarkov-Kontsevich-Pantev. He discussed the change of quantum cohomology of toric orbifolds under ips and explained how the Gamma-integral structure of quantum cohomology are functorially related under toric blow-ups. 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 Calabi-Yau 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 non-compact manifolds with a normal crossing boundary. Tonkonog reported on deep results relating the ad hoc SYZ gluing constructions of Landau-Ginzburg 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 local-to-global 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 Calabi-Yau 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.
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 Calabi-Yau manifold, its connections to Bridgeland stability and quantum Weil-Petersen geometry on the moduli space for B-model and the role of Floer-theoretic deformations of singular SYZ fibers and immersed Lagrangians in HMS; and
- understanding an important class of Calabi-Yau 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.
Kansas State University
Fukaya Categories and the Riemann-Hilbert Correspondence
Mirror symmetry understood vaguely as an equivalence of the category of Lagrangian A-branes and the category of holomorphic B-branes should have a generalization in which coisotropic A-branes are allowed.
The classical Riemann-Hilbert correspondence and its generalizations should have an interpretation in terms of this hypothetical bigger category of A-branes.
In mathematical language, the RH-correspondence 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 Riemann-Hilbert correspondence for differential, q-difference and elliptic difference equations. If time permits, he will explain the corresponding non-abelian Hodge theory in dimension one and its relation to periodic monopoles.
The talk is a part of a joint project with Maxim Kontsevich.
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 D-branes).
University of California, Berkeley
Disk Potentials and Mirror Symmetry
It is well known that the mirror of a Fano manifold is a Landau-Ginzburg 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.
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.
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.
Toric Blow-ups and the Gamma Integral Structure
In this talk, Iritani will discuss the change of quantum cohomology of toric orbifolds under flips and how the Gamma-integral structure of quantum cohomology are functorially related under toric blow-ups.
Toward Local-to-Global 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 Calabi-Yau families admitting toric degenerations.