2022 Simons Collaboration on Homological Mirror Symmetry Annual Meeting

The meeting highlighted major advances and applications obtained by the methodology developed by the collaboration in the past seven years. Aside from a comprehensive overview talk by Abouzaid all the talks reported on landmark new results by former and current graduate students and postdocs in the collaboration.

•    Abouzaid surveyed the key areas of symplectic topology, non-commutative geometry and quantum field theory that underwent extensive developments through the works of the collaboration members. Landmarks here included the proof of mirror symmetry without corrections and near immersed Lagrangians, local to global approaches to constructing mirrors, the categorical reconstruction of Hodge theoretic mirror symmetry, the newly discovered enumerative meaning of the arithmetic of periods, non-commutative gluing theorems, the proof of mirror symmetry for singular spaces, and the integrality of non-commutative Hodge structures of symplectic The emphasis was on the resolution of longstanding conjectures, the introduction of new perspective on important problems and the opening of completely unexpected new connections.

•    In his talk, Hicks explained how the classical Floer theory of monotone Lagrangian cobordisms allows us to describe iterated mapping cones between general objects of the Fukaya category. The construction utilizes generators in the Fukaya category and exhibits a Lagrangian cobordism from the generating Lagrangians to the desired general Hicks discussed strong applications in HMS where the monotonicity requirement is met, analyzed the unobstructedness beyond the monotone setting and demonstrated how the geometry of Lagrangian cobordisms determines unobstructedness. Sample computations for how these decompositions can be used to algorithmically construct bounding cochains for Lagrangian submanifolds were also presented.

•    In her lecture, Ward described her recent joint work with Keating which completes a major step in the HMS Collaboration’s program of extracting new birational invariants of algebraic varieties from symplectic topology. The exciting new direction initiated by Keating and Ward gives a universal description of the symplectomorphisms mirror to the elements in the Cremona group. In particular, Keating and Ward construct a universal four-dimensional Weinstein domain of infinite type and prove an HMS correspondence between distinguished birational transformations of the projective plane preserving a standard holomorphic volume form and symplectomorphisms of this universal domain. The Keating–Ward universal domain contains every Liouville manifold mirror to a log Calabi–Yau surface as a Weinstein subdomain. In particular, Viterbo restriction to these subdomains recovers a mirror correspondence between the automorphism group of any open log Calabi–Yau surface and the symplectomorphism group of its mirror. This exciting progress also leads to a precise conjectural HMS correspondence between the full group of volume-preserving birational transformations of the plane and a group of symplectomorphisms of non-exact symplectic deformations of the Keating–Ward domain.

•    Yu reported on a breakthrough progress, joint with Keel, on proving positivity and integrality of the mirror structure constants for a log Calabi–Yau variety. In particular, Yu presented a new construction of the structure constants of the mirror algebra of theta functions for a smooth affine log Calabi–Yau variety. The construction utilizes delicate counts of non-Archimedean analytic disks in the skeleton of the Berkovich analytification of the Calabi–Yau variety. This generalizes previous constructions of Gross–Hacking–Keel and Keel–Yu which relied on extra toric assumptions. The new proof is based on an analytic modification of the target space as well as the theory of skeletal curves. The toolbox Keel–Yu developed for carrying out this proof has a much wider range of applicability, and several generalizations and comparisons of virtual fundamental classes were discussed.

•    In her lecture, Argüz described exciting new progress understanding the relationship between Gross–Siebert intrinsic mirrors and Fock–Goncharov dual cluster varieties. A sequence of recent works of Argüz, Argüz–Gross, and Argüz–Bousseau bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry. The precise result is that the mirror to a cluster variety is a degeneration of the Fock–Goncharov dual cluster The proof and the algorithmic construction of the requisite degenerations are based on a delicate comparison of the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich and the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions.

•    Yuan reviewed a powerful new construction of the mirror space of a local SYZ singularity. The construction utilizes the family Floer theory developed by Abouzaid, Fukaya and other members of the collaboration in the past seven years. The mirror rigid analytic fibration (including the singular fibers) arises as a modification of the singular non-Archimedean model of Kontsevich–Soibelman and also agrees with the construction of Gross–Hacking–Keel–Siebert. In the process, Yuan identifies the conceptual obstruction that prevented the discovery of this construction for many years. Surprisingly, the non-Archimedean topology requires a modification of the conventional Maurer–Cartan picture since the dual singular fiber can be larger than the Maurer–Cartan set. The dual singular fiber also has an intrinsic geometric description relying on the classification of two types of quantum corrections and the theory of Chambert–Loir and Ducros. The new approach leads to the first unconditional mathematically precise T-duality interpretation of mirror symmetry and provides string supporting evidence for a cornerstone conjecture on the behavior of B-model non-commutative Hodge structures.

•    Wilkins described his new method of equivariant localization construction of Borman–Sheridan classes. The method is based on a surprising, upgraded Atiyah–Bott localization mechanism that can be applied to pseudocycle bordisms. In this construction the bordism relates the homotopy quotient of a space by a circle action with its fixed point set via a correction involving the non-triviality of the induced circle bundle. Wilkins explained how this technology can apply to circle equivariant moduli space of holomorphic curves. This turns out to be quite involved since genericity prevents such spaces from being a homotopy Still the argument can be modified to produce a conceptual localization construction of Borman–Sheridan classes.

•    Xu described his spectacular new joint work with Bai that proves the integral Arnold conjecture on the number of periodic orbits of a Hamiltonian vector field. Due to the existence of curves with nontrivial automorphisms, moduli spaces of stable maps behave like orbifolds rather than manifolds. Xu explained how the classical approach of Fukaya and Ono for extracting integral invariants from the moduli spaces can be made rigorous. The putative Fukaya–Ono invariants, which morally count curves with a trivial automorphism group, depend crucially on the stable complex structure on the moduli. Xu explained a new rigorous construction of such integral invariants in the case of genus zero symplectic Gromov–Witten theory and showed how the new invariants can be used to prove the hardest version of the Arnold conjecture.

Mohammed Abouzaid
Columbia University

A Decade of Mirror Symmetry

Since the beginning of the HMS Simons collaboration, the areas of mirror symmetry has seen extensive developments, including the proof of long-standing conjectures, the introduction of new perspective on important problems and the opening of completely unexpected new connections. Mohammed Abouzaid will survey some if these developments, with a view towards where the potential future of the subject.
 

Hülya Argüz
University of Georgia

Fock–Goncharov Dual Cluster Varieties and Gross–Siebert Mirrors

Cluster varieties come in pairs: for any X-cluster variety there is an associated Fock–Goncharov dual A-cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. Hulya Arguz will explain how to bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry and show that the mirror to the X-cluster variety is a degeneration of the Fock–Goncharov dual A-cluster variety and vice versa. To do this, we investigate how the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich compares with the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Pierrick Bousseau.
 

Jeffrey Hicks
University of Edinburgh

Floer Theory and Lagrangian Cobordisms

A Lagrangian cobordism, as introduced by Arnold, is a Lagrangian submanifold with a notion of “ends” in the stabilization of a symplectic manifold X. The ends are Lagrangian submanifolds of X. The Floer theory of Lagrangian cobordisms was first studied by Biran and Cornea, who showed that the ends of a monotone Lagrangian cobordism are related by an iterated mapping cone in the Fukaya category.

Many known HMS correspondences rely on matching generators of categories. Provided we can meet the monotonicity requirement, Lagrangian cobordisms allow us to bootstrap to more general objects of the Fukaya category by exhibiting a Lagrangian cobordism from the generating Lagrangians to the desired object. In this talk, Jeffrey Hicks will discuss a few applications in HMS where this monotonicity requirement is met, what one can say beyond the monotone setting (unobstructedness) and how the geometry of Lagrangian cobordisms determines unobstructedness.
 

Abigail Ward
MIT

Symplectomorphisms Mirror to Birational Transformations of the Projective Plane

Abigail Ward will construct a non-finite type four-dimensional Weinstein domain M_{univ} and describe an HMS correspondence between distinguished birational transformations of the projective plane preserving a standard holomorphic volume form and symplectomorphisms of M_{univ}. The space M_{univ} is universal in the sense that it contains every Liouville manifold mirror to a log Calabi-Yau surface as a Weinstein subdomain; under Viterbo restriction to these subdomains, Ward will recover a mirror correspondence between the automorphism group of any open log Calabi-Yau surface and the symplectomorphism group of its mirror. We also give a conjectural HMS correspondence between the full group of volume-preserving birational transformations and a group of symplectomorphisms of non-exact symplectic deformations of M_{univ}. This is joint work in progress with Ailsa Keating.
 

Nicholas Wilkins
MIT

S^1 Localization via Pseudocycles, Applied to Borman-Sheridan Classes

S^1-localisation due to Atiyah-Bott is a powerful tool to understand S^1-equivariant cohomology, but historically it could not be leveraged against, for example, the S^1-quotient of a moduli space parameterized by ES^1, as such a moduli space in general is not defined as a homotopy quotient for regularity reasons. In this talk, Wilkins will discuss a new way to interpret S^1-localisation, constructed by way of a pseudocycle bordism between pseudocycles (assigned to an S^1-manifold and its fixed point set) in equivariant homology. Wilkins will then demonstrate how to lift this method of S^1-localisation to moduli spaces, and through this, prove a conjecture about Borman-Sheridan classes due to Seidel.
 

Guangbo Xu
Texas A&M University

Integral Count of Pseudoholomorphic Curves

Due to the existence of curves with nontrivial automorphisms, moduli spaces of stable maps behave like orbifolds rather than manifolds. Therefore, invariants in symplectic geometry obtained from counting curves are generally rational numbers but not integers. In late 1990s, Fukaya and Ono sketched a new way to extract integral invariants from the moduli spaces. These invariants, which morally count curves with a trivial automorphism group, depend crucially on the stable complex structure on the moduli. Following Fukaya-Ono proposal, in a recent preprint (2201.02688), we rigorously constructed such integral invariants in the case of genus zero symplectic Gromov-Witten theory. In this talk, Guangbo Xu will explain the original idea of Fukaya-Ono, our technical construction, and potential extensions. This talk is based on the joint work with Shaoyun Bai.
 

Hang Yuan
Northwestern University

Family Floer Mirror Space for Local SYZ Singularities

Hang Yuan will give an SYZ construction of singularities by the family Floer approach. The dual analytic fibration (including the singular fibers) will be explicitly presented in the talk. It modifies a non-Archimedean singular model of Kontsevich-Soibelman and also agrees with the work of Gross-Hacking-Keel-Siebert. Yuan will review the family Floer mirror construction and explain why it is necessary to challenge the conventional Maurer-Cartan picture for the sake of the non-Archimedean topology. By our new example, the dual singular fiber can be larger than the Maurer-Cartan set. The dual singular fiber also has an intrinsic geometric explanation by a classification of two types of quantum correction and the theory of Chambert-Loir and Ducros. We will make a mathematically precise T-duality statement. We give evidence to convince people of it, even if people don’t know anything about the family Floer business. If time allowed, we will describe a version of SYZ converse and justify some computations for the folklore conjecture for the critical values of the mirror Landau-Ginzburg superpotential.
 

Tony Yue Yu
California Institute of Technology

Mirror Structure Constants via Non-Archimedean Analytic Disks

For any smooth affine log Calabi-Yau variety U, Tony Yue Yu will construct the structure constants of the mirror algebra to U via counts of non-Archimedean analytic disks in the skeleton of the Berkovich analytification of U. This generalizes our previous construction with extra toric assumptions. The technique is based on an analytic modification of the target space as well as the theory of skeletal curves. Consequently, we deduce the positivity and integrality of the mirror structure constants. If time permits, Yu will discuss further generalizations and virtual fundamental classes. Joint work with S. Keel.