Simons Collaboration on Homological Mirror Symmetry Annual Meeting 2021

The new discoveries and results obtained by the collaboration members and research affiliates in the past year focus on novel and deep crossover applications of the mirror correspondence.

    •    Pomerleano reported on his recent insight into the symplectic version of the Frobenius structure conjecture of Gross-Hacking-Keel. He explained how the cases of the conjecture recently proven by Keel and Yu can be combined with strong finite generation results for symplectic cohomology to extend the cluster mirror construction of Gross-Hacking-Keel-Kontsevich and to prove homological mirror symmetry for a large class of affine log Calabi-Yau varieties.

    •    In her public lecture, Aganagić explained how Auroux’s method for constructing mirrors of complements of anticanonical divisors can be used to prove homological mirror symmetry for moduli spaces of Dirac monopoles with prescribed defects. She also showed how this mirror symmetry can be used to produce new invariants of knots and links and described her striking mirror symmetry solution of the knot categorification problem. In her research talk, Aganagić elaborated on this novel methodology recasting it as a far reaching extension of Sheridan’s theorem—the homological mirror correspondence is mediated by an explicit associative algebra that computes both the requisite equivariant category of coherent sheaves on the moduli of monopoles and the Fukaya-Seidel category of its mirror. She used this to construct an explicit extension of Heegaard-Floer homology, which categorifies quantum link invariants colored by an arbitrary simply laced Lie algebra.

    •    In his lecture, Kontsevich described the solution of a key step in the program for extracting new birational invariants from Gromov-Witten theory. The new progress concerns the discovery of an explicit formula expressing the genus zero Gromov-Witten invariants of a blow-up in terms of the Gromov-Witten invariants of the original variety and the Gromov-Witten invariants of the center of the blow-up. The formula has a complicated structure, but Kontsevich explained the characteristic properties which completely determine this structure. He outlined a strategy for proving the formula and explained recent work with Katzarkov and Yu that carries out this strategy. In his talk, Katzarkov explained how the new blow-up formula feeds into non-commutative Hodge theory to produce birational invariants over non-algebraically closed fields. He presented recent joint results with Kontsevich and Pantev that utilize the blow-up formula and a study of WKB asymptotics to extract quantum dimension spectrum invariants from variations of non-commutative Hodge structures and to use these data to prove strong non-rationality results. An essential technical ingredient here is a recent proof of special cases of the exponential type conjecture by Katzarkov, Kontsevich and Pantev. During the meeting, Pomerleano explained a new approach in his ongoing joint work with Seidel which will yield a proof of the conjecture for general Landau-Ginzburg models.

    •    Seidel surveyed the picture of homological mirror symmetry relative to an anticanonical divisor and its structural implications for the Fukaya categories of Calabi-Yau hypersurfaces. In particular, he showed how one can prove strong polynomiality and analyticity statements for the dependence on the Novikov parameter and also discussed finite determination results for the Fukaya A∞-structures. The approach to these structural results is rooted in the homological mirror correspondence and uses Seidel’s new formalism of non-commutative Lefschetz pencils.

    •    Lee described a new joint work with Hosono, Lian and Yau that builds mirror pairs of singular Calabi-Yau varieties arising from double covers. He presented non-trivial enumerative mirror tests for the construction, including proofs of the matching of Euler characteristics, Hodge numbers and orbifold Gromov-Witten invariants. Lee also discussed the categorical invariants of these pairs and indicated how the Auroux-Efimov-Katzarkov approach to proving singular mirror symmetry can be adapted to the fractional Calabi-Yau setting.

    •    Keating described exciting recent progress in understanding how homological mirror symmetry intertwines geometric and non-geometric symmetries on the complex and symplectic sides. She presented a joint work with Hacking, which uses mirror symmetry to construct generators and compute relations of the symplectic mapping class group of the Milnor fibers of cusp singularities. The construction leads to a completely new class of non-finiteness results for symplectic mapping classes.

    •    Abouzaid reviewed his joint work with Blumberg on arithmetic symplectic invariants. He described major recent progress in Floer homotopy theory and some unexpected applications to symplectic topology and algebraic geometry, e.g., the proof of a characteristic p version of Arnold’s conjecture. He also discussed the interaction of these new concepts with homological mirror symmetry and described new powerful results with McLean and Smith in which virtual fundamental classes in Morava K-theory are used to construct global Kuranishi charts on moduli of genus zero pseudo-holomorphic curves.

Mohammed Abouzaid
Columbia University
Prospects for Floer Homotopy Theory

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Abouzaid will review recent progress in applying Floer homotopy theory to symplectic topology and algebraic geometry, and discuss the interaction of these ideas with homological mirror symmetry.
 

Mina Aganagic
University of California, Berkeley
Homological Mirror Symmetry (HMS) and Knot Categorification

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In this talk, Aganagic will describe an application of homological mirror symmetry (HMS) to solve an old problem from a different branch of mathematics. The problem is to categorify Chern–Simons link invariants in a manner which is uniform with respect to the choice of a Lie algebra and originates from geometry. The solution to the knot categorification problem comes with a new relation between HMS and representation theory. There is a family of mirror pairs (X, Y), labeled by a choice of a simply laced Lie algebra, a collection of its representations, and a weight in their tensor product. Mirror symmetry should be manifest in that the derived category of equivariant coherent sheaves on X and the derived Fukaya–Seidel category of its mirror Y are both equivalent to a derived category of modules of the same associative algebra A, which turns out to be a cousin of the algebra considered by Khovanov and Lauda and by Rouquier. The action of braiding and the branes corresponding to the caps and the caps which close off braids into links all have explicit geometric and algebraic descriptions. The symplectic geometry side of mirror symmetry is a new theory generalizing Heegard–Floer theory. The generalization corresponds to replacing gl(1|1) by an arbitrary simply laced Lie algebra. Extension to nonsimply laced Lie algebras should arise by folding.
 

Ailsa Keating
University of Cambridge
Symplectomorphisms of Some Weinstein Four-Manifolds

Let \(M\) be the Weinstein four-manifold mirror to \(Y\backslash D\) for \((Y,D)\) a log Calabi–Yau surface; this is usually the Milnor fiber of a cusp singularity. We introduce two families of symplectomorphisms of \(M\): Lagrangian translations, which we prove are mirror to tensors with line bundles; and nodal slide recombinations, which we prove are mirror to automorphisms of \(Y\). Together with spherical twists, these are expected to generate the symplectic mapping class group of \(M\). Time permitting, some applications will be given. This is based on joint work with Paul Hacking.
 

Ludmil Katzarkov
University of Miami
Spectra and Applications

Katzarkov will introduce new kinds of spectra related to noncommutative Hodge structures. Applications to birational geometry will be considered.
 

Maxim Kontsevich
Institut des Hautes Études Scientifiques
Blow-up Equivalence

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A couple of years ago, Kontsevich proposed a program relating Gromov–Witten invariants and birational geometry. The key step in this program is a hypothetical formula expressing genus 0 GW invariants of a blow-up of a smooth variety X at smooth center Y, in terms of those of X and Y. The exact shape of the formula turns out to be quite complicated, involving WKB solutions of certain universal family of linear ODEs depending only on one natural number, the codimension of Y in X. In his talk, Kontsevich will give the exact formulation of the blow-up formula (which is joint work in progress with Ludmil Katazrkov and Tony Yu Yue). Hypothetically, this formula extends to the equivalence between Fukaya categories of the blow-up and of the disjoint union of X and several copies of Y.
 

Tsung-Ju Lee
Harvard University
Mirror Symmetry on Singular Calabi–Yau Spaces

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Mirror symmetry for singular Calabi–Yau spaces were discovered by Hosono, Lian, Takagi and Yau in their recent work on K3 surfaces. In this talk, Lee will construct pairs of singular Calabi–Yau varieties arising from double covers and give some numerical evidence that indicates they are mirror pairs, including Euler characteristics, Hodge numbers and orbifold Gromov–Witten invariants. Lee will also discuss these pairs from categorical perspectives.
 

Daniel Pomerleano
University of Massachusetts, Boston
The Frobenius Property and HMS for log CY Varieties

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Pomerleano will discuss a symplectic version of the Frobenius structure conjecture, recently proven by Keel and Yu. He will then explain how this result, together with recent finiteness results for wrapped invariants of affine log Calabi–Yau varieties, allows one to prove HMS for a wide class of examples.
 

Paul Seidel
Massachusetts Institute of Technology
Mirror Symmetry and Noncommutative Linear Systems

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Seidel will try to survey the picture of HMS relative to an anticanonical divisor, and what this implies for the Fukaya categories of Calabi–Yau hypersurfaces.

5:00-6:15 PM

A public lecture on meeting themes will be presented the evening before the official start of the conference. Participants are encouraged to register and attend should their travel and other schedules align.

Simons Foundation Lectures are free public colloquia related to basic science and mathematics. These high-level talks are intended for professors, students, postdocs and business professionals, but interested people from the metropolitan area are welcome as well.

More information is available at the lecture’s page.

Participation is optional; separate registration is required.
 

Mina Aganagic
University of California, Berkeley

(Homological) Knot Invariants From Mirror Symmetry

Quantum invariants of knots (also known as Chern-Simons knot invariants) have many applications in mathematics and physics. For example, Khovanov showed in 1999 that the simplest such invariant, the Jones polynomial, arises as the Euler characteristic of a homology theory. The knot categorification problem is to find a general construction of knot homology groups and to explain their meaning: What are they homologies of?

Mirror symmetry is another important strand in the interaction between mathematics and physics. Homological mirror symmetry, formulated by Kontsevich in 1994, naturally produces hosts of homological invariants. Sometimes, it can be made manifest, and then its striking mathematical power comes to the fore. Typically, though, it leads to invariants that have no particular interest outside of the problem at hand.

In this lecture, Mina Aganagic will present how she recently showed there is a vast new family of mirror pairs of manifolds, for which homological mirror symmetry can be made manifest. They do lead to interesting invariants. In particular, they solve the knot categorification problem.

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