### Floer-theoretic and algebro-geometric aspects of SYZ mirror symmetry, September 29-October 2nd, 2020

Speakers:

Hülya Argüz

Pierrick Bousseau

Mandy Cheung

Tristan Collins

Yoel Groman

Mark Gross

Jeff Hicks

Yang Li

Nick Sheridan

Zack Sylvan

Sushmita Venugopalan

**Organizers:**Denis Auroux, Catherine Cannizzo, Sheel Ganatra, Heather Lee, Daniel Pomerleano

This workshop will be held **online** on **September 29 – October 2, 2020**.

We are planning for four half-days of talks, with 3 talks on each day (2 on the last day), running from 8am-1:30pm US Pacific (PDT) / 11am-4:30pm US Eastern (EDT) / 4pm-9:30pm UK (GMT+1) / 5pm-10:30pm European time (GMT+2). There will be ample breaks and opportunities for discussion and social interactions.

We hope to bring together mathematicians who work on different aspects of mirror symmetry inspired by the Strominger-Yau-Zaslow perspective, ranging from differential geometry to symplectic geometry to algebraic geometry.

**Format:** Each talk will have two parts. The first part will be expository and aimed at non-specialists (20-30 minutes), followed by a Q&A break (10-15 minutes); and then a research talk (30-40 minutes), and more time for questions and discussion. The talks will be delivered over zoom, and recorded; the recordings will be made available to the research community.

The workshop is co-sponsored by the Simons Collaboration on Homological Mirror Symmetry and the Western Hemisphere Virtual Symplectic Seminar.

**The Zoom link for the lectures will be posted here on Monday September 28.** Registration is not required to attend the lectures.

### Tentative schedule

Tuesday September 29

8:00-9:15am PDT / 11:00-12:15pm EDT 16:00-17:15 GMT+1 / 17:00-18:15 GMT+2 |
Sushmita Venugopalan: Tropical Fukaya algebras |

9:30-10:45am PDT / 12:30-1:45pm EDT 17:30-18:45 GMT+1 / 18:30-19:45 GMT+2 |
Jeff Hicks: Revisiting HMS for toric varieties |

11:00-11:30am PDT / 2:00-2:30pm EDT 19:00-19:30 GMT+1 / 20:00-20:30 GMT+2 |
Tea break |

11:30-12:45pm PDT / 2:30-3:45pm EDT 19:30-20:45 GMT+1 / 20:30-21:45 GMT+2 |
Yang Li: Weak SYZ conjecture |

1:00-1:30pm PDT / 4:00-4:30pm EDT 21:00-21:30 GMT+1 / 22:00-22:30 GMT+2 |
Further discussion |

Wednesday September 30

8:00-9:15am PDT / 11:00-12:15pm EDT 16:00-17:15 GMT+1 / 17:00-18:15 GMT+2 |
Mark Gross: The canonical scattering diagram |

9:30-10:45am PDT / 12:30-1:45pm EDT 17:30-18:45 GMT+1 / 18:30-19:45 GMT+2 |
Hülya Argüz: The higher dimensional tropical vertex |

11:00-11:30am PDT / 2:00-2:30pm EDT 19:00-19:30 GMT+1 / 20:00-20:30 GMT+2 |
Tea break |

11:30-12:45pm PDT / 2:30-3:45pm EDT 19:30-20:45 GMT+1 / 20:30-21:45 GMT+2 |
Mandy Cheung: Some examples of family Floer mirror |

1:00-1:30pm PDT / 4:00-4:30pm EDT 21:00-21:30 GMT+1 / 22:00-22:30 GMT+2 |
Further discussion |

Thursday October 1

8:00-9:15am PDT / 11:00-12:15pm EDT 16:00-17:15 GMT+1 / 17:00-18:15 GMT+2 |
Yoel Groman: Symplectic cohomology of non-exact embeddingsand SYZ mirror symmetry |

9:30-10:45am PDT / 12:30-1:45pm EDT 17:30-18:45 GMT+1 / 18:30-19:45 GMT+2 |
Pierrick Bousseau: Skein algebras from higher genus mirror symmetry |

11:00-11:30am PDT / 2:00-2:30pm EDT 19:00-19:30 GMT+1 / 20:00-20:30 GMT+2 |
Tea break |

11:30-12:45pm PDT / 2:30-3:45pm EDT 19:30-20:45 GMT+1 / 20:30-21:45 GMT+2 |
Tristan Collins: SYZ mirror symmetry for del Pezzo surfaces andrational elliptic surfaces |

1:00-1:30pm PDT / 4:00-4:30pm EDT 21:00-21:30 GMT+1 / 22:00-22:30 GMT+2 |
Further discussion |

Friday October 2

8:00-9:15am PDT / 11:00-12:15pm EDT 16:00-17:15 GMT+1 / 17:00-18:15 GMT+2 |
Nick Sheridan: The Gamma and SYZ conjectures |

9:30-10:45am PDT / 12:30-1:45pm EDT 17:30-18:45 GMT+1 / 18:30-19:45 GMT+2 |
Zack Sylvan: Mirrors to points near SYZ singularities |

11:00-11:30am PDT / 2:00-2:30pm EDT 19:00-19:30 GMT+1 / 20:00-20:30 GMT+2 |
Further discussion |

### Titles and abstracts

**Hülya Argüz: The higher dimensional tropical vertex**

**Abstract:** This talk reports on our joint work with Mark Gross. We show that punctured log Gromov-Witten invariants of Abramovich-Chen-Gross-Siebert, in the case of log Calabi-Yau varieties obtained from blow-ups of toric varieties along hypersurfaces on the toric boundary, can be captured from a combinatorial algorithm encoded in the canonical scattering diagram defined by Gross-Siebert. This extends previous work of Gross-Pandharipande-Siebert to higher dimensions.

**Background material:** a talk by Mark Gross on the tropical vertex, and the corresponding paper; the paper on the higher-dimensional tropical vertex.

**Pierrick Bousseau: Skein algebras from higher genus mirror symmetry**

**Abstract:** The Kauffman bracket skein algebra of a topological surface is a generally non-commutative algebra defined using links in the 3-manifold obtained by taking the product of the surface with an interval. On the other hand, a q-deformation of the mirror construction of Gross-Hacking-Keel produces non-commutative algebras with canonical bases from counting higher genus curves in log Calabi-Yau surfaces (Y,D).

I will explain that the algebra obtained when (Y,D) is a cubic surface with a triangle of lines is the skein algebra of the 4-punctured sphere, and that the canonical basis coincides with the geometrically defined bracelets basis of the skein algebra. As an application, we prove the q-positivity of the structure constants of the bracelets bases of the skein algebras of the 4-punctured sphere and of the 1-punctured torus.

**Background material:** two videos of talks on the q-deformed mirror construction: video 1, video 2

**Mandy Cheung: Some examples of family Floer mirror**

**Abstract:** Fukaya proposed using family Floer homology to understand the mirror structure by considering the instanton corrections from the holomorphic curves/ discs.On the other hand, Gross, Hacking, and Keel gave an algebro-geometric analogues of holomorphic disks in terms of scattering diagrams and theta functions to construct the mirror. It is natural to ask if the two constructions coincide. Together with Yu-shen Lin, we consider the relation between the family Floer mirror construction and open Gromov-Witten invariants. This enables us to compute the family Floer mirror explicitly and we found that the mirror coincides with the one proposed by Gross, Hacking and Keel in this case.

**Background material:** video of a related talk

**Tristan Collins: SYZ mirror symmetry for del Pezzo surfaces and rational elliptic surfaces**

**Abstract:** I will discuss some aspects of SYZ mirror symmetry for pairs (X,D) where X is a del Pezzo surface or a rational elliptic surface and D is an anti-canonical divisor. In particular I will explain the existence of special Lagrangian fibrations and mirror symmetry for (suitably interpreted) Hodge numbers. If time permits, I will describe a proof of SYZ mirror symmetry for del Pezzo surfaces.

**Background material:** The talk is based on this paper. See also this paper for background.

**Yoel Groman: Symplectic cohomology of non-exact embeddings and SYZ mirror symmetry**

**Abstract:** Local symplectic cohomology is a closed string invariant defined for arbitrary compact subsets of closed or geometrically bounded symplectic manifold. It draws roots back in the nineties in the study of quantitative symplectic topology by Floer, Hofer and Cieliebak. Mirror symmetry heuristics suggest that the for subsets of an SYZ fibration associated with appropriate domains in the base, the local symplectic cohomology is mirror to local analytic functions and polyvector fields of the corresponding subset in the mirror, the action valuation being mirror to the sup norm. In fact, this can be taken as the definition of the local mirror providing a closed string alternative to family Floer theory, insofar as reconstruction is concerned. This idea is particularly effective for dealing with singular fibers in reconstructing the mirror. I will discuss some results in this direction. If time permits I will discuss how for mirrors thus constructed one relates the A and B models of Hodge theoretic mirror symmetry. Based on joint work in progress with U. Varolgunes.

**Background material:** part 3 of these notes of a talk by U. Varolgunes.

**Mark Gross: The canonical scattering diagram**

**Abstract:** I will discuss joint work with Bernd Siebert which explains how to associate to a log Calabi-Yau pair (X,D) with D a simple normal crossings anti-canonical divisor with zero-dimension stratum the “canonical scattering diagram”. At the level of SYZ, this should be viewed as describing all Maslov index zero disks with boundary on the fibres of a putative SYZ fibration X-D → B. In actuality, we define this scattering diagram using punctured Gromov-Witten invariants, a generalization of logarithmic invariants developed with Abramovich, Chen and Siebert.

**Background material:** Video of a previous talk.

**Jeff Hicks: Revisiting HMS for toric varieties**

**Abstract:** Some of the first examples of mirror symmetry for non-Calabi-Yau manifolds come from toric varieties. The mirrors are constructed by first noticing that the algebraic torus X is self-mirror. One obtains that B-model toric variety by compactifying the algebraic torus, which is mirror to equipping the A-model with a “mirror superpotential” W. The data of the mirror superpotential modifies the geometry of the Fukaya category by dictating the behavior of non-compact Lagrangian submanifolds at infinity. The work of Abouzaid shows that the subcategory of “tropical Lagrangian sections” of the Fukaya-Seidel category Fuk(X, W) is derived equivalent to the subcategory of line bundles of coherent sheaves on the mirror space.

In the expository portion of this talk, we will review this story for the example of the projective line. In the second half, I will present some ongoing work with Andrew Hanlon. We prove that the partially wrapped Fukaya category of X stopped at the FLTZ skeleton is equivalent to coherent sheaves on the mirror toric variety. Our method uses Lagrangian surgeries to build Lagrangian correspondences mirror to the inclusion of the toric divisor and a generation criterion of Ganatra, Pardon, and Shende.

**Background material:** An expository talk on homological mirror symmetry and Lagrangian surgery; Slides from an earlier version of the talk

**Yang Li: Weak SYZ conjecture**

**Abstract:** One possible interpretation of the SYZ conjecture is that for a polarized family of CY manifolds near the large complex structure limit, there is a special Lagrangian fibration on the generic region of the CY manifold. Generic here means a set with a large percentage of the CY measure, and the percentage tends to 100% in the limit. I will discuss my recent progress on this version of the SYZ conjecture, with emphasis on how differential geometers think about this problem, and give some hint about where nonarchimedean geometry comes in.

**Background material:** The talk is based on arxiv:2007.01384, and some ideas are already in arxiv:1912.02360. The talk is likely to not go beyond what is contained in the introduction. For backgrounds on nonarchimedean geometry, the survey of Boucksom et al arxiv:1502.05724 is quite nice, but will not be assumed.

**Nick Sheridan: The Gamma and SYZ conjectures**

**Abstract:** In the first part of the talk I will give some background on the Gamma Conjecture, which says that mirror symmetry does *not* respect integral cycles: rather, the integral cycles on a complex manifold correspond to integral cycles on the symplectic mirror, multiplied by a certain transcendental characteristic class called the Gamma class. In the second part of the talk I will explain a new geometric approach to the Gamma Conjecture, which is based on the SYZ viewpoint on mirror symmetry. We find that the appearance of ζ(k) in the asymptotics of period integrals arises from the codimension-k singular locus of the SYZ fibration. This is based on joint work with Abouzaid, Ganatra, and Iritani.

**Background material:** Notes from a previous talk

**Zack Sylvan: Mirrors to points near SYZ singularities**

**Abstract:** Family Floer theory is a tool introduced by K. Fukaya and extended by M. Abouzaid for studying non-Hamiltonian perturbations of a Lagrangian. In recent years, it has been one of the main approaches for relating SYZ and homological mirror symmetry. In the first half, I’ll discuss some of the philosophy of this approach, and I’ll explain the main difficulty in applying it to singlular SYZ fibrations. In the second half, I’ll explain a resolution to this difficulty for the spaces ∏*x _{i}* = 1 + ∑

*y*⊂ ℂ

_{j}^{m}×(ℂ

^{*})

^{n}, which appear as neighborhoods of SYZ singularities. In particular, I’ll explain how to cook up a torus-like closed Lagrangian brane for every point of the mirror. This is joint work in progress with M. Abouzaid.

**Background material:** video of a previous talk (focused on the mirror of the structure sheaf).

**Sushmita Venugopalan: Tropical Fukaya algebras**

**Abstract:** A multiple cut operation on a symplectic manifold produces a collection of cut spaces, each containing relative normal crossing divisors. We explore what happens to curve count-based invariants when a collection of cuts is applied to a symplectic manifold. The invariant we consider is the Fukaya algebra of a Lagrangian submanifold that is contained in the complement of relative divisors. The ordinary Fukaya algebra in the unbroken manifold is homotopy equivalent to a “broken Fukaya algebra” whose structure maps count “broken disks” associated to rigid tropical graphs. Via a further degeneration, the broken Fukaya algebra is homotopy equivalent to a “tropical Fukaya algebra” whose structure maps are sums of products over vertices of tropical graphs. This is joint work with Chris Woodward.

**Background material:** The talk is based on this paper. See also the video and slides of a previous talk on this topic. Further background can be found in the papers *Floer theory and flips*, by Francois Charest and Chris Woodward, and

*Compactness results in Symplectic Field theory*, by Bourgeois, Eliashberg, Hofer, Wysocki and Zehnder.