### Mirror Symmetry and Enumerative Geometry, UC Berkeley, March 26-30, 2018

Speakers:

Jim Bryan

Tobias Ekholm

Sheel Ganatra

Yoel Groman

Andrew Harder

Hansol Hong

Hiroshi Iritani

Martijn Kool

Yankı Lekili

Melissa Liu

Sven Meinhardt

Ivan Smith

Balázs Szendrői

Dmitry Tonkonog

Sara Tukachinsky

Dmitry Vaintrob

Renato Vianna

Zhengyu Zong

**Organizers:** D. Auroux, M. Abouzaid, L. Katzarkov, T. Pantev

This conference will take place at UC Berkeley on **March 26-30, 2018**.

This event is partially supported by the Simons Collaboration on Homological Mirror Symmetry.

Note that MSRI will be holding a workshop on Structures in Enumerative Geometry the preceding week (March 19-23).

→ Registration form (deadline: February 15)

Venue and practical information

The conference will take place on the UC Berkeley campus, in **Evans Hall, Room 60** (ground floor). Note that Friday 3/30 is a campus holiday and the building will be locked; access information will be provided; if 60 Evans is locked we will meet in 1015 Evans (10th floor) instead.

See interactive campus map; printable campus map (beware North is left).

The closest airports are San Francisco (SFO) and Oakland (OAK). Berkeley can be reached from either one by public transportation using BART (about 1 hour from SFO, 40 minutes from OAK); you can walk or take a taxi or bus from the Downtown Berkeley BART stop.

**Housing:** Hotel reservations have been made at the Faculty Club (on campus) and at the Rose Garden Inn (20 minute walk from Evans Hall) for participants who have requested a room. Individual reservation confirmations have been sent, please contact the organizers if you did not receive yours.

**Travel reimbursements:** Participants receiving travel reimbursements will need to submit all receipts (for airfare, the receipt must show name, amount paid and method of payment), and complete UC Berkeley’s travel forms, which will be handed out to you at the conference.

If you are not a US citizen or permanent resident, we will also need to make a copy of your passport and I-94 form. **J-1 visa holders:** you will need to provide a letter from your home institution allowing you to attend this conference and receive travel reimbursement. Please request it from your university’s international scholars office. **B-1/B-2/visa waiver:** please make sure to enter the US on B-1 or W/B (business) status, not B-2 or W/T (tourist); tourist status would restrict our ability to reimburse your expenses.

### Preliminary schedule

Monday March 26 (in 60 Evans)

9:30-10:30 | Tobias Ekholm: Large N duality, SFT stretching, and the skein relation (Video) |

11:00-12:00 | Dmitry Tonkonog: Quantum periods theorem for Landau-Ginzburg potentials (Video) |

2:00-3:00 | C.-C. Melissa Liu: Genus zero open-closed mirror symmetry for toric Calabi-Yau 3-orbifolds (Video) |

3:30-4:30 | Yoel Groman: The wrapped Fukaya category for semi-toric SYZ fibrations (Video) |

Tuesday March 27 (in 60 Evans)

9:30-10:30 | Jim Bryan: Donaldson-Thomas theory of the banana manifold, modular forms, and elliptic genera (Video) (Slides) |

11:00-12:00 | Hiroshi Iritani: A Fock sheaf associated with the Gromov-Witten potentials of the local P^{2} (Video) |

2:00-3:00 | Yankı Lekili: Homological mirror symmetry for K3 surfaces via moduli of A_{∞} structures (Video) |

3:30-4:30 | Sara Tukachinsky: Genus zero open Gromov-Witten theory via A_{∞} structures (Video) |

Wednesday March 28 (in 60 Evans)

9:30-10:30 | Balázs Szendrői: Hilbert schemes of points on singular surfaces (Video) |

11:00-12:00 | Andrew Harder: Filtrations on cohomology and mirror symmetry (Video) |

Thursday March 29 (in 60 Evans)

9:30-10:30 | Ivan Smith: Lagrangian cobordism and tropical curves (Video) |

11:00-12:00 | Sheel Ganatra: Liouville sectors, localizing Fukaya categories, and stop removal (Video) |

2:00-3:00 | Dmitry Vaintrob: Curves with boundary, circle actions and motivic little disks (Video) |

3:30-4:30 | Renato Vianna: Lifting Lagrangians from Donaldson divisors (Video) |

Friday March 30 (in 1015 Evans or 60 Evans, TBD)

9:30-10:30 | Martijn Kool: Virtual cobordism classes of moduli of sheaves on surfaces (Video) |

11:00-12:00 | Zhengyu Zong: On the BKMP Remodeling Conjecture for toric Calabi-Yau 3-orbifolds (Video) |

2:00-3:00 | Sven Meinhardt: Donaldson-Thomas theory and Hall algebras (Video) |

3:30-4:30 | Hansol Hong: Flop on A-side and stability conditions (Video) |

### Titles and abstracts

Jim Bryan: Donaldson-Thomas theory of the banana manifold, modular forms, and elliptic genera

**Abstract:** The Gromov-Witten/Donaldson-Thomas partition function is a fundamental invariant of a Calabi-Yau threefold which is notoriously difficult to compute. The Banana Manifold is a certain compact Calabi-Yau threefold X fibered by Abelian surfaces. We compute the Donaldson-Thomas partition function of X for fiber classes. The resulting 4 variable series has remarkable automorphic properties and a surprising relation to elliptic genera: The corresponding Gromov-Witten potential functions are all genus 2 Siegel modular forms, and after a change of variables, the partition function is the generating function for the equivariant elliptic genera of the Hilbert scheme of points in the plane.

Tobias Ekholm: Large N duality, SFT stretching, and the skein relation

**Abstract:** Large N duality relates open Gromov-Witten theory in the cotangent bundle of the 3-sphere to closed Gromov-Witten theory in the resolved conifold. We show how to understand this duality using Symplectic Field Theory stretching. We also study the dynamics of the stretching in more detail. We consider moduli spaces of holomorphic curves with boundary on the 3-sphere and show that the class of the link, given by the boundaries of the curves, in the skein module remain constant under 1-parameter deformations. This gives a direct relation between the HOMFLY polynomial and open Gromov-Witten theory.

Yoel Groman: The wrapped Fukaya category for semi-toric SYZ fibrations

**Abstract:** I will introduce the wrapped Fukaya category of a semi-toric SYZ fibration with Lagrangian section satisfying a tameness condition at infinity. Examples include the Gross fibration on the complement of an anti-canonical divisor in a toric Calabi-Yau 3-fold. I will discuss a technique of computing the Floer homology of a Lagrangian section using the closed open map. I will use this to prove homological mirror symmetry.

Hansol Hong: Flop on A-side and stability conditions

**Abstract:** Chan-Pomerleano-Ueda proved HMS between the smoothing X and the crepant resolution Y of the (local) conifold singularity. We study the operation on the Fukaya category of X which is expected to be mirror to the flop on Y. On the other hand, there is another mirror construction using special Lagrangian spheres in X and their formal deformation theory, which gives a noncommutative resolution of the conifold. I will explain how this noncommutative mirror behaves when we perform the operation on X, and give an interpretation related to stability conditions. This is a joint work with Y.-W. Fan, S.-C. Lau, and S.-T. Yau.

Hiroshi Iritani: A Fock sheaf associated with the Gromov-Witten potentials of the local P^{2}

**Abstract:**Aganagic, Bouchard and Klemm showed that the genus-g Gromov-Witten potential of the local P^{2} should be a quasi-modular function, based on Witten’s interpretation of topological string partition functions as wave functions in a quantum mechanical system. Using Givental’s quantization formalism, we show that the Gromov-Witten potentials of the local P^{2} extend to a single-valued section of a certain Fock sheaf over the global Kaehler moduli space, and recover the physics picture of Aganagic-Bouchard-Klemm and Witten. This also establishes the crepant resolution conjecture for C^{3}/Z_{3}. This is based on joint work with Tom Coates.

Martijn Kool: Virtual cobordism classes of moduli of sheaves on surfaces

**Abstract:** We conjecture a formula for the virtual cobordism classes of moduli spaces of rank 2 sheaves on surfaces S satisfying q=0 and p_{g}>0. The formula only depends on the Seiberg-Witten invariants of S, the Chern numbers of S, and two universal functions. Upon various specializations, the latter are related to Jacobi forms and Borcherds lifts thereof. Specialization to virtual Euler characteristics recovers a formula from Vafa-Witten. The common strategy to test these conjectures in examples is to write them in terms of Donaldson invariants, which can be calculated by Mochizuki’s formula. Joint with Lothar Göttsche.

Yankı Lekili: Homological mirror symmetry for K3 surfaces via moduli of A_{∞} structures

**Abstract:** Around 2010 in joint work with Perutz, as a by-product of our proof of homological mirror symmetry for the once-punctured torus, we identified moduli of elliptic curves P(4,6) with moduli of A_{∞} structures on a finite-dimensional graded algebra. Generalizations of this story that covers other moduli of curves were subsequently pursued by Polishchuk. Again, inspired by mirror symmetry (for the n-punctured torus), in joint work with Polishchuk, we exhibited Smyth’s modular compactifications of M_{1,n} as moduli of A_{∞} structures (and revealed new properties of these moduli spaces that were previously unknown). My talk will begin by surveying these results and pinpointing their common features.

In an ongoing work with Ueda, I am working on a higher dimensional generalization of this story. I will give sample examples in dimension two, identifying certain lattice polarized moduli of K3 surfaces with moduli of A_{∞} structures on certain finite-dimensional algebras. A proof of homological mirror symmetry for these K3 surfaces is on the horizon. (This gives a different proof than Seidel’s and Sheridan-Smith’s, and covers some non-overlapping examples of HMS.)

C.-C. Melissa Liu: Genus zero open-closed mirror symmetry for toric Calabi-Yau 3-orbifolds

**Abstract:** We study genus zero open-closed orbifold Gromov-Witten invariants counting holomorphic disks in a symplectic toric Calabi-Yau 3-orbifolds with boundary in Lagrangians of Aganagic-Vafa type. I will describe an open mirror theorem which expresses generating functions of orbifold disk invariants in terms of Abel-Jacobi maps of the mirror curve, based on joint work with Bohan Fang and Hsian-Hua Tseng. This generalizes the open mirror theorem for symplectic toric Calabi-Yau 3-manifolds conjectured by Aganagic-Vafa, Aganagic-Klemm-Vafa and proved in joint work with Fang. I will also relate the mirror map to period integrals of differentials of the first kind and the third kind, based on joint work with Bohan Fang and Zhengyu Zong.

Sven Meinhardt: Donaldson-Thomas theory and Hall algebras

**Abstract:** 10 years ago Kontsevich and Soibelman published a remarkable paper containing a definition of motivic Donaldson-Thomas invariants which generalizes the original definition of integral Donaldson-Thomas invariants given by Thomas into many directions. They also formulate the Integrality Conjecture which is important to specialize (quotient of) motives to integers. After a short description of their approach, I will describe the natural categorification of it. As a byproduct we obtain the Cohomological Hall (co)algebra. Under a suitable symmetry condition, this (co)algebra has a very nice structure which categorifies and proves the Integrality Conjecture of Kontsevich and Soibelman. All of this is joint work in progress with Ben Davison.

Ivan Smith: Lagrangian cobordism and tropical curves

**Abstract:** We describe constructions of and obstructions to Lagrangian cobordisms between configurations of fibres in a Lagrangian torus fibration, arising from tropical geometry and family Floer theory. The results are independent of but motivated by homological mirror symmetry, and a relation to rational equivalence on a mirror analytic space. This talk reports on joint work with Nick Sheridan.

Balázs Szendrői: Hilbert schemes of points on singular surfaces

**Abstract:** The purpose of the talk will be to explore the geometry and topology of Hilbert schemes of points of surfaces with ADE singularities, guided by the long-established results for the case of smooth surfaces. I will in particular report on a partly conjectural formula for their Euler characteristics, obtained in joint work with Adam Gyenge and Andras Nemethi. The basic representation of the affine Lie algebra corresponding to the surface singularity via the McKay correspondence, and its crystal basis theory, play an important role in our approach. The resulting function is modular, confirming a new case of S-duality.

Dmitry Tonkonog: Quantum periods theorem for Landau-Ginzburg potentials

**Abstract:** I will explain the theorem stating that the quantum period of a Fano variety equals the classical period of the Landau-Ginzburg potential of any monotone Lagrangian torus contained in it. I will also discuss applications of the theorem to symplectic topology and enumerative geometry, including the classification of potentials of tori in CP2, and a symplectic approach towards the quantum Lefschetz hyperplane theorem.

Sara Tukachinsky: Genus zero open Gromov-Witten theory via A_{∞} structures

**Abstract:** Fix a Lagrangian submanifold of a closed symplectic manifold, of arbitrary dimension. Following ideas of Fukaya-Oh-Ohta-Ono (2009) and Fukaya (2011), we endow the ring of differential forms on it with an A_{∞} structure. Cohomological conditions on the Lagrangian give rise to a canonical choice of a bounding chain, which in turn gives rise to genus-0 open Gromov-Witten invariants. Time permitting, we will discuss an open analog of the WDVV equation satisfied by such invariants. This is joint work with Jake Solomon.

Dmitry Vaintrob: Curves with boundary, circle actions and motivic little disks

**Abstract:** I will talk about a joint project with Alexandru Oancea where we show that in a homotopy-theoretic sense, the Deligne-Mumford topological operad is obtained by trivializing circle action on the Segal conformal operad of framed surfaces (i.e. holomorphic surfaces with boundary, together with analytic boundary parametrization): this is a higher-genus version of a result of Drummond-Cole. Time permitting, I will also talk about more recent and closely related work about motivic structure on framed surfaces, implying in particular Galois actions on cohomology of framed little disks (and framed surfaces), as well as a new proof of the Deligne conjecture for framed little disks.

Renato Vianna: Lifting Lagrangians from Donaldson divisors

**Abstract:** A classical construction due to Paul Biran allows to lift a Lagrangian submanifold L from a Donaldson divisor Y to a Lagrangian L’ in an ambient symplectic manifold X. Biran and Khanevsky have shown that if the minimal Chern number of Y is greater than 1, then the count of Maslov index 2 holomorphic disks with boundary on the lifted Lagrangian L’ is equivalent to the similar count of disks with boundary on L plus one extra disk. We study this enumerative geometry problem in the case when the minimal Chern number of Y is 1. This reveals several new, previously unexplored connections it has with relative closed-string Gromov-Witten theory of the pair (X,Y). We explore applications, in particular, we use that to distinghish (up to action of Symp(X)) lifts of previously known Lagrangians. In the above setting, we use the recent work of Tonkonog to recover a formula for the quantum periods of Y in terms of quantum periods of X, which is the same formula computed by Coates-Corti-Galkin-Kasprzyk using Givental-Coates’ Quantum Lefschetz Hyperplane Theorem. This is joint work with Luis Diogo, Dmitry Tonkonog and Weiwei Wu.

Zhengyu Zong: On the BKMP Remodeling Conjecture for toric Calabi-Yau 3-orbifolds

**Abstract:** The Remodeling Conjecture proposed by Bouchard-Klemm-Marino-Pasquetti (BKMP) relates the all genus open and closed Gromov-Witten invariants of a semi-projective toric Calabi-Yau 3-manifold/3-orbifold to the Eynard-Orantin invariants of its mirror curve. It is an all genus open-closed mirror symmetry for toric Calabi-Yau 3-manifolds/3-orbifolds. In this talk, I will talk about the proof of the Remodeling Conjecture in arXiv: 1604.07123 which is a joint work with Bohan Fang and Melissa Liu. The key idea of the proof is to realize both A-model and B-model higher genus potentials as quantizations of two isomorphic semi-simple Frobenius structures.

### List of participants

Noah Arbesfeld

Denis Auroux

Younghan Bae

Rodrigo Barbosa

Nathaniel Bottman

Jim Bryan

Nuno Cardoso

Elliot Cheung

Benjamin Cooper

Clara Dolfen

Tobias Ekholm

Sheel Ganatra

Michail Gerapetritis

Eugene Gorsky

Yoel Groman

Daniel Halpern-Leistner

Andrew Hanlon

Andrew Harder

Hansol Hong

Hiroshi Iritani

Dogancan Karabas

Yusuf Baris Kartal

Sheldon Katz

Ludmil Katzarkov

Yakov Kononov

Martijn Kool

Oleksandr Kravets

Ties Laarakker

Tim Large

Hwayoung Lee

Sukjoo Lee

Yanki Lekili

Oliver Leigh

Yin Li

Dahisy Lima

Yu-Shen Lin

Chiu-Chu Melissa Liu

Henry Liu

Yu-Hsiang Liu

Sven Meinhardt

Nina Morishige

Tony Pantev

Renata Picciotto

Semen Rezchikov

Sumeyra Sakalli

Ivan Smith

Changjian Su

Balazs Szendroi

Sara Tukachinsky

Dmitry Vaintrob

Renato Vianna

Jun Wang

Ivan Yakovlev

Wai-Kit Yeung

Jingyu Zhao

Zijun Zhou

Zhengyu Zong