### Mirror Symmetry and Related Topics, Miami, January 29-February 3, 2018

Speakers:

Olivia Dumitrescu

Sergei Gukov

Maxim Kontsevich

Lenny Ng

Mohammed Abouzaid

Denis Auroux

Alexander Efimov

Kenji Fukaya

Alexander Goncharov

Dmitry Kaledin

Mikhail Kapranov

Paul Seidel

Carlos Simpson

Yan Soibelman

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

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

**Important:** the **conference venue** is different from past years: the conference took place at the Marriott Courtyard Coral Gables.

The scientific program featured several short mini-courses as well as research talks. Mini-courses will be given by Olivia Dumitrescu, Sergei Gukov, Maxim Kontsevich, and Lenny Ng.

There is a nominal registration fee ($30 for students, $50 for faculty) to cover the cost of coffee breaks.

**Venue: ** The conference took place at the Marriott Courtyard Coral Gables, from Monday morning until Saturday lunchtime.

**Airport:** Miami International Airport is about 7 miles away. The most convenient way to reach the conference venue is to take a taxi. Participants expecting reimbursement: please keep all your original boarding passes since they may be needed for reimbursement.

**Conference dinner:** The conference dinner was held on Tuesday January 30 starting at 8:00pm at *Texas de Brazil* in the South Beach area of Miami Beach. Bus pick up at the hotel at 7:00 pm. The banquet fee is $50 per person.

### Schedule of talks and videos

Monday January 29

9:30-10:30 | L. Ng: Knot contact homology, string topology, and topological strings I (Video) |

11:00-12:00 | M. Kontsevich: Holomorphic Floer theory I (Video) |

2:00-3:00 | K. Fukaya: Lagrangian Floer theory in divisor complement (Video – no sound) |

3:30-4:30 | Y. Soibelman: Fukaya categories with parabolic structure, periodic monopoles and non-abelian Hodge theory in dimension one (Video) |

Tuesday January 30

9:30-10:30 | M. Kontsevich: Holomorphic Floer theory II (Video) |

11:00-12:00 | L. Ng: Knot contact homology, string topology, and topological strings II (Video) |

2:00-3:00 | M. Kontsevich: Holomorphic Floer theory III (Video) |

3:15-4:15 | D. Auroux: Towards relative HMS for hypersurfaces and their complements (Video) |

4:30-5:30 | A. Goncharov: Geometric quantisation of cluster varieties and mirror symmetry (Video) |

8:00 | Conference Dinner |

Wednesday January 31

9:30-10:30 | L. Ng: Knot contact homology, string topology, and topological strings III (Video) |

11:00-12:00 | O. Dumitrescu: Interplay between Higgs bundles, opers and TQFT I (Video) (Slides) |

Thursday February 1

9:30-10:30 | S. Gukov: What is 3d-3d correspondence? (Video) |

11:00-12:00 | M. Abouzaid: Immersed Lagrangians near SYZ singular fibres (Video) |

2:00-3:00 | O. Dumitrescu: Interplay between Higgs bundles, opers and TQFT II (Video) |

3:15-4:15 | P. Seidel: The closed-open string map for Lefschetz fibrations (Video) |

4:30-5:30 | M. Kapranov: N-spherical functors (Video) |

Friday February 2

9:30-10:30 | S. Gukov: 3d-3d correspondence in action (Video) |

11:00-12:00 | O. Dumitrescu: Interplay between Higgs bundles, opers and TQFT III (Video) |

2:00-3:00 | S. Gukov: VOA[M4] (Video) |

3:30-4:30 | C. Simpson: Towards Harder-Narasimhan filtrations for section categories over flat surfaces (Video) |

Saturday February 3

9:30-10:30 | D. Kaledin: On the notion of enhanced category (Video) |

11:00-12:00 | A. Efimov: Algebraicity and algebraizability (Video) |

### Titles and Abstracts

M. Abouzaid: Immersed Lagrangians near SYZ singular fibres

**Abstract:** I will describe a construction of immersed Lagrangians near SYZ singular fibres in three dimensions, with the property that the moduli space of simple objects recovers the mirror space. The construction generalises to higher dimensions.

D. Auroux: Towards relative HMS for hypersurfaces and their complements

**Abstract:** The goal of this talk is to propose an enhanced conjectural picture of homological mirror symmetry for hypersurfaces in toric varieties and their complements. We will start by reviewing the proposal of a mirror Landau-Ginzburg model to a hypersurface H in a toric variety V from earlier joint work with Abouzaid and Katzarkov. We will then propose conjectural statements of HMS for H, V, and the complement V-H, and various functors between the appropriate categories. (When viewing H as the B-side and the Landau-Ginzburg mirror as the A-side, this builds on joint work in progress with Abouzaid; the other direction is mostly speculative).

O. Dumitrescu: Interplay between Higgs bundles, opers and TQFT

**Abstract:** The rainbow is one of the most beautiful phenomena in nature. It has inspired art, mythology, and has been a pleasure and challenge to the mathematical physicists for centuries. You might have wondered what awaited you if you went over the rainbow. Is the world on the other side of the rainbow the same as what we know? Sir George Biddel Airy discovered the rainbow integral and explained the classical analysis of rainbows. 150 years later, Kontsevich found that the same formula determined the multiplication table of cohomology classes on the compactified moduli spaces of Riemann surfaces. These stories are a simple example of a mathematical theory of “quantum curves.” I will present a general framework of quantum curves and I will relate it to a conjecture of Davide Gaiotto (2014) giving a particular construction of opers from Higgs bundles in one Hitchin component. The conjecture has been solved by a joint work of Fredrickson, Kydonakis, Mazzeo, Mulase, Neitzke and myself (2016). In the first talk, I will present a holomorphic description of the limiting oper, and its geometry.

The count of lattice points of the character variety ie representations from the fundamental group a punctured Riemann Surface to a finite group, G, is a 2D TQFT for the center of the group algebra of G. I will present a new axiomatic formulation of a 2D TQFT whose formalism is based on the edge-contraction operations on graphs drawn on a Riemann surface (cellular graphs). Assigning to each cellular graph an element of a symmetric tensor algebra of a Frobenius algebra then the edge-contraction axioms make this assignment graph independent, and that it is equivalent to the TQFT corresponding to the Frobenius algebra. The edge-contraction operations are used in enumeration problems, such as Hurwitz numbers and lattice points on the moduli space of curves. Finally, I will present a generalization of this formalism to Cohomological Field Thories. This is work in progress with Motohico Mulase.

A. Efimov: Algebraicity and algebraizability

**Abstract:** We will recall the notion of algebraizability of perfect complexes on a formal neighborhood of a closed subscheme: these are perfect complexes which are generated (via cones, shifts and direct sumands) by the restrictions of perfect complexes on the ambient scheme. We will state several results stating that in some situations algebraizability is equivalent to algebraicity of certain power series. A special case is covered in the recent preprint arXiv:1711.00756.

K. Fukaya: Lagrangian Floer theory in divisor complement

**Abstract:** This is a work in progress with A. Daemi. I will explain to develop Lagrangian Floer theory for certain class of noncompact symplectic manifold which is not necessarily convex or concave at infinity, that is a complement of divisor. We explain various idea for the construction and also possible generalization and conjectures, for example relation to (partially) wrapped Floer theory.

A. Goncharov: Geometric quantisation of cluster varieties and mirror symmetry

**Abstract:** Cluster varieties come in pairs (X,A), where A is a cluster variety and X is the dual cluster Poisson variety. The space X has a q-deformation, and the space A comes with a 2-form of motivic nature. We use the motivic avatar of the 2-form to define geometric quantisation of cluster varieties – based on joint work with V.Fock. We discuss how it fits into conjectural mirror type symmetry between the q-deformed X-space (B-model) and the A-space with the 2-form serving as the B-field.

S. Gukov: 1. What is 3d-3d correspondence?

**Abstract:** What do dilogarithms, algebraic curves, 3-manifold invariants and BPS invariants all have in common? A unifying framework for these, called 3d-3d correspondence, allows to express different invariants of a given 3-manifold as different partition functions of a specific 3d theory associated to that 3-manifold. The main goal of this first lecture will be to explain the “Big Picture” and connections with other subjects.

2. 3d-3d correspondence in action

**Abstract:** In this lecture we shall focus on concrete examples of 3-manifold invariants, old and new, computed via 3d-3d correspondence. In particular, we shall see how the Heegaard Floer homology and WRT invariants of a 3-manifold M3 both reside in the same quiver gauge theory T[M3].

3. VOA[M4]

**Abstract:** While the first two talks are supposed to be introductory, this one will take us closer to the edge of the current research (joint with Boris Feigin, which builds on earlier work with Gadde, Putrov, and Dedushenko). Using somewhat similar ideas to those in the first two lectures, one can associate a vertex operator algebra (a.k.a. 2d CFT or a chiral algebra) to a smooth 4-manifold. We will discuss various constructions of this algebra for certain classes of 4-manifolds and non-trivial equivalences of VOAs that correspond to Kirby moves.

D. Kaledin: On the notion of enhanced category

**Abstract:** It is well-known that localizing a category with respect to a class of morphisms gives an “enhanced” category that is, at the very least, enriched in homotopy types. Several precise definitions of the full structure are available in the literature, e.g. complete Segal spaces of Rezk, or infinity-categories a la Joyal as developed by Lurie. All the definitions are equivalent in some sense, but they all do rely on a specific choice of a model. I am going to discuss how to re-package the same data in a model-independent way.

M. Kapranov: N-spherical functors

**Abstract:** The concept of a spherical functor involves the cone of the (co)unit of the adjunction. If we take into account iterated (second, third etc.) adjoints, we can construct two natural complexes of functors. Requiring the total functors of these complexes to be equivalences defines a concept of N-spherical functors, which reduces to the usual one (spherical functors) for N=2. Such functors are related to 2N-periodic chains of iterated orthogonals. Joint work in progress with T. Dyckerhoff and V. Schechtman.

M. Kontsevich: Holomorphic Floer theory

**Abstract:** I will talk on various aspects of the joint project with Y. Soibelman on Fukaya quantization of complex symplectic manifolds. The first lecture will be devoted to the generalized Riemann-Hilbert correspondence. Roughly speaking, the derived category of holonomic modules over a non-perturbative quantization of a complex algebraic symplectic manifold should be equivalent to an appropriate version of the Fukaya category. In the second lecture I’ll describe the gluing of the classical limit, and the construction of the twistor family of categories (examples will be given in Soibelman’s talk). The third lecture will be devoted to semi-infinite topology of path integrals, with the application to resurgence equations with small parameter (as in previous lectures), and to a more general set-up, including heat kernels and Ecalle-Voronin theory.

L. Ng: Knot contact homology, string topology, and topological strings

**Abstract:** Knot contact homology is a package of knot invariants obtained by studying holomorphic curves in the cotangent bundle with boundary on the knot conormal. In this minicourse, we will explore some recent developments tying knot contact homology to string topology (and then to partially wrapped Floer homology and loop spaces) and to topological string theory (and then to mirror symmetry in the spirit of Aganagic-Vafa).

The first lecture will introduce knot contact homology and some properties, including connections to representations of the knot group. In the second lecture, I will discuss an interpretation of knot contact homology via string topology and how one can apply this to prove that the knot conormal is a complete knot invariant. In the third lecture, I will talk about the augmentation variety and how it relates to topological strings on the resolved conifold via Gromov-Witten disk potentials, and a conjectural way to quantize the augmentation variety to recover the recurrence relation for colored HOMFLY polynomials (this last part is work in progress with Tobias Ekholm).

P. Seidel: The closed-open string map for Lefschetz fibrations

**Abstract:** The “closed string” part of the Floer theory of Lefschetz fibrations has been the object of mild puzzlement. The geometric counterpart of Hochschild homology, and the open-closed string map, are straightforward to understand. Since the associated categories are not Calabi-Yau, there is a different geometric object corresponding to Hochschild cohomology. In this talk, I will explain this object and its closed-open string map. These constructions will require us to revise the definition of the relevant Fukaya categories as well.

C. Simpson: Towards Harder-Narasimhan filtrations for section categories over flat surfaces

**Abstract:** We report on our joint work with F. Haiden, L. Katzarkov, and P. Pandit on the program of extending the Bridgeland-Smith construction of stability conditions to the case of SL(3) spectral curves. We consider Fukaya-Seidel categories of sections over graph Lagrangians with coefficients in a constant category, in our case of type A2-CY2, on a contractible flat Riemann surface. Sections of the coefficient category, over graphs (with linear edges) in the base Riemann surface, involve putting triangles at the threefold vertices of the underlying graph. We discuss some ideas for defining the deformed section category, and then for defining a stability condition by deformation of an object to minimize the “slope and size of the highest slope piece”. We sketch our current view of this process and discuss some of the remaining unsolved problems.

Y. Soibelman: Fukaya categories with parabolic structure, periodic monopoles and non-abelian Hodge theory in dimension one.

**Abstract:** Classically, non-abelian Hodge theory (NAHT for short) of Simpson establishes, in particular, an equivalence between the category of locally constant sheaves on a projective curve (equivalently, bundles with connections) and the category of semistable Higgs bundles of degree zero on the same curve. In the case of curves with punctures one has a similar result as long as one utilizes the notion of parabolic structure at the punctures.

I am going to explain how to re-interpret these kind of results so that they will be applicable not only to differential equations, but also to q-difference and elliptic difference equations. In the latter two cases the role of harmonic objects (which in conventional NAHT interpolate between Higgs bundles and local systems) is played by doubly and triply periodic monopoles with singularities. Altogether the corresponding categories can be combined into a “twistor family of categories” where the general fiber can be interpreted in terms of a partially wrapped Fukaya category with parabolic structure of a complex symplectic surface.

The talk is based on a part of the project “Holomorphic Floer theory”, joint with Maxim Kontsevich.

### List of participants

Mohammed Abouzaid

Denis Auroux

Rodrigo Barbosa

Enrique Becerra

Pierrick Bousseau

Catherine Cannizzo

Nuno Cardoso

Juan Carlos Castro Contreras

Marco Castronovo

Dongwook Choa

Daniel Chupin

Patrick Clarke

Colin Diemer

Charles Doran

Olivia Dumitrescu

Alexander Efimov

Tigran Eritsyan

Yu-Wei Fan

Kenji Fukaya

Benjamin Gammage

Yuan Gao

Elizabeth Gasparim

Michail Gerapetritis

Alexander Goncharov

Lino Grama

Roberta Guadagni

Sergei Gukov

Andrew Hanlon

Andrew Harder

Jeff Hicks

Justin Hilburn

Hansol Hong

Jesse Huang

Matei Ionita

Maxim Jeffs

Dmitry Kaledin

Mikhail Kapranov

Yusuf Baris Kartal

Ilya Karzhemanov

Maxim Kontsevich

Oleksandr Kravets

Priya Kshirsagar

Nikon Kurnosov

Jia-Choon Lee

Sangwook Lee

Sukjoo Lee

Valery Lunts

Ernesto Lupercio

Jie Min

Benedict Morrissey

Jacob Mostovoy

Lenhard Ng

Ignacio Otero Rubio

Yu Pan

Tony Pantev

Dmitrii Pedchenko

Daria Poliakova

Yingdi Qin

Semen Rezchikov

Larry Richardson

Manuel Rivera

Jan-David Salchow

Renato SalmerÃ³n

Paul Seidel

Hyung-Seok Shin

Carlos Simpson

Yan Soibelman

Leonardo Soriani

Ted Spaide

Yuhan Sun

Zack Sylvan

Dmitry Tonkonog

Renato Vianna

Abigail Ward

Ivan Yakovlev

Dingyu Yang

Zhongyi Zhang