Simons Collaboration on Homological Mirror Symmetry

Preliminary schedule

Monday March 21 (in 60 Evans)

9:45-10:45	T. Ekholm: I. Chern-Simons theory, topological string, and knot contact homology    (Slides)    (Video)
11:15-12:15	P. Seidel: Second order ODEs and the mirror map    (Video)
1:45-2:45	Y. Lekili: Generating the Fukaya categories of Hamiltonian G-manifolds    (Video)
3:10-4:10	J. Pascaleff: Symplectic geometry of scattering diagrams for log CY surfaces    (Slides)    (Video)
4:15-5:15	S.-C. Lau: SYZ for affine A-type local Calabi-Yau manifolds    (Slides)    (Video)

Tuesday March 22 (in 60 Evans)

9:45-10:45	T. Ekholm: II. The augmentation variety and the Gromov-Witten disk potential    (Slides)    (Video)
11:15-12:15	C.-C. Liu: I. Open Gromov-Witten invariants and the large N duality    (Video)
1:45-2:45	K. Fukaya: Wall-crossing and mirror symmetry    (Video)
3:10-4:10	D. Tonkonog: The wall-crossing formula for mutations of Lagrangian tori    (Video)
4:15-5:15	M. Akhtar: Mirror symmetry and the classification of Fano varieties    (Video)

Wednesday March 23 (in 60 Evans)

9:45-10:45	T. Ekholm: III. Higher genus curves, topological recursion, and other 3-manifolds    (Slides)    (Video)
11:15-12:15	C.-C. Liu: II. Open-closed mirror symmetry and wall-crossing    (Video)
1:45-2:45	D. Nadler: Wrapped microlocal sheaves on pairs of pants    (Video)
3:10-4:10	A. Keating: Homological mirror symmetry for singularities of type Tpqr.    (Video)
4:15-5:15	free time for discussion

Thursday March 24 (in 60 Evans)

9:45-10:45	P. Hacking: Mirror symmetry for cluster varieties I    (Video)
11:15‑12:15	M. Abouzaid: Family Floer homology I    (Video)
1:45-2:45	K. Fukaya: Lagrangian surgery and bifurcation of holomorphic disks of deformation of hyper Kahler twist of type I singular fiber in elliptic K3 surface    (Video)
3:10-4:10	H. Williams: Skeletal surgery and the combinatorics of exact Lagrangian surfaces    (Video)
4:15-5:15	R. Vianna: Infinitely many monotone Lagrangian tori in del Pezzo surfaces    (Slides)    (Video)

Friday March 25 (in 1015 Evans)

9:45-10:45	P. Hacking: Mirror symmetry for cluster varieties II    (Video)
11:15-12:15	M. Abouzaid: Family Floer homology II    (Video)
1:45-2:45	M. Gross: Mirror symmetry    (Video)
3:10-4:10	F. Haiden: Spaces of stability conditions of Fukaya categories of surfaces    (Video)
4:15-5:15	free time for discussion

Titles and abstracts

M. Abouzaid: Family Floer homology (TBA)

M. Akhtar: Mirror symmetry and the classification of Fano varieties

Abstract: The classification of Fano varieties is an important long-standing problem in algebraic geometry. A new approach to this problem via mirror symmetry was recently proposed by Coates-Corti-Galkin-Golyshev-Kasprzyk. Their philosophy was that Fano varieties can be classified by studying their Laurent polynomial mirrors. This talk will survey the results of a collaborative effort to apply this philosophy to the classification of Fano orbifold surfaces. We will describe a conjectural picture which suggests that classifying suitable deformation classes of certain Fano orbifold surfaces is equivalent to classifying Fano lattice polygons up to an appropriate notion of equivalence. Central to this framework is the notion of mirror duality (between a Fano orbifold surface and a Laurent polynomial) and the closely related operations of algebraic and combinatorial mutations. We will also discuss how combinatorial mutations allow us to find mirror dual Laurent polynomials in practice and will give experimental evidence supporting our conjectures.

T. Ekholm: I. Chern-Simons theory, topological string, and knot contact homology

Abstract: We present background material. Chern-Simons theory, topological string, and conifold transition will be discussed from the physics perspective. We also introduce knot contact homology which is a Floer homological theory called Legendrian contact homology applied to the unit conormal of a knot or link and describe how to compute it combinatorially.

II. The augmentation variety and the Gromov-Witten disk potential

Abstract: Knot contact homology associates a differential graded algebra over the complex group ring of the second relative homology group of the unit conormal inside the unit cotangent bundle of the three sphere. The coefficients can be viewed as the ring of functions on a complex 3-torus and the locus where the algebra admits a chain map is called the augmentation variety. We relate the augmentation variety of knot contact homology to the Gromov-Witten disk potential of a version of the Lagrangian conormal in the resolved conifold.

III. Higher genus curves, topological recursion, and other 3-manifolds

Abstract: We discuss work in progress that aims at generalizing the relation between knot contact homology and Gromov-Witten theory described above. We consider Gromov-Wiiten theory of higher genus curves. This relates to a version of Legendrian symplectic field theory that implicitly determines the curve count. More precisely, the equation for the augmentation variety lifts to an operator equation that determines a D-module related to the recursion relation for the colored HOMFLY polynomial. We furthermore use the SFT equation together with the sheaves of linearized contact homologies over the augmentation variety to demonstrate a version of topological recursion that determines counts of higher genus curves inductively form counts of lower genus curves. Finally, we discuss the case of other 3-manifolds (quotients of the three sphere) for which the counterpart of conifold transition is known. Here, the local Calabi-Yau manifold after transition has one Kähler parameters for each free homotopy class of loops. The corresponding knot contact homology is in this case an algebra over the orbit contact homology algebra of the unit cotangent bundle which in degree zero has one generator for each free homotopy class of loops as well.

K. Fukaya: Lagrangian surgery and bifurcation of holomorphic disks of deformation of hyper Kahler twist of type I singular fiber in elliptic K3 surface

M. Gross: Mirror symmetry

Abstract: I will introduce a variant of log Gromov-Witten invariants called “punctured invariants” developed in joint work with Abramovich, Chen and Siebert. I will then explain how to use these invariants, in joint work with Siebert, to give a general construction of mirrors by building the coordinate ring of the mirror explicitly in terms of these invariants. The ring is easy to define, but proof of associativity requires delicate issues of gluing stable log maps. This construction can be viewed as a natural generalization of the mirror construction of myself, Hacking and Keel for log Calabi-Yau surfaces.

P. Hacking: Mirror symmetry for cluster varieties

Abstract: We will first explain a geometric description of cluster varieties, as blowups of toric varieties carrying a holomorphic symplectic form with log poles at infinity. Then we will describe mirror symmetry for cluster varieties using the scattering diagram encoding counts of holomorphic discs ending on SYZ fibers. This is joint work with Gross, Keel, and Kontsevich.

F. Haiden: Spaces of stability conditions of Fukaya categories of surfaces

Abstract: I will report on joint work with L. Katzarkov and M. Kontsevich (arXiv:1409.8611) in which we identify spaces of half-translation surfaces, equivalently Riemann surfaces with quadratic differential, with spaces of stability structures on Fukaya-type categories of punctured surfaces. The result is non-trivial already in the simplest examples, which involve the representation theory of quivers and an old result of Nevanlinna. Time permitting, I will discuss some analogies and future directions.

A. Keating: Homological mirror symmetry for singularities of type Tpqr

Abstract: We present some homological mirror symmetry statements for the singularities of type $T_{p,q,r}$. Loosely, these are one level of complexity up from so-called “simple” singularities, of types A, D and E. We will consider some symplectic invariants of the real four-dimensional Milnor fibres of these singularities, and explain how they correspond to coherent sheaves on certain blow-ups of the projective space $Bbb{P}^2$, as suggested notably by Gross-Hacking-Keel. We hope to emphasize how the relations between different “flavours” of invariants (e.g., versions of the Fukaya category) match up on both sides.

S.-C. Lau: SYZ for affine A-type local Calabi-Yau manifolds

Abstract: Affine A-type local Calabi-Yau manifolds are building blocks for compact K3 surfaces and Shoen’s Calabi-Yau threefolds. They admit a natural Lagrangian fibration whose open Gromov-Witten potential has rich arithmetic content. Their SYZ mirrors are conic fibrations over Abelian varieties, which can be constructed and computed using the technique of wall-crossing. This is a joint work with Atsushi Kanazawa.

Y. Lekili: Generating the Fukaya categories of Hamiltonian G-manifolds

Abstract: Let G be a compact Lie group and k be a field of characteristic p≥0 such that H*(G) does not have p-torsion. We show that a free Lagrangian orbit of a Hamiltonian G-action on a compact, monotone, symplectic manifold X split-generates an idempotent summand of the monotone Fukaya category over k if and only if it represents a non-zero object of that summand. Our result is based on: an explicit understanding of the wrapped Fukaya category of T*G through Koszul twisted complexes involving the zero-section and a cotangent fibre; and a functor canonically associated to the Hamiltonian G-action on X. Several examples can be studied in a uniform manner including toric Fano varieties and certain Grassmannians. Time permitting, I will also discuss how our result leads to examples of symplectic (possibly Kähler) manifolds with a non-formal A_∞ structure on their quantum cohomology.
This is joint work with Jonny Evans (London).

C.-C. Liu: I. Open Gromov-Witten invariants and the large N duality

Abstract: The large N duality relates Gromov-Witten theory on the resolved conifold to the Chern-Simons theory on the 3-sphere. Under the large N duality, open Gromov-Witten invariants of the resolved conifold correspond to colored HOMFLY polynomial of knots and links. In this talk, I will discuss mathematical aspects of the large N duality and an open version of the Gopakumar-Vafa formula.

II. Open-closed mirror symmetry and wall-crossing

Abstract: I will describe open-closed mirror symmetry for a symplectic toric Calabi-Yau threefold in terms of its three mirror families: the Hori-Vafa mirror, the equivariant Landau-Ginzburg mirror, and the mirror curve. The main examples in the talk will be the resolved conifold and local $Bbb{P}^2$. I will also describe wall-crossing phenomena in the open-closed string moduli.

D. Nadler: Wrapped microlocal sheaves on pairs of pants

Abstract: Inspired by partially wrapped Fukaya categories, we will introduce a parallel notion of wrapped microlocal sheaves. We will explain some of their fundamental properties, notably their duality with traditional microlocal sheaves, in analogy with expected dualities for Fukaya categories. Our main application will be the calculation of wrapped microlocal sheaves on the n-dimensional pair of pants, confirming their mirror symmetry with the Landau-Ginzburg B-model with superpotential $W = x_1… x_{n+2}$.

J. Pascaleff: Symplectic geometry of scattering diagrams for log CY surfaces

Abstract: Gross, Hacking, and Keel constructed mirrors to log Calabi-Yau surfaces by writing down a scattering diagram. On the other hand, according to Fukaya, a mirror to a symplectic manifold with a Lagrangian torus fibration can be constructed by gluing together spaces of local systems on the Lagrangian tori. I will describe how these two approaches are related by giving a symplectic argument for how the GHK scattering diagram controls the pseudoisotopy of local systems supported on certain Lagrangian tori. Some applications to symplectic cohomology will also be discussed.

P. Seidel: Second order ODEs and the mirror map

Abstract: The complements of anticanonical divisors carry holomorphic volume forms. We will consider the dependence of these on the divisor (in a pencil), and the corresponding mirror question. (This is essentially the same talk as in Miami in January, but disguised by changing the title.)

D. Tonkonog: The wall-crossing formula for mutations of Lagrangian tori

Abstract: Given a monotone Lagrangian two-torus and a Lagrangian disk attached to it, there is a geometric mutation procedure that constructs a new monotone Lagrangian torus out of this configuration. For example, Vianna’s collections of tori in del Pezzo surfaces are obtained from a single torus by a sequence of such mutations. I will prove the wall-crossing formula, predicted by Kontsevich-Soibelman and Auroux, that states how the enumerative geometry of holomorphic Maslov index 2 disks changes under this mutation. The proof is performed by reducing the problem to a local computation, due to Seidel, in a certain model exact manifold. This is joint work with James Pascaleff.

R. Vianna: Infinitely many monotone Lagrangian tori in del Pezzo surfaces

Abstract: In 2014, we showed how the Chekanov torus arises as a fiber of an almost toric fibration and how this perspective enable us to describe an infinite range of monotone Lagrangian tori. More precisely, for any Markov triple of integers $(a,b,c)$ — satisfying $a^2 + b^2 + c^2 = 3abc$ — we get a monotone Lagrangian torus $T(a^2,b^2,c^2)$ in $Bbb{CP}^2$. Using neck-stretching techniques we are able to get enough information on the count of Maslov index 2 pseudo-holomorphic disks that allow us to show that for $(d,e,f)$ a Markov triple distinct from $(a,b,c)$, $T(d^2,e^2,f^2)$ is not Hamiltonian isotopic to $T(a^2,b^2,c^2)$.
In this talk we will describe how to get almost toric fibrations for all del Pezzo surfaces (endowed with monotone symplectic form), in particular for $Bbb{CP}^2 # koverline{Bbb{CP}}{}^2$ for $4 le k le 8$, where there is no toric fibrations. From there, we will be able to construct infinitely many monotone Lagrangian tori. We are able to prove that these tori give rise to infinitely many symplectomorphism classes in $Bbb{CP}^2 # koverline{Bbb{CP}}{}^2$ for $0 le k le 8$, $k ne 2$, and in $Bbb{P}^1timesBbb{P}^1$. We conjecture the same to hold for $Bbb{CP}^2 # 2overline{Bbb{CP}}{}^2$, but we can’t reach a proof with our technique.
Some Markov like equations appear. These equations also appear in the work of Hacking-Prokhorov regarding degeneration of surfaces to weighted projective spaces and in the work of Karpov-Nogin regarding 3-block collections of exceptional sheaves in del Pezzo surfaces.
Time permitting, we discuss how we can get infinitely many exact tori in $X setminus Sigma$ the complement of an anti-canonical divisor $Sigma$, and why they remain in distinct Hamiltonian isotopy classes even after attaching the positive half of the symplectization to the boundary. In particular, we get infinitely many exact tori on the Milnor fibers $mathcal{T}_{3,3,3}, mathcal{T}_{2,4,4}, mathcal{T}_{2,3,6}$, described by Keating. The above discussion is based on ideas developed by Shende-Treumann-Williams.

H. Williams: Skeletal surgery and the combinatorics of exact Lagrangian surfaces

Abstract: We study Weinstein 4-manifolds formed from cotangent bundles of surfaces by handle attachment. We describe an iterative mutation operation on their Lagrangian skeleta and use it to study exact Lagrangians obtained by iterated surgery on the original zero section. Relationships between these Lagrangians are reflected in the Fukaya category, or the category of microlocal sheaves, by structures from cluster algebra. In particular, the problem of producing and distinguishing such Lagrangians maps to a combination of combinatorial-geometric questions about curve configurations on surfaces and algebraic questions about mutation graphs of cluster algebras. This is joint work with Vivek Shende and David Treumann. In the other direction, we discuss applications to the cluster theory of character varieties and integrable systems (also joint with Eric Zaslow).

List of participants

Mohammed Abouzaid
Mohammad Akhtar
Denis Auroux
Catherine Cannizzo
Ricardo Castano-Bernard
ManWai Cheung
Colin Diemer
Luis Diogo
Ron Donagi
Philippe Durand
Tobias Ekholm
Kenji Fukaya
Benjamin Gammage
Sheel Ganatra
Honghao Gao
Yuan Gao
Mark Gross
Roberta Guadagni
Paul Hacking
Fabian Haiden
Andrew Hanlon
Andrew Harder
Jeff Hicks
Dogancan Karabas

Yusuf Baris Kartal
Ludmil Katzarkov
Ailsa Keating
Tyler Kelly
Gabriel Kerr
Yoosik Kim
Oleksandr Kravets
Enka Lakuriqi
Siu Cheong Lau
Oleg Lazarev
Heather Lee
Yanki Lekili
Jun Li
Xiaobin Li
Dahisy Lima
Chiu-Chu Melissa Liu
Andrew Macpherson
Matthew Mahowald
Cheuk Yu Mak
Svetlana Makarova
Mike Miller
Jie Min
Lang Mou
Mihai Munteanu

John Murray
David Nadler
Khoa Nguyen
Liya Ouyang
Tony Pantev
James Pascaleff
Alexander Pavlov
Daria Poliakova
Helge Ruddat
Jan-David Salchow
Paul Seidel
Ivan Smith
Ruifang Song
Haoyu Sun
Tom Sutherland
Hiro Tanaka
Dmitry Tonkonog
Umut Varolgunes
Sara Venkatesh
Renato Vianna
Juan Jose Villarreal
Harold Williams
Weiwei Wu
Jingyu Zhao