Summer school 2016 of the IMJ-PRG
Paris, June 27 - July 8, 2016
Symplectic topology, sheaves and mirror symmetry.
Notes by Kevin Sackel: lecture 1, lecture 2, complements.
The basic idea of family Floer cohomology in the non-exact setting, following Fukaya, is that the Floer cohomology groups of family of Lagrangian submanifolds with non-trivial flux form a (coherent analytic) sheaf over an underlying parameter space. I will explain a different point of view wherein there is an enlargement of the Fukaya category which admits as additional objects certain analytic sheaves on this parameter space. The focus will be on a situation where technical difficulties are avoided by excluding holomorphic disc bubbling.
Complementary lecture by Jingyu Zhao: Convergence of the differential in family Floer homology
Following Mohammed Abouzaid's lectures, I will first recall the differential for the family Floer complex between a tautologically unobstructed Lagrangian and fibres of a Lagrangian torus fibration of the symplectic manifold.
Using the works of Fukaya and Groman-Solomon, we explain that the family Floer differential constructed above can be viewed as a convergent function on an affinoid domain of the rigid analytic mirror.
Notes by Kevin Sackel: lecture 1, lecture 2, lecture 3, complements 1, complements 2.
The goal of this lecture series will be to illustrate homological mirror symmetry by focusing on some simple examples. The basic example will be the cylinder $T^*S^1 = \mathbb{C}^*$; we will then discuss what happens when this example is modified by adding or removing points. Along the way we will encounter wrapped and partially wrapped Fukaya categories of these spaces and their mirrors. While the basic examples may seem elementary, they illustrate general features of homological mirror symmetry and provide test cases for work in progress on hypersurfaces in toric varieties.
Complementary lectures by Sheel Ganatra
In these complementary theory sessions, we will study some structural aspects of Fukaya categories that are pertinent to proving homological mirror symmetry in some of the situations arising in Denis Auroux’s lectures. Potential topics include: (a) Fukaya categories of Landau-Ginzburg models (LG) $(E,W)$ (as a special type of partially wrapped Fukaya category) and their relationship to ordinary and wrapped Fukaya categories, (b) mechanisms for finding (split-)generators for Fukaya categories, (c) miscellaneous topics and/or examples.
Notes by Manuel Rivera: lectures 1,2,3
1. We will start with a discussion of linear algebra in the higher categorical setting: derived, $DG-$ and $A_\infty$ categories. Stable $k$-linear $\infty$-categories. Derived symplectic linear algebra.
2. Shifted symplectic structures on derived stacks, after Pantev-Toën-Vaquié-Vezzosi. Examples of symplectic and Lagrangian derived stacks. Topological field theories from shifted symplectic structures.
3. We will discuss some recent results of Brav-Dyckerhoff about topological Fukaya categories: Calabi-Yau structures on $DG-$categories and shifted symplectic structures on topological Fukaya categories of surfaces.
Complementary lectures by Grégory Ginot
Notes by Fabio Gironella: lectures 1,2,3
1. Homological algebra and sheaves
2. Micro-support of sheaves, involutivity, Morse theory
3. Operations, $\mathbb{R}$-constructible sheaves
Complementary lectures by Pierre Schapira:
1. Complements and examples
2. Examples of microsupports, links with D-modules
Notes (by the speaker)
Notes by Vincent Humilière: lectures 1,2,3 + complements
1. $\mu hom$, simple and pure sheaves, the Kashiwara-Schapira stack
2. Quantization of exact Lagrangians in cotangent bundle (I)
3. Quantization of exact Lagrangians in cotangent bundle (II)
We will associate a sheaf to a given compact exact Lagrangian submanifold of a cotangent bundle and see how to deduce that this Lagrangian has the homotopy type of the base.
Complementary lecture by Nicolas Vichery:
Examples of quantization of Lagrangian submanifolds, quantization of Hamiltonian isotopies
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Notes by Ailsa Keating: lecture 1
Notes by Denis Auroux: lecture 2
Notes (by Sheridan), solutions to exercises (by Maydanskiy)
Notes taken by Dingyu Yang: lecture 1, lecture 2, lecture 3, complements 1, complements 2.
Notes by Amiel Peiffer-Smadja: lectures 1,2,3
1. Lagrangian Floer cohomology:
We will introduce the basics of Lagrangian Floer cohomology, using the Arnold conjecture as motivation. This will involve discussing the Novikov field, transversality, compactness and gluing.
2. Product structures:
We will introduce the Fukaya category, and give example computations. We will discuss obstructions, and how to deal with them.
3. Triangulated structure:
This talk will focus on the triangulated structure of the derived Fukaya category. Examples will include the relationship between Lagrangian surgery and cones, and Seidel's long exact sequence for a Dehn twist.
Complementary lectures by Maksim Maydanskiy:
In these supplementary sessions we will take up, in the form of exercise solutions, some additional material on Lagrangian Floer cohomology and Fukaya category as described the first two talks of N. Sheridan. With audience participation, we will select for discussion a few topics from the following: interpretation of holomorphic strips as gradient flowlines of the action functional, explicit examples of Gromov convergence, gradings and computations of Floer cohomology in (real) dimension 2, relations of holomorphic discs to displacement energy, identification of Morse-Witten and Floer complexes (in the cotangent bundle case), Stasheff associahedra and moduli of holomorphic discs.
We apply microlocal category methods to a contact non-squeezing conjecture proposed by Eliashberg, Kim and Polterovich. Let $V_R$ be the product of the $2n$-dimensional open ball of radius $R$ and the circle with a circumference of one. Let $R>r$ be some positive numbers. If $\pi r^2$ is no less than 1, then it is impossible to squeeze $V_R$ into $V_r$ via compactly supported contact isotopies.
When we consider Hochschild or cyclic cohomology of the $A_\infty$ category $Fuk(X)$ for a symplectic manifold $X$, it has various structures such as L infinity structure, involutive bi Lie infinity structure. In case $X$ is a compact symplectic manifold such structure becomes formal (that is all the operations are trivial) in various situations. In this talk I want to explain such phenomenon as well as the construction of those strutures.
The Gromov-Witten invariants play a central role in the mirror symmetry conjecture which in turn gives predictions for these invariants. Many such predictions for closed Gromov-Witten invariants have been established mathematically. Similar predictions exist for open and real Gromov-Witten invariants and I will discuss some of the difficulties related to understanding the open invariants and recent advances in the real case.
I will sketch how to create a generating family quadratic at infinity for any exact Lagrangian in $\mathbb{R}^{2n}$ equal to standard $\mathbb{R}^n$ outside a compact set. I will then if time permits also discus some consequences of this.
Since its discovery in the 60s, Yang-Baxter equation (YBE) has been studied extensively as the master equation in integrable models in statistical mechanics and quantum field theory. In 2000, Polishchuk discovered a connection between the solutions to Yang-Baxter equations (classical, associative and quantum) and the Massey products in a Calabi-Yau 1-category, and using this he was able to construct geometrically some of the trigonometric solutions of the YBE coming from simple vector bundles on cycles of projective lines. We first prove a homological mirror symmetry statement, hence see these trigonometric solutions to YBE via the Fukaya category of punctured tori. Next, we consider Fukaya categories of higher genus square-tiled surfaces to give a geometric construction of all the trigonometric solutions to associative Yang-Baxter equation parametrized by the associative analogue of the Belavin-Drinfeld data. This is based on joint work with Polishchuk.
I will survey recent work devoted to singularities of Lagrangian skeleta, with a focus on applications to mirror symmetry.
We consider the symplectic cohomology of the total space of a Lefschetz fibration. Under suitable assumptions, this can be equipped with a connection (an operator of differentiation with respect to the Novikov variable). We will show that with respect to this operator, the Borman-Sheridan class satisfies a nonlinear first order differential equation (a Riccati equation).
I will sketch an argument that the wrapped Fukaya category localizes to a cosheaf of categories on a Lagrangian skeleton, locally modeled on the cosheaf dual to the Kashiwara-Schapira sheaf. This is joint work with Sheel Ganatra and John Pardon.
Recent work of Haiden, Katzarkov and Kontsevich leads to a classification of objects in derived (wrapped) Fukaya categories of punctured surfaces. We will describe an attempt to partially extend this classification to closed surfaces of higher genus, and discuss possible applications of such an extension. This is joint speculation-in-progress with Denis Auroux.
I am planning to overview my preprint arXiv:1511.08961 'On the microlocal category' where a $dg$-category over $\mathbb{Q}$ is associated to a compact symplectic manifold whose symplectic form has integral periods. The properties of this category are similar to those of the Fukaya category.
Let $L$ be an exact Lagrangian submanifold of a cotangent bundle $T^*M$. If a topological obstruction vanishes, a local system of $R$-modules on $L$ determines a constructible sheaf of $R$-modules on $M$ -- this is the Nadler-Zaslow construction. I will discuss a variant of this construction that avoids Floer theory, and that allows $R$ to be a ring spectrum. The talk is based on joint work with Xin Jin.
I will outline a construction of a microlocal category of a symplectic manifold using déformation quantization rather than sheaf theory. This is supposed to be an analog of the De Thames definition of cohomology of manifolds as opposed to the sheaf theoretical definition. I will concentrate on the local computation that amounts to a version of the Poincaré Lemma (it is also some sort of a stationary phase statement). I will address the local to global gluing question only briefly.
Beginning with a cubic, planar graph, I will define a Legendrian
surface in the cosphere bundle of three-space, equivalently the
first jet bundle of the two-sphere. Its
wavefront projects generically two-to-one onto the base two-sphere,
but is one-to-one over the graph. The Lagrangian defines a
singular support condition for both a category of constructible
sheaves and for a Fukaya category. I will describe the moduli space
of objects of this category, and study two applications.
First, the moduli space can be defined over a finite field,
in which case the number of points can be related to the chromatic polynomial of
the dual graph. Using this observation, I will show that none of the
Lagrangian surfaces admits a smooth, exact Lagrangian filling in six-space.
Instead, I will describe the construction of fillings which are not exact, and
in fact obstructed. Second, the moduli space sits as a Lagrangian submanifold of
a symplectic period domain. It has a cover which is exact in a cotangent.
Generalizing Aganagic-Vafa mirror symmetry,
I will exhibit this Lagrangian in examples as the graph of the differential
of a superpotential written as an integral linear combination of dilogarithms in
special coordinates. The superpotential conjecturally encodes the
Ooguri-Vafa invariants, and with them the open Gromov-Witten
invariants of the obstructed Lagrangian.
This is joint work with David Treumann.
We present a flexibility result with applications in the computation of holomorphic invariants of Lagrangian and Legendrian submanifolds. Legendrian contact homology for 1-dimensional Legendrian knots in standard contact $\mathbb{R}^3$ can be understood combinatorially via the Chekanov dga of the Lagrangian projection of the knot, or equivalently in terms of its Legendrian projection. Similarly, recent work of Ng, Rutherford, Shende, Sivek and Zaslow shows that the $A_\infty$ category of augmentations of the Legendrian contact homology dga can be understood in terms of a certain category of sheaves in the front plane. A major obstruction to extending these results to higher dimensions is the fact that generically Legendrian fronts have terrible singularities, rendering the combinatorics intractable. This is just one of many examples where one would like a Lagrangian or Legendrian front to have singularities that are as simple as possible. Other examples include a desire to globally understand family Floer homology or an interest in studying the space of 1-dimensional Legendrian knots (the fronts of families depending on many parameters will also generically have terrible singularities). We prove an $h$-principle which shows that whenever the obvious homotopy theoretic obstruction to simplifying the singularities of a Lagrangian or Legendrian front vanishes, then the simplification can be achieved by means of an ambient Hamiltonian isotopy. In many cases this homotopy theoretic obstruction can be easily shown to vanish, for example for even dimensional Legendrian spheres in standard Euclidean space.
The Hilbert schemes $Hilb^n(C)$ of a singular curve $C$ parametrize $n$-dimensional quotients of the structure sheaf of. We study the motivic Hilbert zeta function $Z_C(t)$ which is the generating function for $Hilb^n(C)$ in the Grothendieck ring of varieties. When $C$ is planar, work of Oblomkov-Shende, Maulik, and others showed that after passing to Euler characteristic, a refinement of $Z_C(t)$ is the HOMFLY polynomial of the link of the singularity.
With D. Ranganathan and R. Vakil, we study the structure of $Z_C(t)$ for general curve singularities. We show that $Z_C(t)$ is a rational function for any curve with isolated singularities and study to what extent it exhibits a functional equation and topological invariance. Inspired by physics, with A. Takeda we explore a conjectural relation between $Z_C(t)$ and knot invariants for general isolated curve singularities and relate this to the study of a certain generalization of the Hitchin fibration.
Persistence modules are simply functors from a poset (often $\mathbb{R}$) to Vect. They have been used in symplectic geometry going back to the symplectic capacity of Viterbo and Floer-theoretic analogs (Oh, Schwartz) although the terminology of persistence, barcodes etc... arose later in Computer Science when persistence modules were studied as tools for topological data analysis. In particular Viterbo's symplectic capacity of domains in $\mathbb{R}^{2n}$ and Sandon's contact capacity of domains in $\mathbb{R}^{2n} \times S^1$ are persistences of certain homology classes in the persistence module formed by generating function (GF) homology groups. While Sandon's capacity $c_S(-)$ allows to re-prove non-squeezing of $B(R) \times S^1$ into itself for integral $R$ (a result due to Eliashberg-Kim-Polterovich 2006), by introducing new filtration-decreasing morphisms between GF homology groups one can set up a functor from a sub-category of $\mathcal{D} \times \mathbb{Z}$ to Vect, where $\mathcal{D}$ is the category of bounded domains with inclusion. Persistences in this persistence module then yield a sequence $m_\ell(-)$, $\ell \in \mathbb{N}$ of integer-valued contact invariants for prequantized balls, such that $m_1 = c_S - 1$ and $m_\ell(B(R) \times S^1)$ is the greatest integer strictly less than $\ell R$. This provides an alternate proof of non-squeezing at large scale, i.e. of $B(R) \times S^1$ into itself for any $R>1$ (proved by Chiu 2014).
One perspective on knot invariants is to study the contact/symplectic geometry of the conformal bundle of the knot. Sheaves with singular support along this conormal give rise to a categorical knot invariant. The front projection of the Legendrian conormal bundle also defines a sheaf category, but on a different space. We define a functor between the two sheaf categories and show it is an equivalence. We also study how microlocal rank-1 sheaves transform under this equivalence.
It is well known that the Bernstein-Gelfand-Gelfand category $O$ for a Lie algebra $g$ behaves roughly like the Fukaya category of the cotangent bundle to the flag variety $G/B$. In "Morse theory and tilting sheaves" Nadler defined a geometric construction of tilting modules in category $O$ by taking certain constructible sheaves known as "Morse kernels" on the flag variety and flowing them along a $C^*$-action whose ascending manifold stratification coincides with the Schubert stratification. Braden-Licata-Proudfoot-Webster have defined a version of category $O$ for any suitable holomorphic symplectic variety. Other than cotangent bundles the simplest family of holomorphic symplectic varieties are the hypertoric varieties of Bielawski-Dancer. I will describe a construction that generalizes Nadler's result to hypertoric category $O$. In this case, the "Morse kernels" are certain GKZ integrable systems. This construction arose from joint work with Bullimore, Dimofte, and Gaiotto on mirror symmetry of 3d $N=4$ gauge theories.
In Riemannian and semi Riemannian geometry, metric tensor has a warped product structure in foliated chart related to conformal gradient fields, which helps to obtain global structural information about manifold. This correspondence between existence of a conformal gradient field and warped product structure has not been yet provedfor Finsler metrics. In this proposed poster, I will present a warped product Finsler structure from Hamiltonian point of view and explore the functions which have conformal gradient and the structural information obtained in presence of such a vector field.
This piece of work is concerned with the use of local systems of rank higher than 1 as coefficient spaces for Lagrangian Floer Theory in the monotone case, particularly when pseudoholomorphic discs of Maslov index 2 with boundary on a Lagrangian cannot be excluded. We use this technique to establish that the Chiang Lagrangian in $CP^3$, when equipped with an appropriate local system of rank 2, can be ensured to have non-zero Floer cohomology with itself over characteristic 2. This is an extension of work by Evans and Lekili ("Floer cohomology of the Chiang Lagrangian." Selecta Mathematica 21.4 (2015): 1361-1404) and provides a negative answer to their question whether the Chiang Lagrangian and $RP^3$ can be Hamiltonianly displaced from each other.
Utilizing Biran-Cornea Lagrangian cobordism theory and Mau-Wehrheim-Woodward functor, we give an alternative proof of Seidel's long exact sequence and Wehrheim-Woodward fibered Dehn twist long exact sequence in the monotone setting. Our approach also confirms Huybrechts-Thomas’s prediction of projective Dehn twist modulo determination of connecting morphism. Emphasis is put on the functorial perspective of the approach and possible generalization to other long exact sequences. This is a joint work with Weiwei Wu.
A generating family for a Legendrian submanifold is a family of functions that encodes information about the Reeb chords of the Legendrian. I use generating families to construct an A-infinity algebra using Morse flow trees, and show this algebra is invariant up to A-infinity quasi-isomorphism under Legendrian isotopy.
We discuss semicanonical bases from the point of view of Cohomological Hall algebras via the "dimensional reduction" from 3-dimensional Calabi-Yau categories to 2-dimensional ones. Also, we discuss the notion of motivic Donaldson-Thomas invariants (as defined by M. Kontsevich and Y. Soibelman) in the framework of 2-dimensional Calabi-Yau categories. In particular we propose a conjecture which allows one to define Kac polynomials for a 2-dimensional Calabi-Yau category (this is a theorem of S. Mozgovoy in the case of preprojective algebras).
Kaputsin and Orlov have suggested to enlarge the Fukaya category with objects arising from coisotropic submanifolds for the HMS conjecture to be true. However, not much is know about coisotropics so far. We show that every stable, fibered, displaceable coisotropic submanifold is a torus fibration over a symplectically uniruled base. To the coisotropic $C$ we associate a Lagrangian $L_C$ and a stable hypersurface $H_C$ . We then analyse the pearly differential of the Floer complex of $L_C$ and use neck stretching around $H_C$ to produce a non constant sphere though every point in the symplectic base of the coisotropic.
In recent work of Shende, Nadler, Zaslow et al, many spaces of classical interest such as character varieties of surfaces, wild character varieties, augmentation varieties appearing in knot theory etc. can be phrased as particular cases of moduli spaces of constructible sheaves on stratified spaces. One common feature of such spaces is that they all carry natural Poisson or symplectic structures, a fact that has been explained ad hoc in the above cases. In my work with Shende, using derived symplectic geometry we give a general method for constructing such symplectic structures, and show that the existence of these structures is related to orientations of the corresponding category of constructible sheaves. More generally, we define a larger class of locally arboreal spaces, of which all the spaces above are particular cases, and prove that all such spaces carry natural shifted symplectic structures.
References:
V. Shende and A. Takeda, Orientations on locally arboreal spaces, in preparation
V. Shende, D. Treumann, H. Williams and E. Zaslow, Cluster varieties and Legendrian knots, arXiv:1512.08942
T. Pantev, B. Toën, M. Vaquié, G. Vezzosi, Shifted symplectic structures , arXiv:1111.3209
Recently, we generalize Schwarz's theorem to $C^0$-case on aspherical closed surfaces and prove that the contractible fixed points set (and consequently the fixed points set) of a nontrivial Hamiltonian homeomorphism is not connected.
It is expected that homological mirror symmetry for the complements of a divisor in a variety is obtained from the categorical localization of homological mirror symmetry for the ambient variety. We give this picture in the coherent-constructible correspondence which is a version of homological mirror symmetry for toric varieties given by Fang-Liu-Treumann-Zaslow. In our description, we use the concept of wrapped constructible sheaves which is recently introduced by Nadler.