Institute for Geometry and Physics Miami-Cinvestav-Campinas Collaboration in Geometry & Physics in the Americas

Conference on Homological Mirror Symmetry – January 27-February 1, 2014

Organizers: Denis Auroux, Ludmil Katzarkov, Maxim Kontsevich, Elizabeth Gasparim, Ernesto Lupercio, Tony Pantev

This event will be held at the University of Miami (Coral Gables, Florida).

The workshop will start on Monday morning (January 27) and will end on Saturday (February 1) mid-day. There will be three mini-courses of three lectures each, individual research talks, as well as time for informal discussions.

The workshop is partially supported by the NSF FRG ``Wall-crossings in Geometry and Physics'' (grants DMS-1265228, 1265196, 1264662, 1265230, 1262531).

Talks: The conference will be held in the University Center, Flamingo Ballroom C & D (2nd Floor). See the campus map for directions (the arrows show the path from the hotel); there's also a newer campus map.

Accommodation: Most participants will be staying at the Holiday Inn Coral Gables/University of Miami, located right next to the University of Miami campus. The hotel address is: 1350 South Dixie Highway, Coral Gables FL 33146; phone: 1-305-667-5611.

Airport: Miami International Airport is about 7 miles from campus. The most convenient way to reach the hotel or the campus is to take a taxi. Invited participants: When booking flights, please keep in mind NSF and University of Miami rules. Only US-based airlines are permitted (more specifically your plane ticket mush be issued by a US air carrier and show US carrier flight numbers). Please keep all your original boarding passes since they will be needed for reimbursement.

Monday January 27

Tuesday January 28

Wednesday January 29

Thursday January 30

Friday January 31

Saturday February 1

Minicourses:

• Dominic Joyce (Oxford):
  

  Talk 1: "Darboux theorems" for shifted symplectic derived schemes and stacks.

  Abstract: Pantev, Toën, Vezzosi and Vaquié (arXiv:1111.3209) introduced the notion of k-shifted symplectic structure on a derived scheme or derived stack, for all integers k, where 0-shifted symplectic structures on derived schemes are just classical algebraic symplectic structures on classical smooth schemes. They prove that derived moduli stacks of (complexes of) coherent sheaves on a Calabi-Yau m-fold have a (2-m)-shifted symplectic structure. So the case k = -1 is relevant to Donaldson-Thomas theory of Calabi-Yau 3-folds. The semi-classical truncation of a -1-shifted symplectic structure on a derived scheme is a symmetric obstruction theory on the underlying classical scheme in the sense of Behrend, a basic tool in Donaldson-Thomas theory.

  We prove a "Darboux Theorem" for k-shifted symplectic derived schemes for all k < 0. When k = -1, this says that a -1-shifted symplectic derived scheme (which includes moduli schemes of simple (complexes of) coherent sheaves on a Calabi-Yau 3-fold) is Zariski locally equivalent to the critical locus of a regular function on a smooth scheme. Related results in the complex analytic setting were proved by Joyce-Song for coherent sheaves using gauge theory, and claimed by Behrend-Getzler for complexes. We extend our results to give standard 'Darboux forms' for smooth atlases of shifted symplectic derived Artin stacks.

  This talk is joint work with with O. Ben-Bassat, C. Brav, V. Bussi, based on arXiv:1305.6302 and arXiv:1312.0090.

  Talk 2: D-critical loci, categorification of Donaldson-Thomas theory using perverse sheaves.

  Abstract: First we define "d-critical loci" and "d-critical stacks", which are new (classical) geometric structures on moduli schemes and moduli stacks of (complexes of) coherent sheaves on a Calabi-Yau 3-fold, and so tools for use in Donaldson-Thomas theory. A d-critical locus (X,s) is a scheme X with an extra geometric structure s which records information on how X may be written locally as a critical locus. We construct a truncation functor from -1-shifted symplectic derived schemes to d-critical loci, and deduce that moduli schemes of simple (complexes of) coherent sheaves on a Calabi-Yau 3-fold are d-critical loci. Intersections of complex or algebraic Lagrangians in a complex or algebraic symplectic manifold also have the structure of d-critical loci. We extend the whole picture to Artin stacks.

  A d-critical locus (X,s) has a "canonical bundle", which for moduli schemes is the determinant line bundle of the natural obstruction theory. An "orientation" is a choice of square root of this canonical bundle; this is essentially the same as "orientation data" in the work of Kontsevich-Soibelman.

  We prove that an oriented d-critical locus (X,s) carries a natural perverse sheaf P_X,s (also a D-module, and a natural mixed Hodge module), such that if (X,s) is locally modelled on Crit (f : U –> C) then P_X,s is locally modelled on the perverse sheaf of vanishing cycles of f. The pointwise Euler characteristic of P_X,s is the Behrend function of X. For a D-T moduli scheme, the graded dimension of the hypercohomology H*(P_X,s) is the corresponding Donaldson-Thomas invariant. Thus, this provides a categorification of Donaldson-Thomas invariants. Recent work of Kiem-Li uses gauge theory to do more or less the same thing for moduli schemes of sheaves on a C-Y 3-fold. We again extend the whole picture to Artin stacks.

  This talk is based on arXiv:1304.4508, arXiv:1211.3259, and arXiv:1312.0090. It is joint work with with O. Ben-Bassat, C. Brav, V. Bussi, D. Dupont and B. Szendroi.

  Talk 3: More about perverse sheaves; motivic Donaldson-Thomas theory; Calabi-Yau 4-fold counting invariants; future projects.

  Abstract: First we discuss a conjecture on morphisms of the perverse sheaves constructed in Talk 2. Proving this conjecture has applications to constructing Kontsevich-Soibelman style "cohomological Hall algebras" using perverse sheaves, and to defining "Fukaya categories" of Lagrangians in a complex or algebraic symplectic manifold.

  Secondly, we prove that an oriented d-critical locus (X,s) carries a natural motive M_X,s, such that if (X,s) is locally modelled on Crit ( f : U –> C) then M_X,s is locally modelled on the motivic vanishing cycle of f, and we extend these to Artin stacks. Applied to Calabi-Yau 3-fold moduli schemes and stacks, these are essentially Kontsevich and Soibelman's motivic Donaldson-Thomas invariants.

  Thirdly, I explain a project to define new, deformation-invariant, D-T style invariants "counting" coherent sheaves on a Calabi-Yau 4-fold, starting from the "Darboux theorem" for -2-shifted symplectic derived schemes in Talk 1. If D-T invariants are "holomorphic Casson invariants", these new CY4 invariants are "holomorphic Donaldson invariants".

  Fourthly, I discuss other interesting directions to go from here, such as matrix factorization categories.

  The second part is joint with V. Bussi and S. Meinhardt, based on arXiv:1305.6428 and arXiv:1312.0090, and the third part is joint with Dennis Borisov (work in progress).

• Maxim Kontsevich (IHES):

  Talk 1: Fukaya categories with coefficients

  Abstract: Consider a Calabi-Yau X fibered over a Fano Y. Fukaya categories of fibers form a constructible sheaf of triangulated categories on Y. The global Fukaya category of X can be interpreted as a kind of derived global section (along singular Lagrangian subsets of Y), perturbed by holomorphic discs in Y. I argue that this construction makes sense for general constructible sheaves of categories on Kahler manifolds, without any symplectic interpretation of fibers. Moreover, stability conditions on stalk categories with values in the canonical bundle of Y give rise to the stability conditions on the global category, via the Lagrangian mean curvature flow.

  Talk 2: Non-archimedean methods in mirror symmetry

  Abstract: I'll explain how to use Berkovich spectra of rigid analytic spaces over non-archimedean fields as an alternative to the Gross-Siebert and Gross-Hacking-Keel approaches. For the B-model it gives a generalization of cluster varieties and a notion of algebraic stability data. For the A-model one can define the counting of anaytic discs with boundary on small tori for non-archimedean varieties over fields of arbitrary characteristic.

  Talk 3 is joint with Yan Soibelman, see below

• Andy Neitzke (UT Austin): Open string mirror symmetry and Hitchin's equations

Research talks:

• Mohammed Abouzaid (Columbia): TBA
• Denis Auroux (UC Berkeley): HMS for hypersurfaces in (C*)^N

  Abstract: The mirrors of hypersurfaces in (C*)^N are toric Landau-Ginzburg models. In this limited setting, we define a fiberwise wrapped Fukaya category of the Landau-Ginzburg model and show (by constructing a mirror to the structure sheaf) that the derived category of the hypersurface embeds into it. (This is joint work in progress with Mohammed Abouzaid.)

• George Dimitrov (Vienna): Bridgeland stability conditions and exceptional collections.

  Abstract: The talk concerns the interplay between these two notions. Applications to both stability conditions and exceptional collections will be presented.

  We define "σ-exceptional collection", where σ denotes a stability condition. Any full σ-exceptional collection (if such exists) generates σ in a procedure described by E. Macrì. We focus on constructing σ-exceptional collections from a given σ on D^b(A), where A is hereditary, hom-finite category, linear over an algebraically closed field. One difficulty is due to exceptional objects X,Y in A with non-vanishing Ext¹(X,Y) and Ext¹(Y,X). We introduce a property on A, called regularity-preserving, which makes this difficulty manageable and show examples of this property. An application is a description of the entire space of stability conditions on the tame quiver with three vertices. This is a joint work with L. Katzarkov.

  An application to exceptional collections (on quivers) is in a joint work with F. Haiden, L. Katzarkov, M. Kontsevich. We will explain briefly this result as well.

• Alexander Efimov (Steklov Institute): Noncommutative Hitchin system

  Abstract: First, we construct a family of associative algebras, which are formally smooth (Quillen-smooth), and whose representation spaces are (affine charts of) moduli spaces of semi-stable vector bundles on a smooth projective curve, with trivialization of a single fiber. Then we construct a noncommutative version of Hitchin system, which induces the usual one (for GL_n) by taking its "trace".

  Our construction is closely related with some generalization (actually, extension) of the DG Lie algebra of Hochschild cochains of a small DG category.

• Penka Georgieva (Princeton): A WDVV-type relation for real/open Gromov-Witten invariants

  Abstract: The classical problem of enumerating rational curves in projective spaces is solved using a recursion formula for Gromov-Witten invariants. In this talk, I will describe a similar relation for real Gromov-Witten invariants with conjugate pairs of constraints. An application of this relation provides a complete recursion for counts of real rational curves with such constraints in odd-dimensional projective spaces. I will outline the proof and discuss some vanishing and non-vanishing results. This is joint work with A. Zinger.

• Alexander Goncharov (Yale): TBA
• Fabian Haiden (Vienna): Geometricity and stability of objects in Fukaya categories of surfaces.

  Abstract: In this talk I will discuss two related problems involving the (partially) wrapped Fukaya categories of open surfaces. The first is the classification of objects in these categories up to isomorphism. The second is constructing Bridgeland stability conditions from flat structures with singularities. A solution to the former using a topological approach to these categories, recently implemented in work of Dyckerhoff-Kapranov, will be presented. Part of joint work with Katzarkov and Kontsevich.

• Mikhail Kapranov (Yale): Generalized orders and invariants of structured surfaces.

  Abstract: Crossed simplicial groups were introduced by Fiedorowicz and Loday as generalizations of Connes' cyclic category Λ appearing in cyclic homology. They are categories Q containing the category Δ of simplices, and besides Λ, include dihedral, quaternionic etc. categories.

  It turns out that a class of such categories Q is in bijection with groups G that can be structure groups of 2-dimensional topology: various higher order Spin and Pin groups existing in 2d, as well as universal covers of SO(2) and O(2). Objects of such categories Q can be seen as finite sets with generalized orders: cyclic order, dihedral order etc. We show that Q gives rise to a combinatorial concept of structured graphs which play the same role for G-structured surfaces as ribbon graphs do for oriented surfaces. As one application, a 2-Segal Q-object gives rise to a datum on the moduli space of G-structured surfaces. Joint work with T. Dyckerhoff.

• Gabriel Kerr (Kansas State): Homological mirror symmetry from the perspective of VGIT.

  Abstract: This talk will investigate HMS of toric stacks [U/G] by examining two Fukaya type categories mirror to the quasi-affine space U. The first category is defined as a wrapped Fukaya category associated to a non-standard wrapping Hamiltonian. The second category is associated to a Lagrangian skeleton. We sketch a relationship between these categories and explain how they are affected by a variation of GIT through examples.

• Maxim Kontsevich (IHES) and Yan Soibelman (Kansas State): Fukaya-Seidel category in infrared (after Gaiotto, Moore and Witten)

  Abstract: We will explain some mathematical structures involved in the recent approach of Gaiotto-Moore-Witten to Fukaya-Seidel categories. We are going to describe an A-infinity category with semi-orthogonal decomposition starting with a planar configuration of points.

  It gives a description of the Fukaya-Seidel category of the pair (X,W) in terms of "local categories" associated with critical values of the potential W.

  If time permits, we will discuss some speculations about the relation of this story to:

  1. counting of coherent sheaves on a Calabi-Yau 3-fold "with the boundary on a coisotropic 5-cycle";
  2. spectral networks;
  3. counting of coassociative cycles with boundary in G₂-manifolds and holomorphic Chern-Simons functional.

• Alexander Kuznetsov (Steklov Institute): Higher blowups.
• Chiu-Chu Melissa Liu (Columbia): All genus open-closed mirror symmetry for toric Calabi-Yau 3-orbifolds.
• Yong-Geun Oh (Wisconsin): Nondisplaceable Lagrangian tori in S² × S²

  Abstract: In this talk, using the idea of toric degeneration and bulk deformation of Lagrangian Floer homology, we produce a continuum of Lagrangian tori in S² × S² which are nondisplaceable by Hamiltonian isotopy. This is based on the joint works with Fukaya, Ohta and Ono.

• Pranav Pandit (Vienna): Buildings, Spectral Networks, and the Asymptotics of Monodromy.

  Abstract: The talk will focus on how the asymptotic behavior of the Riemann-Hilbert correspondence (and, conjecturally, the non-abelian Hodge correspondence) on a Riemann surface is controlled by certain harmonic maps from the Riemann surface to affine buildings. This is part of joint work with Katzarkov, Noll and Simpson, which revisits, from the perspective afforded by the theory of harmonic maps to buildings, the work of Gaiotto, Moore and Neitzke on spectral networks, WKB problems, BPS states and wall-crossing.

• Tony Pantev (U.Penn.): Meromorphic Higgs bundles and Calabi-Yau threefolds

  Abstract: I will describe two geometric setups which relate Higgs bundles on curves to Calabi-Yau threefolds. In the first setup the Hitchin system with poles is identified with a Calabi-Yau integrable system, and in the second setup the moduli of parabolic Higgs bundles identified with a moduli of perverse coherent sheaves on a Calabi-Yau threefold. I will discuss the dependence of these identifications on the particular geometric model for the Calabi-Yau geometry and the effect the choice of a model has on the description of stability. This is a joint work with D.-E. Diaconescu and R. Donagi.

• James Pascaleff (UT Austin): Equivariant Lagrangian branes and representations

  Abstract: Classical Borel-Weil theory constructs representations of semisimple groups in spaces of sections of homogeneous vector bundles on flag varieties. In this talk, we describe a construction in symplectic geometry which is meant to serve as the mirror dual to Borel-Weil construction. Building on the fundamental work of Seidel-Solomon, we define a notion of "equivariant Lagrangian brane" in an exact symplectic manifold M. We may then obtain representations of a Lie algebra g on Floer cohomology of equivariant Lagrangian branes. We will make our construction completely explicit in the case of sl2. This is joint work with Yanki Lekili.

• Helge Ruddat (Mainz): Conifold Transitions, Tropical geometry and Mirror Symmetry.

  Abstract: Morrison conjectured that mirror symmetry dualizes conifold transitions of Calabi-Yau threefolds. Since mirror symmetry is a phenomenon at a maximally unipotent boundary point of the Calabi-Yau moduli space, in order to prove the conjecture, one needs a theory combining conifold transitions with maximal degenerations.

  This would also allow to include the beautiful relationship between SYZ fibrations and conifold transitions as first studied by Gross and Ruan. I will report on joint work with Siebert where we produce such a theory by giving a comprehensive account of conifold transitions in the Gross-Siebert program. We exhibit tropical homology groups that control the obstructions à la Friedman-Tian and Smith-Thomas-Yau. We also draw the connection to recent work by Matessi and Castano-Bernard. We expect that the techniques lead to a proof of Morrison's conjecture.

• Renato Vianna (UC Berkeley): An exotic monotone Lagrangian torus in CP²

  Abstract: The aim of the talk is to present the ideas behind the construc- tion of a exotic monotone Lagrangian torus in CP² that bounds 10 families of (Maslov index 2) holomorphic discs. The number of such holomorphic discs is an invariant among monotone Lagrangian tori, which imply that this exotic torus is the first example of a Lagrangian torus not Hamiltonian isotopic to the classical Clifford torus or to the monotone torus found by Chekanov in 1995.

  Time permitting, we will see that the development of such ideas allow us to conjecture the existence of an infinite range of monotone Lagrangian torus in CP² not Hamiltonian isotopic to each other.

• Tony Yue Yu (Paris): Gromov compactness in non-Archimedean analytic geometry

  Abstract: An "algebraic" analog of the SYZ fibration in mirror symmetry is the deformation retraction of a non-Archimedean analytic space to its skeleton. While the SYZ fibration remains largely conjectural, the retraction to skeleton is well-grounded by the work of Berkovich. In this talk, after a heuristic introduction to the basic objects and theories, I will discuss several aspects of non-Archimedean analytic enumerative geometry: the construction of the moduli space of k-analytic stable maps, the Gromov compactness theorem and the tropicalization of the moduli space. Using formal models, we settle these problems of analytic nature with tools from algebraic geometry. Using the functor of vanishing cycles, we relate the tropical curves with intersection theories on the special fiber. If time permits, I will discuss the motivations and the expected relations with complex mirror symmetry.