Launching Workshop: Mirror symmetry and Irregular singularities
04.02. - 06.02.2014
Universität Mannheim, Germany
Aim
This workshop is the first activity of the newly established research network SISYPH. It intends to
bring together members of the network and some exterior participants. We seek to report on some recent progress on
subjects such as mirror symmetry, related Hodge theoretic questions and singularities
of differential systems arising in this context. A large part of the talks will be concerned
with ideas for future research and collaborations within the network.
Program
Tuesday, 04.02.
13.30 - 14.15, Thierry Mignon (Montpellier): Computation of Gromov-Witten invariants for
hypersurfaces in orbifolds
Abstract:
The aim is to compute the orbifold Gromov-Witten
invariants of a generic hypersurface in a toric orbifold, or
-to begin with- a weighted projective space. The standard
method is to use Orbifold Quantum Lefschetz theorem proved by
Tseng. This method could fail even if the hypersurface is
defined via a line bundle which is positive on every curve (for
instance for P(1, 2, 2) in P(1, 1, 2, 2) and L = O(1) one can
not apply orbifold Quantum Lefschetz). For varieties, Gathmann
used relative Gromov-Witten invariants and degeneration formula
to compute the Gromov-Witten invariants of a generic
hypersurface in a smooth projective variety. This method does
not use ampleness ; it should be possible to generalize it to
orbifolds, and compute new Gromov-Witten invariants. This is
joint work with Etienne Mann (Montpellier) and Cristina
Manolache (London).
15.00 - 15.45, Philip Boalch (Paris): Mirror symmetry, Langlands duality and wild nonabelian Hodge theory
Abstract:
Moduli spaces of meromorphic connections on G-bundles on curves are known to be fibred by special Lagrangian tori. I'll discuss some questions relating this to the other words in the title, and try to survey what is known.
16.15 - 17.00,
Claus Hertling (Mannheim): Stokes structures and Torelli problems for
singularities.
Abstract:
Slogan/hope: Start with a mu-homotopy class of an
isolated hypersurface singularity. The base space of a certain
global versal unfolding should be an atlas of distinguished
bases (up to sign) of its Milnor lattice. This is made precise
and proved for the simple singularities (Looijenga+Deligne
1974) and the simple elliptic singularities
(Hertling+Roucairol, essentially 2007). One can hopefully
interpret it as a global Torelli type conjecture for those
functions in the global versal unfolding which have mu
different critical values, as there the distinguished bases (up
to sign) encode the Stokes structures of the Brieskorn
lattices. It should be related to global Torelli type
conjectures for the isolated hypersurface singularities
themselves. It is related to other questions on their Stokes
structures.
Wednesday, 05.02.
9.15 - 10.00, Thomas Reichelt (Bonn): Differential graded D-modules and rational homotopy
theory
Abstract:
A simply-connected topological space X admits two dual decompositions, the CW-complex which encodes the homology of X and a Postnikov tower which encodes its homotopy groups.
Sullivan built from the latter a differential graded algebra, the minimal model of X, from which one can easily
deduce the rational homotopy groups of the space. For a proper smooth map Aznar defined the corresponding
notion of a minimal model of a connection. This allowed him to construct a Gauss-Manin connection
measuring rational homotopy groups. I want to propose a possible extension of this theory to D-modules and
discuss some applications in singularity theory and mirror symmetry.
11.00 - 11.45, Clélia Pech (London): Mirror symmetry for Lagrangian Grassmannians
Abstract:
This talk reports on joint work with Konstanze Rietsch. After recalling different Landau-Ginzburg models for the usual Grassmannians, respectively from Batyrev-Ciocan-Fontanine-Kim-van Straten and March-Rietsch, I will explain our construction of a mirror for Lagrangian Grassmannians LG(n). The superpotential is expressed in natural coordinates of a dense open part X^ of a 'dual maximal orthogonal Grassmannian', related to the Schubert basis of LG(n). I will also explain the construction of a cluster structure on the mirror X^, and how it should help construct an isomorphism between the Dubrovin connection on LG(n) and the Gauss-Manin connection of X^.
12.00 - 14.00, Lunch
14.00 - 14.45, Claude Sabbah (Palaiseau): Aspects of the irregular Hodge filtration
Abstract:
Given a regular function f on a smooth
quasi-projective variety U, the de Rham complex of U relative
to the twisted differential d+ df can be equipped canonically
with a filtration (the irregular Hodge filtration) for which
the associated hypercohomology spectral sequence degenerates at
E_1. A logarithmic version of this de Rham complex (relative to
a suitable compactification of U) has been introduced by M.
Kontsevich, who showed the independence of the dimension of the
corresponding cohomologies with respect to the differential ud
+ vdf, for u,v arbitrary complex numbers. This leads to bundles
on the projective line of the (u:v) variable, on which we
construct a natural connection for which the Harder-Narasimhan
filtration satisfies the Griffiths transversality property and
standard limiting properties at v=0. This is joint work with
Hélène Esnault (Berlin) and Jeng-Daw Yu (Taipei).
15.30 - 16.15, Helge Ruddat (Mainz): Speculations on Mirror Symmetry for Riemann Surfaces
Abstract:
There has been quite some evidence that some form of
mirror symmetry is valid for curves of higher genus. In known
constructions, the dual geometry is derived from a
higher-dimensional Landau–Ginzburg model. We present some ideas
of how an intrinsic form of the mirror construction could be
formulated and expect that Higgs bundles play a central role.
16.30 - 17.15, Jérémy Guéré (Paris): Mirror symmetry for the quantum singularity theory of chain polynomials
Abstract:
We will talk about the quantum invariants for singularities, which were introduced by Fan, Jarvis, and Ruan in 2007. Our dream is to compute them in every genus and for every singularities, but it goes through the computation of a `virtual (cohomological) class', which is very hard to get without the so-called concavity hypothesis. I will present the first explicit computation of every genus-zero quantum invariants for a range of singularities called chain polynomials, which do not respect concavity. To this purpose, I developed a machinery based on my recent notion of recursive complexes, which is fun and easy to use. As a remarkable consequence, we are going to understand in which precise sense the virtual class is viewed as a partial extension of Euler class to K-theory. Last but not least, this work yields a mirror symmetry theorem for chain singularities, at the level of local systems. You could find the full article on arXiv:1307.5070.
Thurstday, 06.02.
9.00 - 9.45 Etienne Mann (Montpellier): Categorification of Gromov-Witten invariants
Abstract:
This is an ongoing project with Marco Robalo. We plan to find a morphism of modular operads between the (M_g,n)_{g,n} and the endomorphism operad of the derived category of X. This statement should be understood as "the categorification of Gromov-Witten invariants". We also want to deduce some applications of that project to the crepant resolution conjecture and Dubrovin's conjecture.
10.00 - 10.45, Christian Sevenheck (Mannheim): Hypergeometric systems and mixed Hodge modules
Abstract: Since the work of Givental, hypergeometric functions and hypergeometric differential equations are
ubiquitous in toric mirror symmetry. While the case of Calabi-Yau hypersurfaces in toric varieties has been studied thoroughly and can be understood within the framework
of classical Hodge theory, the quantum D-module of a Fano or nef variety necessarily carries irregular singularities. Correspondingly, one needs an extension nowadays called non-commutative Hodge theory. It is best understood using the formalism of mixed Hodge modules and the operation of Fourier-Laplace transformation. We intend to report on work in progress (together with Thomas Reichelt) concerning the description of these kind of Hodge structures for the case of hypergeometric D-modules.
11.00 - 12.00, Discussion
Participants
- Christian Barz (Universität Mannheim)
- Philip Boalch (École normale supérieure, Paris)
- Antoine Douai (Université de Nice Sophia-Antipolis)
- Michel Granger (Université d'Angers)
- Jérémy Guéré (Université de Paris 6)
- Martin Guest (Waseda University, Tokyo)
- Marco Hien (Universität Augsburg)
- Claus Hertling (Universität Mannheim)
- Etienne Mann (Université de Montpellier)
- Giovanni Morando (Universität Augsburg)
- Thierry Mignon (Université de Montpellier)
- Clélia Pech (King's College, London)
- Luca Prelli (Universidade de Lisboa, Portugal)
- Thomas Reichelt (MPIfM Bonn)
- Helge Ruddat (Universität Mainz)
- Claude Sabbah (Ecole Polytechnique, Palaiseau)
- Mathias Schulze (Universität Kaiserslautern)
- Timo Schürg (Universität Augsburg)
- Christian Sevenheck (Universität Mannheim)