conference on
Special Geometry, Mirror Symmetry
and Integrable Systems
at waseda university (online)
Place | Date | Speakers (see below for program) |
Waseda University and RIMS, Kyoto University
This conference is an RIMS review seminar: an activity in the RIMS Project 2020 "Differential Geometry and Integrable Systems - Mathematics of Symmetry, Stability, Moduli"
It will be held as an online* conference with lectures in the Europe/Japan time zone. |
29 November 2021 - 2 December 2021 |
Murad Alim (Hamburg) Yalong Cao (RIKEN) Vicente Cortes (Hamburg) Liana David (Romanian Academy) Kazuyuki Hasegawa (Kanazawa) Keizo Hasegawa (Osaka) Claus Hertling (Mannheim) Shinobu Hosono (Gakushuin) Hiroshi Iritani (Kyoto) Katsushi Ito (Tokyo Inst. Tech.) David Lindemann (Aarhus) Atsuhira Nagano (Kanazawa) Yuichi Nohara (Meiji) Alexis Roquefeuil (IPMU) Ian Strachan (Glasgow) Andrew Swann (Aarhus) |
* Participants will receive Zoom log in information each day by e-mail. In order to receive Zoom log in information, please send an e-mail to Prof. Martin Guest (martin at waseda.jp) stating your name, university affiliation, and position/student status. Speakers do not need to do this - they are automatically registered.
PROGRAM OF THE CONFERENCE (all times are Japan time). TITLES AND ABSTRACTS ARE LISTED AT THE END OF THIS PAGE.
Monday 29 November
17:00-17:10: opening
17:15-18:00 Vicente Cortes
18:15-19:00 Keizo Hasegawa
19:15-20:00 David Lindemann
20:15-21:00 Yuichi Nohara
Tuesday 30 November
17:00-17:45 Hiroshi Iritani
18:00-18:45 Alexis Roquefeuil
break
19:15-20:00 Kazuyuki Hasegawa
20:15-21:00 Andrew Swann
Wednesday 1 December
17:00-17:45 Yalong Cao
18:00-18:45 Ian Strachan
break
19:15-20:00 Liana David
20:15-21:00 Claus Hertling
Thursday 2 December
17:00-17:45 Katsushi Ito
18:00-18:45 Atsuhira Nagano
break
19:15-20:00 Murad Alim
20:15-21:00 Shinobu Hosono
Organizing Committee: Martin Guest (Waseda), Shinobu Hosono (Gakushuin), Yoshihiro Ohnita (OCAMI)
Supported by the RIMS Research Project 2020 and by JSPS Grant-in-Aid for Scientific Research 18H03668 (Martin Guest)
LINKS AND RELATED ACTIVITIES:
RIMS International Joint Usage/Research Center
TITLES AND ABSTRACTS
Murad Alim
Title: Quantum Geometry of BPS structures
Abstract:BPS invariants of certain physical theories correspond to Donaldson-Thomas
(DT) invariants of an associated Calabi-Yau geometry. BPS structures refer to
the data of the DT invariants together with their wall-crossing structure. On
the same Calabi-Yau geometry another set of invariants are the Gromov-Witten
(GW) invariants. These are organized in the GW potential, which is an asymptotic
series in a formal parameter and can be obtained from topological string theory.
Bridgeland showed that the GW potential can be extracted from a Tau-function
which solves a Riemann-Hilbert problem associated to BPS structures. I will
show that there is also a path going in the other direction. Studying the resurgence
of the GW potential leads to DT invariants. DT wall-crossing phenomena correspond
to Stokes jumps of the Borel resummation of the GW potential. In an explicit
example, I will furthermore discuss a corresponding hyperkähler geometry
and integrable hierarchy. This is based on joint work with Arpan Saha, Iván
Tulli and Jörg Teschner.
Yalong Cao
Title: Gopakumar-Vafa type invariants for Calabi-Yau 4-folds
Abstract: Gopakumar-Vafa type invariants on Calabi-Yau 4-folds (which are non-trivial
only for genus zero and one) are defined by Klemm-Pandharipande from Gromov-Witten
theory, and their integrality is conjectured. In this talk, I will explain how
to give a sheaf theoretical interpretation of them using counting invariants
on moduli spaces of one dimensional stable sheaves. Based on joint works with
D. Maulik and Y. Toda.
Vicente Cortes
Title: Quaternionic Kaehler manifolds with ends of finite volume and
instanton corrections thereof
Abstract: I will mainly explain a construction of complete quaternionic Kaehler
manifolds with interesting fundamental groups, which are not locally symmetric.
The examples include manifolds which are diffeomorphic to a cylinder over a
compact locally homogeneous manifold such that one of the two ends is of finite
volume. This is joint work with Danu Thung and Markus Roeser, see arXiv:2105.00727.
If time permits, I will also mention relations to joint work with Ivan Tulli
on nonperturbative quantum corrections to quaternionic Kaehler metrics at the
end of the talk, see arXiv:2105.09011.
Liana David
Title: T-duality for transitive Courant algebroids
Abstract: I will start with a brief review of the theory of Courant algebroids
and Dirac generating operators. Then I will develop a T-duality for transitive
Courant algebroids, which is a generalization Cavalcanti and Gualtieri's T-duality
for exact Courant algebroids. I will show that the T-duality between two transitive
Courant algebroids E and ¥tilde{E} induces a map between the spaces of sections
of their corresponding canonical weighted spinor bundles, which intertwines
the canonical Dirac generating operators of E and ¥tilde{E}. I will present
a general existence result for a T-dual Courant algebroid, under assumptions
generalizing the cohomology integrality conditions for the T-duality in the
exact case. If time allows, I will explain that the T-dual of a heterotic Courant
algebroid is again heterotic. This is joint work with Vicente Cortes.
Kazuyuki Hasegawa
Title: The quaternionic/hypercomplex-correspondence and its converse
Abstract : This talk is based on a published work (Osaka J. Math., 58 (2021),
213-238 (arXiv:1904.06056)) and an ongoing research collaborated with V. Cortes.
I will explain the quaternionic/hypercomplex-correspondence, that is, a construction
of a hypercomplex manifold for a given quaternionic manifold with a certain
U(1)-action of the same dimension. This construction generalizes the quaternionic
Kaehler/hyper-Kaehler-correspondence. As a converse construction, we can associate
a quaternionic manifold to a given hypercomplex manifold with a rotation symmetry
of the same dimension in our ongoing research. This is a generalization of the
hyper-Kaehler/quaternionic Kaehler-correspondence. In both constructions, we
obtain examples which can not be applied to the QK/HK- and HK/QK-correspondences.
In this talk, such examples are also explained.
Keizo Hasegawa
Title: Sasaki and CR Lie groups — construction and classification
problems
Abstract: Sasaki or CR Lie groups are the Lie groups which admit (left)-invariant
Sasaki or CR structures respectively. We consider here only CR structures which
are strongly pseudoconvex and of codimension one. So, Sasaki Lie groups consist
a subclass of CR Lie groups. There is a well-known real one parameter family
of CR structures which are non-Sasaki except the standard Sasaki structure on
a 3-sphere, considered as the Lie group SU(2). We have recently obtained a complete
classification of unimodular Sasaki Lie groups (Lie algebras); they are, up
to modifications, sl(2, R), su(2) and Heisenberg Lie groups. We aim to extend
this result to CR Lie groups (Lie algebras). The key ideals of the arguments
are “modification” for Sasaki Lie groups and “Cartan connection”
for CR Lie groups. We discuss basic ideas with some specific examples, illustrating
how they work in the arguments. This talk is based on a joint work with V. Cortes
for Sasaki Lie groups, and a joint work with H. Kasuya for CR Lie groups.
Claus Hertling
Title: Problems in the construction of Frobenius manifolds from F-manifolds
and TE-structures.
Abstract: One way to construct a Frobenius manifold (or a flat F-manifold) builds
it up in four steps. One needs an F-manifold, a TE-structure over it, a solution
of a Riemann-Hilbert problem, and a primitive section. The steps will be explained
in more detail. Results and open problems at the different steps will be discussed.
Part of it is joint work with Liana David.
Shinobu Hosono
Title: Mirror symmetry of a Calabi-Yau manifold fibered by abelian surfaces
Abstract: I will report on recent results on mirror symmetry of a Calabi-Yau
threefold fibered by (1,8)-polarized abelian surfaces based on a collaboration
with Hiromichi Takagi (arXiv:2103.08150). Such a Calabi-Yau manifold is interesting
since its hodge numbers are given by $h^{1,1}=h^{2,1}=2$ (self mirror?) and
also it admits a free group action by Heisenberg group H_8. I will construct
a mirror family of it; and, by describing its parameter space globally, I will
find special boundary points (called LCSLs) where we observe mirror symmetry
to a birational Calab-Yau manifold and also to a Fourier-Mukai partner to the
original Calabi-Yau manifold. By using mirror symmetry, I will calculate Gromov-Witten
invariants (of genus zero, one and two) from each boundary points and find that
they are expressed by quasi-modular forms.
Hiroshi Iritani
Title: Equivariant quantum cohomology
Abstract: The equivariant quantum cohomology (D-module) has the structure of
a difference module with respect to shifts of equivariant parameters. In this
talk, I will discuss the relation to mirror symmetry and possible applications.
Katsushi Ito
Title: TBA equations and WKB periods for higher order ODE
Abstract: We study the WKB periods for the (r+1)-th order ordinary differential
equation (ODE) with polynomial potential, which is obtained by the conformal
limit of the linear problem associated with the A^(1)_r affine Toda field equation.
The ODEs are also regarded as the quantum Seiberg-Witten curves of the (A_r,
A_{N-1})-type Argyres-Douglas theories where N is the order of the polynomial.
The ODE/IM correspondence provides a relation between the Wronskians of the
solutions and the Y-functions which satisfy the thermodynamic Bethe ansatz (TBA)
equation. We show the equivalence between the logarithm of the Y-functions and
the WKB periods, which is confirmed by solving the TBA equation numerically.
We also discuss the wall-crossing of the TBA equations from the minimal chamber
to the maximal chamber. This talk is based on arXiv:2104.13680[hep-th] and others.
David Lindemann
Title: Towards a better understanding of the moduli space of projective
special real manifolds
Abstract: A projective special real (PSR) manifold is a hypersurface H that
is contained in the positive level set of a real hyperbolic homogeneous cubic
polynomial h, such that every point in H is a hyperbolic point of h. Their study
is motivated by supergravity theories, where they appear as the scalar manifolds
in 4+1 dimensional supergravity. Via the so-called r- and q-map constructions
from supergravity, PSR manifolds give rise to projective special Kaehler and
quaternionic Kaehler manifolds, respectively. The q=c¥circ r-map in particular
allows for the explicit examples of locally non-homogeneous complete non-compact
quaternionic Kaehler manifolds of negative scalar curvature. Furthermore, PSR
manifolds appear in the study of the geometry of Kaehler cones of compact Kaehler
3-folds as level sets of the volume form. A general classification of PSR manifolds
would require a classification of hyperbolic cubic polynomials or, equivalently,
hyperbolic symmetric 3-tensors in any dimension, which is currently out of reach.
By restricting the considered PSR manifolds, e.g. their dimension, one can however
obtain partial results. Requiring that the PSR manifold is geodesically complete
with respect to its centro-affine metric allows one to find a nice generating
set of such PSR manifolds in every dimension. This allows for an easier study
of the moduli space of PSR manifolds under linear transformations of their ambient
space. I will give an overview over the most important known classification
results for PSR manifolds, present new results about the asymptotic geometry
of geodesically complete PSR manifolds, and give an outlook in which I will
explain the consequences of these results for the properties of the moduli space
PSR manifolds. Lastly, I will discuss some open problems.
Atsuhira Nagano
Title: Families of K3 surfaces, theta functions and invariants of complex
reflection groups
Abstract: In this talk, the speaker will introduce families of lattice polarized
K3 surfaces which are closely related to complex reflection groups. We will
study the period mappings of these families of K3 surfaces. Hilbert, Siegel
and Hermitian modular forms appear as the inverse correspondences of the period
mappings. We will give exact expressions of the modular forms using theta functions
and invariants of appropriate complex reflection groups. The main result, which
gives a natural extension of the classical expression of the period mapping
of elliptic curves using the Jacobi theta functions, is expected to have some
applications.
Yuichi Nohara
Title: Lagrangian mutations on Grassmannians and cluster transformations
Abstract: The Landau-Ginzburg mirror of the Grassmannian Gr(k, n) of k-planes
is a pair of a Zariski open set of the dual Grassmannian Gr(n-k, n) and a Laurent
polynomial of Plucker coordinates. The coordinate ring of the mirror manifold
admits a cluster algebra structure such that Plucker relations are exchange
relations of the cluster variables. We show that the Floer theoretic wall-crossing
formula for certain Lagrangian tori in the Grassmannian Gr(k, n) gives the exchange
relations in the mirror side. This is a joint work with Kazushi Ueda.
Alexis Roquefeuil
Title: q-oscillatory integrals and confluence in quantum K-theory
Abstract: In the context of mirror symmetry for quantum cohomology of a toric
manifold X, we will focus on two mirrors to Givental's J-function, given by
the I-function and the oscillatory integral. The comparison between these pieces
of data was obtained in details by H. Iritani and involves an interesting characteristic
class of X, called the gamma class. In this talk, we will try to obtain a K-theoretic
analogue of Iritani's comparison. More precisely, we will define a q-analogue
of the oscillatory integral and, when X has Picard rank 2, explain how it is
related to the K-theoretic I-function. Our identity will involve a K-theoretic
version of the gamma class. Finally, starting from K-theoretic data, we will
explain how to use the confluence of q-difference systems to recover every cohomological
analogues. This talk is based on a joint work with T. Milanov.
Ian Strachan
Title: From Frobenius manifolds to hyperKahler geometry via Donaldson-Thomas
invariants
Abstract: In the theory of Frobenius manifolds a connection with a regular and
an irregular singularity, with associated Stokes phenomena, plays a fundamental
role. In this talk the link between Donaldson- Thomas (DT) invariants and such
isomonodromy problems - with an infinite dimensional Lie algebra - is studied.
The DT-invariants control the Stokes factors between sectors, and the various
objects can be combined to form what is called a Joyce structure, and this in
turn defines a (complex) hyperKahler structure on a certain tangent bundle TM.
Finally, borrowing ideas from the deformation quantisation programme, the relationship
between quantum DT-invariants and Moyal-deformations of hyperKahler structures
is studied. (joint work with Tom Bridgeland)
Andrew Swann
Title: Geometric T-duality and the c-map
Abstract: The c-map is a construction from supergravity that takes what is known
as a projective special Kaehler manifold of dimension 2n and produces a quaternionic
Kaehler manifold of dimension 4n+4. Special Kaehler manifolds are bases of holomorphic
completely integrable systems. The talk will give an overview of how the c-map
may be described geometrically and of certain aspects of its naturality. One
essential ingredient is a type of geometric T-duality applied to a circle symmetry
of the integrable system.