conference on

Special Geometry, Mirror Symmetry

and Integrable Systems

at waseda university (online)

Place Date Speakers (see below for program)

Waseda University


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



2 December


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 stating your name, university affiliation, and position/student status. Speakers do not need to do this - they are automatically registered.


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


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


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


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)


Differential Geometry and Integrable Systems - Mathematics of Symmetry, Stability and Moduli - RIMS Research Project 2020

RIMS International Joint Usage/Research Center


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.