Workshop on mirror symmetry and related topics, Kyoto 2016.
Dates: 12 December 2016 – 16 December 2016
Venue: Room127 (conference room), Department of Mathematics (Graduate School of Science
Bldg no.3), Kyoto University

Schedule:12 Mon
13 Tue14 Wed15 Thu
16 Fri9:00-10:00
Hosonofree
Coates1
ChanOh10:15–11:15
Odaka
free
KoderaFang
Ueda

13:30 – 14:30
NoharaGonzález
freeJohnson2
Sanda

14:45 – 15:45
TonitaJohnson1
freeCoates2
Takahashi

16:00 – 17:00
Ono1
Ono2
freeIke(closing)• Dinner Meeting (研究交流会): Monday from 18:00, University Cafeteria in the NorthCampus, 2nd ﬂoor (reservation required)• Banquet: Restaurant Mon, from 18:00 (reservation required)

Titles (in time order)
Shinobu Hosono
Yuji Odaka
Yuichi Nohara
Valentin Tonita
Kaoru Ono
Eduardo González
Paul Johnson
Tom Coates
Ryosuke Kodera

Birational geometry from the moduli spaces of mirror CICYs
Tropical Geometric Compactifications of Moduli spaces
Lagrangian fibrations on Grassmannians and cluster transformations
K-theoretic mirror formulae
Generation criterion for Fukaya category and related topics
The quantum Kirwan morphism and abelianisation of quotientsTBA (tropical approach to Gromov-Witten theory)
Mirror Symmetry and Fano Manifolds
Quantized Coulomb branches of Jordan quiver gauge theories andcyclotomic rational Cherednik algebras

Kwokwai Chan
Bohan Fang
Yuichi Ike
Jeongseok Oh
Kazushi Ueda
Fumihiko Sanda
Atsushi Takahashi

Revisiting the open mirror theorem
The conifold transition of a torus knot and open Gromov-Witten invariants
Categorical localization for the coherent-constructible correspondence
Quasimap theory for relative GIT quotientsCalabi-Yau 3-folds in Grassmannians of exceptional typeAn analog of the Dubrovin conjecture
On entropy of autoequivalences of smooth projective varieties

This workshop is supported by JSPS KAKENHI Kiban-S 16H06337 (Atsushi Takahashi) andKiban-S 16H06335 (Atsushi Moriwaki).
Organizers: Hiroshi Iritani, Yukiko Konishi, Atsushi Takahashi

1Abstracts (in alphabetical order)• Kwokwai Chan
Title: Revisiting the open mirror theorem
Abstract: In this talk, I will ﬁrst review an open analogue of the mirror theorem fortoric Calabi-Yau manifolds obtained in joint works with Cho, Lau, Leung and Tseng;this theorem gives an enumerative interpretation for inverse of mirror maps and also aneﬀective computation of disk counting invariants. Then I will explain a recent attempt inunderstanding this theorem from a diﬀerent perspective.
• Tom Coates
Title: Mirror Symmetry and Fano Manifolds• Bohan Fang
Title: The conifold transition of a torus knot and open Gromov-Witten invariants
Abstract: The large-N duality and the conifold transition of a knot in the 3-sphere producea Lagrangian submanifold in the resolved conifold. When the knot is a torus knot one candeﬁne open invariants w.r.t. this Lagrangian by localization or relative Gromov-Witten
theory. I will describe this procedure and the all genus mirror symmetry where the Bmodel is the topological recursion on the mirror curve. The B-side theory predicts these
open invariants. This talk is based on the joint work with Zhengyu Zong.
• Eduardo González
Title: The quantum Kirwan morphism and abelianisation of quotients
Abstract: Given a reductive G and a polarised projective G variety X there is a cobordism
relating the moduli of stable maps to the GIT quotient X//G with the moduli stack ofgauged maps with target X. This cobordism is itself a moduli of gauged maps enriched witha scaling. By letting the polarisation go to inﬁnity, we will see that this cobordism can beused to deﬁne a morphism from equivariant quantum K-theory (cohomology) of X to theusual quantum K-theory (cohomology) of the quotient, which relates the gauged potentialof X with the graph potential of X//G up to certain quantum corrections appearing as
counts of aﬃne gauged maps. We will see how this can be used to understand the relationof potentials for the GIT quotient of X//G and the GIT quotient X//T by the maximal
torus subgroup of G. This is joint work with C. Woodward and other parts also with P.
Solis.
• Shinobu Hosono
Title: Birational geometry from the moduli spaces of mirror CICYs
Abstract: It is known that birational geometry of Calabi-Yau manifolds naturally appears
when we describe mirror symmetry. In this talk, I will show some interesting examples
which have birational automorphisms of inﬁnite order, and identify them with monodromy(connection) transformations in their mirror families. If time permits, I will mention someobservations on Picard-Lefschetz transformations. This talk is based on collaborations
with Hiromichi Takagi.
• Yuichi Ike
Title: Categorical localization for the coherent-constructible correspondence
Abstract: The coherent-constructible correspondence is a version of homological mirror
symmetry for toric varieties. It states that the derived category of coherent sheaves ona complete toric variety is equivalent to that of constructible sheaves on the real torus2whose microsupports are contained in some Lagrangian. Even if the toric variety is notnecessarily complete, the category of wrapped constructible sheaves (which was recentlyintroduced by Nadler) is the mirror category in the coherent-constructible correspondence.
We prove categorical localization for categories of wrapped constructible sheaves, which canbe regarded as a microlocal counterpart of categorical localization for Fukaya categories.
This is a joint work with Tatsuki Kuwagaki.
• Paul Johnson
Title: TBA (tropical approach to Gromov-Witten theory)• Ryosuke Kodera
Title: Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras
Abstract: Braverman-Finkelberg-Nakajima gave a mathematically rigorous deﬁnition of
the Coulomb branches of 3d N = 4 supersymmetric gauge theories. They are certain
Poisson aﬃne algebraic varieties and admit natural quantizations. In this talk we consider
the quantized Coulomb branches associated with quiver gauge theories of Jordan type.
We prove that they are isomorphic to the spherical parts of cyclotomic rational Cherednik
algebras. This is a joint work with Hiraku Nakajima.
• Yuichi Nohara
Title: Lagrangian ﬁbrations on Grassmannians and cluster transformations
Abstract: For each triangulation of a convex n-gon, one can associate a Lagrangian torus
ﬁbration on the Grassmannian of 2-planes in an n-space. In this talk, we discuss a relation
between potential functions for the Lagrangian torus ﬁbers and cluster transformations on
the Landau-Ginzburg mirror of the Grassmannian.
• Yuji Odaka
Title: Tropical Geometric Compactiﬁcations of Moduli spaces• Jeongseok Oh
Title: Quasimap theory for relative GIT quotients
Abstract: Ciocan-Fontanine, Kim and Maulik invented a new cohomological ﬁeld theory
for GIT qoutient target space which is called quasimap theory. By several evidences,quasimap theory seems to be a suitable counterpart of Gromov-Witten theory in mirror
symmetry. For instance, Ciocan-Fontanine and Kim conjectured so called “wall crossing
formula conjecture” which describe a relationship between quasimap theory and GromovWitten theory and proved it for several interesting target spaces even for any genus.
On the other hand, wall crossing formula conjecture has exactly same form as BCOV
conjecture. In this sense, we can think quasimap theory as mathematically well-deﬁned
B-model theory.
For genus zero case, this behaves more well. In this case, they proved wall crossing formulaconjecture for any target space with a suitable torus action. Especially, wall crossing
formula recovers Givental’s mirror theorem for toric case. Also, wall crossing formula
recovers mirror theorem for ﬂag varieties.
In this talk, I will introduce a new cohomological ﬁeld theory for relative GIT space whichmeans a ﬁber bundle over smooth projective varieties with GIT quotient space as ﬁber.
We call it as quasimap theory either. Then, I will explain wall crossing formula for genus
zero and for relative GIT with torus action on its ﬁber. Moreover, I will introduce atwisted theory. Wall crossing formula can be clearly written for relative GIT with toric3

variety or ﬂag variety as its ﬁber. Finally, I will explain why it can be regarded as a mirror
theorem by showing that wall crossing formula recovers Brown’s theorem for relative GITwith toric variety as its ﬁber.
• Kaoru Ono
Title: Generation criterion for Fukaya category and related topics• Fumihiko Sanda
Title: An analog of the Dubrovin conjecture
Abstract: B. Dubrovin conjectured the equivalence between the semi-simplicity of thequantum cohomology of a Fano manifold and the existence of full exceptional ollection in
the derived category of coherent sheaves on it. He alsoconjectured the Stokes matrix of thequantum D-module can be escribed by the Euler pairings of the full exceptional collection.
Recently, this statement is reﬁned as a Gamma conjecture by Galkin-Golyshev-Iritani. In
this talk, I will speak about an analog of the Dubrovin conjecture for the case that thequantum cohomology is not necessarily semi-simple. This is a joint work in progress withY. Shamoto.
• Atsushi Takahashi
Title: On entropy of autoequivalences of smooth projective varieties
Abstract: Entropy for endofunctors of triangulated categories is deﬁned by DmitrovHaiden-Katzarkov-Kontsevich. Based on the joint work with Kohei Kikuta, the categoricalentropy of an surjective endomorphism of a complex smooth projective variety is shown
to be equal to its topological entropy, which is done by DHKK under a certain technicalcondition. It is natural to expect a generalization of the fundamental theorem by GromovYomdin: the entropy of an autoequivalence of a complex smooth projective variety should
be given by the logarithm of the spectral radius of the induced automorphism of the numerical Grothendieck group. This conjecture holds for elliptic curves (Kikuta’s result) andif the canonical or anti-canonical sheaf is ample.
• Valentine Tonita
Title: K-theoretic mirror formulae
Abstract: I will deﬁne permutation-equivariant K-theoretic Gromov-Witten invariants(introduced by Givental). Roughly speaking ”mirror formulae” means that certain qhypergeometric series associated to a manifold X are generating series of these invariants.
I will write these series for certain X (toric, zero sections of convex bundles) and, time
permitting, discuss the proofs of such results.
• Kazushi Ueda
Title: Calabi-Yau 3-folds in Grassmannians of exceptional types
Abstract: We discuss classiﬁcation of Calabi-Yau complete intersections deﬁned by equivariant vector bundles on homogeneous spaces of simple Lie groups of exceptional typeassociated with maximal parabolic subgroups. Only Grassmannians of types E6 and G2
have Calabi-Yau complete intersections of dimension 3. Those in G2 -Grassmannians are
deformation-equivalent to the famous Pfaﬃan-Grassmannian pairs of Calabi-Yau 3-folds.
They give rise to a pair of non-compact Calabi-Yau 7-folds, which are related by a ﬂop andare derived equivalent. They also give examples of annihilators of the class of the aﬃneline in the Grothendieck ring of varieties. This is a joint work with Atsushi Ito, Makoto
Miura, and Shinnosuke Okawa.

4