Mirror Symmetry and Related Topics 2017
Dates: 11 – 15 December 2017
Venue: Conference Room 127, Department of Mathematics(Graduate School of Science Bldg no.3), Kyoto University
Talks
Mohammed Abouzaid
Renzo Cavalieri
Emily Clader
Sheel GanatraJérémy Guéré
Kentaro Hori
Bumsig Kim
Tatsuki Kuwagaki
Siu-Cheong Lau
Kaoru Ono
Thomas Prince
Konstantze Rietsch
Dusty Ross
Helge Ruddat
Nick Sheridan
Constantin Teleman
Dimitri Zvonkine

Homological Mirror Symmetry without corrections
Graph formulas for tautological classes
Wall-Crossing in Higher-Genus Quasimap Theory
Liouville sectors and localizing Fukaya categories
From DR hierarchies to tautological classes
Mirror Symmetry — from 3d to 2d
Virtual cycles via two-periodic localized Chern characters
Fukaya category of the projective line andmicrolocal sheaf theory
Moduli of Lagrangian immersions in pair-of-pants
decompositions and mirror symmetry
Anti-symplectic involution and twisted sectors
in Lagragian Floer theory
Lagrangian torus fibration models of Fano threefolds
Mirror Symmetry for Grassmannians
Higher-genus global mirror symmetry
Analyticity of canonical Calabi-Yau familiesSymplectic mapping class groups andhomological mirror symmetry
Coulomb branches and gauged sigma-models
Witten’s r-spin class and the cohomology of M g,nSchedule1
9:00-9:45 10:00-11:00 11:30-12:30
Mon disc
disc
disc
Tue disc
Ganatra
Lau
RietschTelemanWed disc
Thu disc
Zvonkine
PrinceFri
disc
GuéréRoss

14:00-15:00
Abouzaid
SheridanRuddatCladerCavalieri

15:30-16:30
Ono
Kuwagaki
Hori
Kim
disc

Acknowledgments: This conference is supported by JSPS KAKENHI Kiban-S16H06335 (Atsushi Moriwaki), Kiban-S 16H06337 (Atsushi Takahashi), Kiban-S
25220701 (Shigeru Mukai), Kiban-C 16K05127 (Hiroshi Iritani).
Organizers: Hiroshi Iritani, Yukiko Konishi, Hiraku Nakajima, Atsushi Takahashi.
https://www.math.kyoto-u.ac.jp/ iritani/mirrorsymmetry2017/mirrorsymmetry 20171

disc=“discussion”

1Title and Abstracts• Mohammed Abouzaid (Columbia University)
Title: Homological Mirror Symmetry without corrections
Abstract: I will explain how to prove HMS for symplectic manifolds admitting
Lagrangian torus ﬁbrations without singularities.
• Renzo Cavalieri (Colorado State University)Title: Graph formulas for tautological classes
Abstract: The tautological ring of the moduli space of curves is a subring of
the Chow ring that, on the one hand, contains many of the classes represented
by “geometrically deﬁned” cycles (i.e. loci of curves that satisfy certain geometric properties), on the other has a reasonably manageable structure. By
this I mean that we can explicitly describe a set of additive generators, which
are indexed by suitably decorated graphs. The study of the tautological ring
was initiated by Mumford in the ’80s and has been intensely studied by several groups of people. Just a couple years ago, Pandharipande reiterated that
we are making progress in a much needed development of a “calculus on thetautological ring”, i.e. a way to eﬀectively compute and compare expressions
in the tautological ring. An example of such a “calculus” consists in describing
formulas for geometrically described classes (e.g. the hyperelliptic locus) via
meaningful formulas in terms of the combinatorial generators of the tautological ring. In this talk I will explain in what sense “graph formulas” give a good
example of what the adjective “meaningful” meant in the previous sentence,and present a few examples of graph formulas. The original work presented isin collaboration with Nicola Tarasca and Vance Blankers.
• Emily Clader (San Fransisco State University)
Title: Wall-Crossing in Higher-Genus Quasimap Theory
Abstract: Quasimaps are a generalization of stable maps that depend on theadditional datum of a positive rational number epsilon. The dependence of thetheory on epsilon is encoded in certain wall-crossing formulas, ﬁrst conjectured
by Ciocan-Fontanine and Kim and since proved in many cases, which are
intimately related to mirror symmetry. I will discuss an alternative proofof Ciocan-Fontanine–Kim’s wall-crossing theorem for all-genus quasimaps tocomplete intersections in projective space, which has the advantage that it
can be adapted to prove an analogous theorem for more general gauged linear
sigma models. This is joint work with Felix Janda and Yongbin Ruan.
• Sheel Ganatra (University of Southern California)
Title: Liouville sectors and localizing Fukaya categories
Abstract: We introduce a new class of Liouville manifolds-with-boundary,called Liouville sectors, and show they have well-behaved, covariantly functorial Fukaya/Floer theories. Stein manifolds frequently admit coverings by
Liouville sectors, which can then be used to study the Fukaya category of the2

total space. Our ﬁrst main result in this setup is a local criterion for generating (global) Fukaya categories. One of our goals, using this framework, isto obtain a combinatorial presentation of the Fukaya category of any Steinmanifold. This is joint work with John Pardon and Vivek Shende.
• Jérémy Guéré (Université Grenoble Alpes)
Title: From DR hierarchies to tautological classes
Abstract: In 2014, Alexandr Buryak constructed, for each cohomological ﬁeldtheory (CohFT), a new integrable hierarchy based on the geometry of the double ramiﬁcation cycle and called the DR hierarchy. He then conjectured that
it is equivalent to a construction of Dubrovin-Zhang for semi-simple CohFTs.
In this talk, we recall the main steps of these two constructions and we will
see how their conjectural equivalence leads to conjectural relations in the tautological rings of the moduli spaces of stable curves. It is a joint work withBuryak, Dubrovin, and Rossi.
• Kentaro Hori (Kavli IPMU)
Title: Mirror Symmetry — from 3d to 2d
Abstract: I will talk about an attempt to understand 2d mirror symmetry
from 3d mirror symmetry.
• Bumsig Kim (Korea Institute for Advanced Study)
Title: Virtual cycles via two-periodic localized Chern characters
Abstract: The localized Chern character of a bounded complex of vector bundles is a bivariant class deﬁned by Baum, Fulton, and MacPherson. They usedsuch classes to prove a general Riemann-Roch theorem for singular varieties.
For a two-periodic complex of vector bundles, Polishchuk and Vaintrob haveconstructed its localized Chern character. This is a generalization of the usual
one. We discuss some properties of PV’s localized Chern characters. In particular, the cosection localization deﬁned by Kiem and Li can be expressed
as a localized Chern character operation. This result is a generalization of
the related work by Chang, Li, and Li. The talk is based on joint work withJeongseok Oh.
• Tatsuki Kuwagaki (Kavli IPMU)
Title: Fukaya category of the projective line and microlocal sheaf theory
Astract: For a cotangent bundle T ∗ X, Nadler-Zaslow showed that the derived
Fukaya category is equivalent to the derived category of constructible sheaves
over the base X. On the other hand, Guillermou’s sheaf quantization produces
a sheaf over X × R from an exact Lagrangian in T ∗ X, which is considered as
the Novikov-enhanced version of NZ-equivalence (some central ideas are dueto Tamarkin). In this talk, I’ll report my work in progress constructing acandidate of sheaf quantization of the great circle in the projective line, whichexhibits some properties which appear in the Fukaya category of the projectiveline.
3• Siu-Cheong Lau (Boston University)
Title: Moduli of Lagrangian immersions in pair-of-pants decompositions andmirror symmetry
Abstract: We construct a quantum-corrected moduli of Lagrangian immersions in a pair-of-pants decomposition and study mirror symmetry. The key ingredient is gluing between local moduli constructed in previous works. Novikov
ring plays an essential role in the construction, even though the exact Lagrangians only correspond to the complex-valued part. It is a joint work with
Cho and Hong.
• Kaoru Ono (Kyoto University, RIMS)
Title: Anti-symplectic involution and twisted sectors in Lagragian Floer theory• Thomas Prince (University of Oxford)
Title: Lagrangian torus fibration models of Fano threefolds
Abstract: Inspired by the work of Gross on topological Mirror Symmetry weconstruct candidate Lagrangian torus ﬁbration models for the 105 familiesof smooth Fano threefolds. In the case the second Betti number is one, weshow that the total space of each ﬁbration is homeomorphic to the expected
Fano threefold, and show that the numerical invariants coincide for all 105.
Our construction relies on a notion of toric degeneration for aﬃne manifolds
with singularities, and the correspondence we obtain between polytopes andFano manifolds is compatible with that appearing in the work of Coates-CortiKasprzyk et al. on Mirror Symmetry for Fano manifolds.
• Konstantze Rietsch (King’s College London)
Title: Mirror Symmetry for Grassmannians
Abstract: In joint work with R. Marsh we wrote down a rational function
on a Grassmannian, the ‘superpotential’, and showed how it can be used todescribe Gromov-Witten invariants of the (Langlands dual) Grassmannian via
a Dubrovin/Givental style of mirror symmetry construction. In this talk I will
brieﬂy report on these results, and then talk about joint work with L. Williams
which makes use of the same superpotential, but in a very diﬀerent way, toconstruct a class of Newton-Okounkov convex bodies.
• Dusty Ross (San Fransisco State University)
Title: Higher-genus global mirror symmetry
Abstract: Over ten years ago, Chiodo and Ruan proved a genus-zero global
mirror theorem, relating the Gromov-Witten invariants of the quintic threefold to the corresponding Fan-Jarvis-Ruan-Witten invariants. Moreover, theysuggested that the genus-zero relationship quantizes to an all-genus statement.
In this talk, I’ll describe recent work with Shuai Guo and Huailiang Chang
to compute higher-genus FJRW invariants and to verify the quantized global
mirror theorem.

4• Helge Ruddat (Johannes Gutenberg Universität Mainz)
Title: Analyticity of canonical Calabi-Yau families
Abstract: Gross and Siebert gave an algorithm to produce from toric degeneration data a canonical formal Calabi-Yau family. Siebert and I prove that thisfamily is in fact the completion of an analytic family. In particular, its nearby
ﬁbres are decent Calabi-Yau manifolds over the complex numbers. Furthermore, the family is semi-universal, i.e. locally the moduli space of Calabi-Yaus.
The key result on the route to analyticity is the computation of canonical coordinates on the base by explicit integration of a holomorphic volume form
over topological cycles that we construct from tropical 1-cycles in the base ofthe SYZ-ﬁbration.
• Nick Sheridan (University of Cambridge)
Title: Symplectic mapping class groups and homological mirror symmetry
Abstract: I will explain how one can get new information about symplectic
mapping class groups by combining two recent results: a proof of homologicalmirror symmetry for a new collection of K3 surfaces (joint work with Ivan
Smith), together with the computation of the derived autoequivalence groupof a K3 surface of Picard rank one (Bayer–Bridgeland). For example, it ispossible to give an example of a symplectic K3 whose smoothly trivial symplectic mapping class group (the group of isotopy classes of symplectic automorphisms which are smoothly isotopic to the identity) is inﬁnitely-generated.
This is joint work with Ivan Smith.
• Constantin Teleman (University of Oxford)
Title: Coulomb branches and gauged sigma-models
Abstract: There has been a recent increase in interest in constructing “Coulombbranches” for 3-dimensional gauge theory with matter, but the mathematicalmeaning of these objects seems less widely known. The Coulomb branch
for pure gauge theory was constructed by Bezrukavnikov, Finkelberg andMirkovic, and characterised by the speaker as a classifying space for gauged
2-dimensional topological ﬁeld theories, speciﬁcally gauged quantum cohomology. Given this universal description, the existence of additional Coulomb
branches seems mysterious. In this lecture, I will describe the construction of
the more general Coulomb branches from the pure one and the gauged mirror of the matter representation, and its related interpretation as a TQFT
classifying space.
• Dimitri Zvonkine (Versailles University and CNRS)
Title: Witten’s r-spin class and the cohomology of M g,nAbstract: In the introductory part of the talk I will introduce the tautologicalcohomology ring of the moduli space M g,n of stable curves and its naturalgenerators. Two natural problems arise: what are the relations between thesegenerators and how to express natural cycles in M g,n via these generators. I
will explain how Witten’s r-spin class helps in solving both problems.
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