Mirror Symmetry and Related Topics, 2019
Abstracts
Dylan Allegretti
Stability conditions, cluster varieties, and WKB analysis
This lecture series will focus on the relationship between two spaces associatedto a quiver with potential. The first is a complex manifold parametrizing Bridgelandstability conditions on a triangulated category, and the second is a cluster variety witha natural Poisson structure. Work of Gaiotto, Moore, and Neitzke in physics suggeststhat these spaces are related in a highly nontrivial way involving Donaldson-Thomasinvariants and the Kontsevich-Soibelman wall-crossing formula. I will explain recent work,
partly in collaboration with Tom Bridgeland, which provides a way of understanding
this relationship rigorously in the case where the quiver with potential arises from a
triangulation of a surface.
Lecture I: Bridgeland stability conditions
I will describe stability conditions on an important class of triangulated categories
constructed from quivers with potential. A particularly interesting set of examples comesfrom considering triangulations of surfaces.
Lecture II: Bridgeland stability conditions, continued
I will discuss the relationship between the stability conditions of the previous lecture
and meromorphic quadratic diﬀerentials on Riemann surfaces.
Lecture III: Voros symbols and cluster varieties
I will describe the relationship between stability conditions and the cluster variety
associated to a triangulated surface. This relationship involves in an essential way thenotion of a Voros symbol from the theory of exact WKB analysis.
Alexey Basalaev
Opposite filtrations in mirror symmetry for toric varieties
Mirror symmetry between a Landau-Ginzburg B–model and Gromov–Witten A–
model is now established for a wide class of toric varieties. In order to establish thismirror symmetry one has to make the certain addition choice on the B–side, know as anopposite filtration. The certain choice of it was made in the proof of mirror symmetryreferenced above. At the same time a B–side Landau–Ginzburg model has a natural
canonical opposite filtration due to M. Saito. In this talk we will identify when thefiltration used in mirror symmetry coincides with the M. Saito canonical filtration.
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Masahiro Futaki
Homological mirror symmetry for the projective space and an equivariant
version
Homological mirror symmetry has been proved for various Fano cases. I will review
some known results and then explain how a partially wrapped Fukaya category arises when
considering an equivariant version of it. This is a joint work in progress with Fumihiko
Sanda (Nagoya University).
Akishi Ikeda
On classification of simply-laced generalized root systems of type A via marked
bordered surfaces
The notion of a simply-laced generalized root system was introduced by Kyoji Saito
in his study of Milnor lattices of isolated singularities. In this talk, we construct generalized root systems of type A from marked bordered surfaces and show that there is abijection between generalized root systems of type A and marked bordered surfaces. Iftime permits, we also see the relationship with Hurwitz spaces and the categorification
via gentle algebras based on the work of Haiden-Katzarkov-Kontsevich.
Shunsuke Kano
Categorical entropy of pseudo-Anosov mapping classes of marked surfaces
Categorical entropy, which is introduced by Dimitrov-Haiden-Katzarkov-Kontsevich,is a categorical analog of topological entropy. Pseudo-Anosov mapping class is the mostgeneral type in mapping classes of surfaces. For a punctured surface with an ideal triangulation, we can associate a 3d CY category as a derived category of some dg algebra. We
construct a canonical lift of a pseudo-Anosov mapping class to a autoequivalence of thatcategory, and compare the categorical entropy of this lift and the topological entropy ofthe original pseudo-Anosov mapping class.
Omar Kidwai
Spectral networks, cluster charts, and Grassmannians
Spectral networks are certain graph-like objects on a Riemann surface C introduced
by Gaiotto-Moore-Neitzke (and in an earlier context, Berk-Nevins-Roberts) that describe
both the (exact WKB) asymptotics of diﬀerential equations and the “BPS spectrum” ofcertain 4d N=2 QFTs. One mathematical application of a spectral network is the socalled “abelianization map”, which provides nice coordinate systems on various moduli
spaces of flat connections on C. But many mathematical properties of spectral networks

3and abelianization remain to be established in generality - in particular the question of theexistence and uniqueness of this abelianization process is open in general. I will describe
recent progress on a particular instance of this question, as it arises in the study of somenetworks for moduli spaces which are (roughly) Grassmannians, and discuss the relation
between the spectral coordinates and the well-known cluster structure on these varieties.
Kohei Kikuta
Metric properties of the space of stability conditions
Toward geometric-group-theoretic study for autoequivalence groups of K3 surfaces,
we consider non-positive curvature properties of some metrics on the space of Bridgelandstability conditions. The results due to Allcock and Bayer-Bridgeland imply CAT(0)property of the pull-back metric via the covering map from the space of stability conditionsof K3 surfaces of Picard rank 1. On the other hand, we can show that a metric defined byBridgeland is neither CAT(0) nor Gromov hyperbolic for general triangulated categories.
We also discuss the hyperbolicity of pseudo-Anosov functors defined by Dimitrov-HaidenKatzarkov-Kontsevich, and compare two dynamical invariants (i.e. translation length andcategorical entropy) of them.
Yanki Lekili
Homological mirror symmetry via categorical resolutions
In the first two lectures, I will survey joint work with A. Polishchuk where weused categorical resolutions to prove homological mirror symmetry in various examples
including punctured surfaces, and higher dimensional pairs-of-pants and their covers.
The Symplectic Geometry of Higher Auslander Algebras
We show that the derived category of Iyama s d-dimensional Auslander algebra oftype A is equivalent to the partially wrapped Fukaya category of the d-fold symmetricproduct of a 2-dimensional disk with stops on its boundary. This allows us to give asimple proof of a derived Morita equivalence between higher Auslander algebras, produce
a paracyclic object equivalent to the d-dimensional Waldhausen S-construction, and more.
Joint work with T. Dyckerhoﬀ, G. Jasso.
Todor Milanov
Primitive forms and Frobenius structures on the Hurwitz spaces
The main goal of my lectures is to extend Saito’s theory of primitive forms andto introduce the notion of a primitive form for a generic family of Hurwitz covers of P 1
with a fixed profile at infinity. I am planning also to discuss two applications. The first4

one, is to give a construction of Dubrovin’s Hurwitz Frobenius manifolds via primitiveforms. The second one is to construct a certain set of oscillatory integrals, which could beviewed as an attempt to give a higher-genus generalization of Givental’s mirror symmetry
proposal for the J-function in Gromov–Witten theory.
Alexis Roquefueil
Quantum K-theory of projective spaces and confluence of q-diﬀerence equations
In 2004, Y.P. Lee defined new invariants, called K-theoretic Gromov–Witten invariants, as the Euler characteristic of some sheaves living on Kontsevich’s moduli space ofstable maps. A natural question is to understand how these invariants are related tothe usual Gromov–Witten invariants. A Riemann–Roch formula relating the two invariants was provided by A. Givental and V. Tonita in 2014, however this formula is quite
technical and has not seen many applications so far. One interesting consequence of
their formula is that Givental’s J-function satisfy functional equations called q-diﬀerence
equations. We suggest a new point of view to compare the invariants, by relating theq-diﬀerence module in quantum K-theory and the quantum diﬀerential module in quantum cohomology. For projective spaces, we show that the confluence principle in thetheory of q-diﬀerence equations provides a natural way to obtain the quantum diﬀerential
module and the cohomological J-function as a limit of the q-diﬀerence module and theK-theoretical J-function.
Ikuo Satake
Coxeter transformation and the Frobenius structure
It is known that the orbit space of the finite reflection group or the elliptic Weylgroup (with codimension 1) has the structure of Frobenius manifold. Here the Coxeter
transformation was used to show the non-degeneracy of the metric of the Frobenius structure. In this talk, we show that the Coxeter transformation is more useful, i.e. it gives thedirect descriptions of the flat coordinates and the potential of the Frobenius structure.
Kazushi Ueda
3d N = 2 Chern-Simons matter theory and K-theoretic quasimap invariantsof Grassmannians
We will discuss our joint work in progress with Yutaka Yoshida on the relation
between correlation functions of topologically twisted 3d N = 2 Chern-Simons mattertheory on S 1 times a Riemann surface and K-theoretic quasimap invariants. The former5has various parameters such as Chern-Simons levels and R-charges, which must be adjusted to particular values to match with the latter. If the time permits, we will also
discuss relations with Bethe/gauge correspondence, S 1 times D2 partition functions, andK-theoretic I-functions.