Gauge Theory, Moduli Spaces and Representation Theory,
Kyoto 2023
In honor of the 60th birthday of Hiraku NakajimaDates: 20-24 February 2023
Place: 420 RIMS, Kyoto University
Website:https://www.math.kyoto-u.ac.jp/~iritani/Nakajima60/Nakajima60_Kyoto.htm
Schedule:
20 Mon21 Tue22 Wed23 Thu24 Fri

9:30-10:30
Yoshioka
Honda
Nekrasov∗
Yang
Negut

11:00-12:00 13:30-14:30
CherkisDimofteKronheimer Gukov
Göttsche
OkounkovVasserot
GorskyBraverman

14:50-15:50
Yamakawa
Finkelberg

16:10-17:10
FukayaHausel∗Shan
Zhao

KimuraMcBreen(∗): Zoom talkTitle:
Alexander Braverman
Sergey Cherkis
Tudor Dimofte
Michael Finkelberg
Kenji Fukaya
Eugene Gorsky
Lothar Göttsche
Sergei Gukov
Tamas Hausel
Ko Honda
Yoshiyuki KimuraPeter Kronheimer
Michael McBreen
Andrei Negut
Nikita Nekrasov
Andrei Okounkov
Peng Shan
Eric Vasserot
Daisuke Yamakawa
Yaping Yang
Kota Yoshioka
Gufang Zhao

Gaiotto conjectures from geometric Langlands point of view
Up and down the bow construction
Vertex algebras in 3d gauge theory
Gaiotto conjectures for basic classical Lie supergroups
Generating function and Floer homology
The aﬃne Springer ﬁber – sheaf correspondence(Reﬁned) Verlinde and Segre formulas for Hilbert schemes of points
Where Quantum Topology Meets Geometric Representation Theory
Mirror symmetry and big algebras
Higher-dimensional Heegaard Floer homology and Hecke algebras
Twist automorphisms and dual canonical bases
Algebraic curves from instanton homology
Microlocal sheaves on aﬃne Slodowy slices
Quivers and quantum loop groups
Lax operators of integrable systems from intersection theory
The Eisenstein Spectrum
Modularity for W -algebras and aﬃne Springer ﬁbers
Critical convolution algebras and quantum loop groups
Stokes local systems and wild character varieties
An application of power operations to quantum groups
Moduli of stable sheaves on an elliptic surface
Quasimaps to quivers with potentials

Organizers: Tomoyuki Arakawa, Hiroshi Iritani, Syu Kato, Hiraku Nakajima, Atsushi Takahashi
Acknowledgments: This workshop is supported by JSPS KAKENHI 17H01086,
21H04993 (Tomoyuki Arakawa) 21H04994 (Atsushi Takahashi) and Kavli IPMU.

Alexander Braverman
Gaiotto conjectures from geometric Langlands point of view
This talk may be viewed as a continuation of M. Finkelberg’s talk, but I will try
to make it formally independent. In the ﬁrst half of the talk I will review the formalism of local quantum geometric Langlands correspondence for general reductive
groups. In the 2nd half of the talk I will discuss certain speciﬁc examples of thiscorrespondence having to do with super-groups whose even part is our original reductive group (these examples are mostly due to D. Gaiotto). I will explain how one
can derive the category equivalences from Finkelberg’s talk from these conjectures.
In addition, I will explain the relationship between the above conjectures and coset
constructions of certain super-analogs of vertex W -algebras.
Sergey Cherkis
Up and down the bow construction
We give an overview of the bow construction of instantons on Asymptotically
Locally Flat hyperKähler manifolds, with the view to its generalizations. From thispoint of view, deﬁning corresponding bow varies constitutes only a ﬁrst step in achain of generalizations of the Nakajima s quiver varieties. In the second half ofthe talk we focus on completeness of the bow construction and on demonstratingthat it establishes an isometry between bow moduli spaces and instanton moduli
spaces. These results were obtained in the joint work with Andres Larrain-Hubachand Mark Stern.
Tudor Dimofte
Vertex algebras in 3d gauge theory
Topological and/or holomorphic twists of 3d supersymmetric gauge theories admit boundary vertex algebras. The vertex algebras can be used to access features ofthe bulk theories — much as the chiral WZW model was used in the 80’s and 90’s
to access Chern-Simons TQFT. For example, it was shown by Costello-CreutzigGaiotto that boundary vertex algebras in 3d N = 4 theories can recover the watershed Braverman-Finkelberg-Nakajima construction of bulk Coulomb branches. I will
review constructions of these boundary vertex algebras, and present recent work on
their role in deﬁning braided tensor categories of line operators in 3d N = 4 theories,
and proving 3d mirror symmetry thereof. (Based on joint work with T. Creutzig,
N. Garner, and N. Geer; and A. Ballin, T. Creutzig, and W. Niu.)
Michael Finkelberg
Gaiotto conjectures for basic classical Lie supergroups
In the framework of the Gaiotto-Witten S-duality of boundary conditions in
N = 4 super Yang-Mills theory, the boundary conditions arising from the basic
classical Lie supergroups are S-dual to the ones arising from certain equivariant
slices. D. Gaiotto has upgraded this S-duality to a (conjectural) equivalence of therepresentation category of the corresponding quantum group and a certain categoryof twisted D-modules on the corresponding aﬃne Grassmannian on the S-dual side.

The Gaiotto equivalences specialize to certain particular cases of the Fundamental Local Equivalence (Gaitsgory-Lurie) and of the local version of the Periods–Lfunctions duality (Ben-Zvi-Sakellaridis-Venkatesh). I will report on our joint workwith A. Braverman, V. Ginzburg and R. Travkin in this direction.
Kenji Fukaya
Generating function and Floer homology
To study Lagrangian submanifolds of the cotangent bundle there are two ways,
one is using Floer homology and the other uses generating functions. The relation
between these two methods are clariﬁed by Viterbo and Milkovic-Oh in the caseof exact embedded Lagrangian submanifolds. Including immersed cases there aremany works in one and two dimensional cases. In this talk based on a joint workwith A. Daemi I will explain a way to study in the case of higher dimensional exact
immersion.
Eugene Gorsky
The aﬃne Springer ﬁber – sheaf correspondence
Given a semisimple element in the loop Lie algebra of a reductive group, we construct a quasi-coherent sheaf on a partial resolution of the trigonometric commuting
variety of the Langlands dual group. The construction uses aﬃne Springer theory
and Coulomb branch algebras of Braverman-Finkelberg-Nakajima. We also consider
a quantization of this construction for homogeneous elements. This is a joint workwith Oscar Kivinen and Alexei Oblomkov.
Lothar Göttsche(Reﬁned) Verlinde and Segre formulas for Hilbert schemes of points
This is joint work with Anton Mellit. Segre and Verlinde numbers of Hilbert
schemes of points have been studied for a long time. The Segre numbers are evaluations of top Chern and Segre classes of so-called tautological bundles on Hilbert
schemes of points. The Verlinde numbers are the holomorphic Euler characteristics
of line bundles on these Hilbert schemes. We give the generating functions for the
Segre and Verlinde numbers of Hilbert schemes of points. The formula is provenfor surfaces with KS2 = 0, and conjectured in general. Without restriction on KS2
we prove the conjectured Verlinde-Segre correspondence relating Segre and Verlinde
numbers of Hilbert schemes. Finally we ﬁnd a generating function for ﬁner invariants, which specialize to both the Segre and Verlinde numbers, giving some kind of
explanation of the Verlinde-Segre correspondence.
Sergei Gukov
Where Quantum Topology Meets Geometric Representation Theory
Tamas Hausel
Mirror symmetry and big algebras
First we recall the mirror symmetry identiﬁcation of the coordinate ring of certain very stable upward ﬂows in the Hitchin system and the Kirillov algebra forthe minuscule representation of the Langlands dual group via the equivariant cohomology of the cominuscule ﬂag variety (e.g. complex Grassmannian). In turn we
discuss a conjectural extension of this picture to non-very stable upward ﬂows interms of a big commutative subalgebra of the Kirillov algebra, which also ringiﬁes
the equivariant intersection cohomology of the corresponding aﬃne Schubert variety.
Ko Honda
Higher-dimensional Heegaard Floer homology and Hecke algebras
Hecke algebras are ubiquitous in number theory and geometric representation
theory. In this talk we describe the appearance of various Hecke algebras suchas the aﬃne Hecke algebra and the double aﬃne Hecke algebra (DAHA) in Floer
theory, through the higher-dimensional analog of Heegaard Floer homology. This is
joint work with Yin Tian and Tianyu Yuan.
Yoshiyuki Kimura
Twist automorphisms and dual canonical bases
Originally, twist automorphisms for unipotent cells are introduced by BerensteinFomin-Zelevinsky in the study of total positivity of Schubert varieties. In the talk,we study the quantum analogue of the twist automorphism of unipotent cells andtheir generalization to (upper) cluster algebras and quantization. In particular, weshow that the quantum twist automorphism preserves the dual canonical bases andtheir generalization (called common triangular bases). This is a joint work with
Hironori Oya; and with Fan Qin and Qiaoling Wei.
Peter Kronheimer
Algebraic curves from instanton homology
Instanton homology assigns to a knot or link K in a 3-manifold Y a vector spaceI(Y, K), and in a version with local coeﬃcients this is a module for a ring of Laurent
polynomials in a variable T . It also carries certain natural operators related to thehomology of Y and K. In this talk we will describe a computation of this instanton
homology, as a module over the polynomial algebra generated by T and these naturaloperators, in the case of the very simplest braids K in S 1 ×S 2 . This builds on earlier
work (with Q coeﬃcients) by Street and Xie-Zhang, and on the work of Munoz forthe case that K and T are absent. Our answer expresses I(Y, K) in these cases as
the structure sheaf of an algebraic curve with an interesting singular structure. Thisis joint work with Tom Mrowka, where one eventual aim is to answer topologicalquestions about immersed surfaces in 4-manifolds. A sample question is whether asmooth quintic surface can contain a rational curve of large degree.
Michael McBreen
Microlocal Sheaves on Aﬃne Slodowy Slices
I will describe certain moduli of wild higgs bundles on the line, and explain
why they are aﬃne analogues of Slodowy slices. I will then describe an equivalence
between microlocal sheaves on a particular such space and a block of representationsof the small quantum group. Joint work with Roman Bezrukavnikov, Pablo Boixeda
Alvarez and Zhiwei Yun.

Andrei Negut
Quivers and quantum loop groups
We propose a general framework that starts with a quiver whose arrows are labeled by arbitrary elements of a ground ﬁeld, and outputs a quantum loop group.
The gist of the construction is to use shuﬄe algebras to motivate the appropriate relations one needs to impose between the generators of such a quantum loop
group (for quivers of ﬁnite type, these are the famous Drinfeld-Serre relations). The
construction is completely explicit in many cases of interest, such as Hall algebrasof curves over ﬁnite ﬁelds, localized equivariant K-theoretic Hall algebras, quantum loop groups associated to symmetric Cartan matrices, and the quiver quantum
toroidal algebras from mathematical physics.
Nikita Nekrasov
Lax operators of integrable systems from intersection theory
Classical (and quantum) integrable systems of Garnier (Gaudin) and elliptic
Calogero-Moser type have been long expected to capture the Seiberg-Witten geometry of A-type quiver N = 2 gauge theories in four dimensions. The traditional
approach to integrable systems casts them in the Lax form. We present the construction of the associated Lax operators from the intersection theory on the moduli
space of framed parabolic sheaves on the projective plane. In gauge theory terms,
this is related to the studies of surface defects in (Omega-deformed) linear or cyclic
quiver N = 2 theory. Based on the works with I. Krichever and A. Grekov.
Andrei Okounkov
The Eisenstein Spectrum
This will be a report about our ongoing work with David Kazhdan devoted tothe spectral analysis of the span of the Eisenstein series.
Peng Shan
Modularity for W -algebras and aﬃne Springer ﬁbers
We will explain a bijection between admissible representations of aﬃne KacMoody algebras and ﬁxed points in aﬃne Springer ﬁbers. We will also explain
how to match the modular group action on the characters with the one deﬁned by
Cherednik in terms of double aﬃne Hecke algebras, and extensions of these relations
to representations of W -algebras. This is based on joint work with Dan Xie andWenbin Yan.
Eric Vasserot
Critical convolution algebras and quantum loop groups
We introduce a new family of algebras attached to quivers with potentials, using
critical K-theory and critical Borel-Moore homology, which generalize the convolution algebras attached to quivers deﬁned by Nakajima. We give some applications
to cohomological and K-theoretical Hall algebras, to shifted quantum loop groups,
to Kirillov-Reshetikhin and to prefundamental representations.
Daisuke Yamakawa

Stokes local systems and wild character varieties
By the Riemann-Hilbert correspondence, regular singular meromorphic connections on a compact Riemann surface are classiﬁed by the corresponding local systems
on the surface minus singular points. In the irregular singular case, connections areclassiﬁed by the corresponding Stokes local systems, which is the rephrasing of the
Riemann-Hilbert-Birkhoﬀ correspondence. The wild character varieties are deﬁned
as the aﬃne GIT quotients of the ”wild representation varieties” parameterizing
framed Stokes local systems, modulo the change of framing. Thus points on the wild
character varieties correspond to closed orbits in the wild representation varieties. I
will review our joint work with Philip Boalch on the wild character/representation
varieties.
Yaping Yang
An application of power operations to quantum groups
I will talk about the quantum group associated to a quiver and an oriented
cohomology theory A∗ , which is constructed via the A∗ -cohomology of the moduli
of representations of the preprojective algebra of the quiver. I will explain power
operations in cohomology theories, which in special cases are Steenrod operations in
mod-p cohomology and Adams operations in K-theory. I will give one applicationof power operations in the setting of quiver with automorphism. In particular, it
recovers Lusztig’s quantum Frobenius homomorphism from the quantum group atroots of 1 to the enveloping algebra. This is based on my joint work with Gufang
Zhao.
Kota Yoshioka
Moduli of stable sheaves on an elliptic surface
Bridgeland studied moduli of stable sheaves on elliptic surfaces by using relative
Fourier-Mukai transforms. In particular a suitable relative Fourier-Mukai transform
induces a birational morphism between a moduli space of stable sheaves and theHilbert scheme of points. On the other hand, the bitarional morphism is not isomorphic in general, and the obtained moduli space parameterizes rank 1 sheaves
with torsions. By considering various Fourier-Mukai transforms, we have a collections of moduli spaces of rank 1 sheaves with torsions. I would like to talk about
relations of these moduli spaces.
Gufang Zhao
Quasimaps to quivers with potentials
This talk concerns non-compact GIT quotient of a vector space, in the presenceof an abelian group action and an equivariant regular function (potential) on thequotient. We deﬁne virtual counts of quasimaps from prestable curves to the criticallocus of the potential. The construction borrows ideas from the theory of gauged
linear sigma models as well as recent development in shifted symplectic geometryand Donaldson-Thomas theory of Calabi-Yau 4-folds. Examples of virtual counts
arising from quivers with potentials are discussed. This is based on work in progress,
in collaboration with Yalong Cao.