Representation theory, gauge theory, and integrable systems

3 Feb 2019, 09:30 → 8 Feb 2019, 17:00 Asia/Tokyo

Lecture Hall(1F), Kavli IPMU

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Dates: February 4 - 8, 2019

Venue: Lecture Hall, Kavli IPMU

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The workshop will focus on recent interface among representation theory, gauge theory and integrable systems. There are numerous connections in recent decades, such as computation of partition functions in gauge theories via representation theory and integrable systems, realization of representations of quantum algebras via moduli spaces in gauge theories, new examples of quantum algebras via gauge theory, and so on.
 

The workshop aims to bring together mathematicians and physicists, both experts and young people in these related areas from overseas and Japan to discuss new developments and investigate potential directions for future research.

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Invited Speakers:

Andrea Appel (University of Edinburgh, UK)

Mikhail Bershtein (Landau Institute for Theoretical Physics, Russia)

Sergei Gukov (Caltech, USA)

Tamás Hausel (IST, Austria)

Justin Hilburn (University of Pennsylvania, USA)

Seyed Morteza Hosseini (Kavli IPMU, Japan)

Hiroaki Kanno (Nagoya University, Japan)

Syu Kato (Kyoto University, Japan)

Taro Kimura (Keio University, Japan) 

Michael McBreen (University of Toronto, Canada)

Dinakar Muthiah (Kavli IPMU, Japan)

Nikita Nekrasov (SCGP, USA)

Du Pei (Caltech, USA)

Changjian Su (University of Toronto, Canada)

Takuya Okuda (University of Tokyo, Japan) 

Masahito Yamazaki (Kavli IPMU, Japan)

Shintaro Yanagida (Nagoya University, Japan)

Yaping Yang (University of Melbourne, Australia)

 

 

Organizers:
Hiraku Nakajima, Francesco Sala, Yuji Tachikawa, Yutaka Yoshida

 

Contact: seminar@ipmu.jp
Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU), the University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa City, Chiba 277-8583, Japan

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Monday 4 February

Registration

Nikita Nekrasov: Some applications of detects in supersymmetric gauge theory

I will explain the formula of Gamayun, Iorgov and Lysovyy relating Painleve VI tau-function to $c=1$ conformal blocks and some of it generalizations, using the blowup formulas for $N_f = 2N_c$ supersymmetric $N=2$ $d=4$ theory. If time permits I will talk about the eigenvalue problem for the elliptic Calogero-Moser system.

1 Some applications of defects in supersymmetric gauge theory

I will explain the formula of Gamayun, Iorgov and Lysovyy relating Painleve VI tau-function to $c=1$ conformal blocks and some of it generalizations, using the blowup formulas for $N_f = 2N_c$ supersymmetric $N=2$ $d=4$ theory. If time permits I will talk about the eigenvalue problem for the elliptic Calogero-Moser system.

Speaker: Nikita Nekrasov

10:30 Coffee and Tea Break

Yaping Yang: Cohomological Hall algebras, Vertex algebras and instantons

A new class of vertex operator algebras, vertex algebras at the corner, are recently introduced by Gaiotto and Rapčák, generalizing the affine W-algebra of $\mathfrak{gl}_N$. In my talk, I will discuss an action of this new vertex algebra on the cohomology of certain spiked instanton moduli spaces on 3CY manifold in the sense of Nekrasov. This action is naturally obtained using the cohomological Hall algebras of Kontsevich-Soibelman. This talk is based on my joint work with Miroslav Rapčák, Yan Soibelman, and Gufang Zhao.

2 Cohomological Hall algebras, vertex algebras and instantons

A new class of vertex operator algebras, vertex algebras at the corner, are recently introduced by Gaiotto and Rapčák, generalizing the affine W-algebra of $\mathfrak{gl}_N$. In my talk, I will discuss an action of this new vertex algebra on the cohomology of certain spiked instanton moduli spaces on 3CY manifold in the sense of Nekrasov. This action is naturally obtained using the cohomological Hall algebras of Kontsevich-Soibelman. This talk is based on my joint work with Miroslav Rapčák, Yan Soibelman, and Gufang Zhao.

Speaker: Yaping Yang

12:00 Lunch break

Masahito Yamazaki: Discrete Painlevé Equation and Four-dimensional Gauge Theories

We discuss special solutions for the Hirota-type bilinear identity for the E8 discrete Painlevé equation and its "lens-generalization". The key identity is provided by transformation formulas for the lens-elliptic gamma function, which were first found via the Seiberg dualities in $4d$ $\mathcal{N}=1$ theories, and studied in connection with the super-master solution of the star-triangle relation.

3 Discrete Painlevé Equation and Four-dimensional Gauge Theories

We discuss special solutions for the Hirota-type bilinear identity for the E8 discrete Painlevé equation and its "lens-generalization". The key identity is provided by transformation formulas for the lens-elliptic gamma function, which were first found via the Seiberg dualities in $4d$ $\mathcal{N}=1$ theories, and studied in connection with the super-master solution of the star-triangle relation.

Speaker: Masahito Yamazaki

15:00 Coffee and Tea Break

Seyed Morteza Hosseini: Five-dimensional topological indices and Bethe ansatz at large $N$

I will provide a general formula for the exact partition function of five-dimensional supersymmetric gauge theories on a four-manifold times a circle. The four-manifold is either toric Kaehler or a product of two Riemann surfaces. Then I will discuss the Bethe ansatz system associated to our partition functions at large $N$.

4 Five-dimensional topological indices and Bethe ansatz at large $N$

I will provide a general formula for the exact partition function of five-dimensional supersymmetric gauge theories on a four-manifold times a circle. The four-manifold is either toric Kaehler or a product of two Riemann surfaces. Then I will discuss the Bethe ansatz system associated to our partition functions at large $N$.

Speaker: Seyed Morteza Hosseini

Tuesday 5 February

Tamás Hausel: Intersection of mirror branes on Higgs moduli spaces

I will discuss a computational approach for the semiclassical limit of mirror symmetry for Higgs moduli spaces for Langlands dual groups, by comparing the equivariant indices of intersections of mirror branes.

5 Intersection of mirror branes on Higgs moduli spaces

I will discuss a computational approach for the semiclassical limit of mirror symmetry for Higgs moduli spaces for Langlands dual groups, by comparing the equivariant indices of intersections of mirror branes.

Speaker: Tamás Hausel

10:30 Coffee and Tea Break

Hiroaki Kanno: 3d holomorphic blocks from the intertwiner of quantum toroidal algebra

The intertwiner of the Fock representation of the quantum toroidal algebra of $\mathfrak{gl}_1$ type can be identified with the refined topological vertex, which is a building block of 5d lift of the Nekrasov instanton partition function. In general the correlation function of the intertwiners satisfies a difference equation of KZ type, where the associated R-matrix is featured. In this talk I will explain how we can derive generalized KZ equation for quantum toroidal algebra in a simplified setting and show that 3d holomorphic blocks arise as solutions to the equation.

6 3d holomorphic blocks from the intertwiner of quantum toroidal algebra

The intertwiner of the Fock representation of the quantum toroidal algebra of $\mathfrak{gl}_1$ type can be identified with the refined topological vertex, which is a building block of 5d lift of the Nekrasov instanton partition function. In general the correlation function of the intertwiners satisfies a difference equation of KZ type, where the associated R-matrix is featured. In this talk I will explain how we can derive generalized KZ equation for quantum toroidal algebra in a simplified setting and show that 3d holomorphic blocks arise as solutions to the equation.

Speaker: Hiroaki Kanno

12:00 Lunch break

Du Pei: Non-unitary modular categories from the Coulomb branch

We propose a new link between the geometry of moduli spaces and the representation theory of vertex operator algebras. The construction goes through a class of four-dimensional quantum field theories that are said to satisfy "property F". Each such theory gives rise to a family of modular tensor categories, whose algebraic structures are intimately related to the geometry of the Coulomb branch. This is based on joint work with Mykola Dedushenko, Sergei Gukov, Hiraku Nakajima and Ke Ye.

7 Non-unitary modular categories from the Coulomb branch

We propose a new link between the geometry of moduli spaces and the representation theory of vertex operator algebras. The construction goes through a class of four-dimensional quantum field theories that are said to satisfy "property F". Each such theory gives rise to a family of modular tensor categories, whose algebraic structures are intimately related to the geometry of the Coulomb branch. This is based on joint work with Mykola Dedushenko, Sergei Gukov, Hiraku Nakajima and Ke Ye.

Speaker: Du Pei

15:00 Coffee and Tea Break

Michael McBreen: Categorical Hikita duality

Symplectic duality predicts various relationships (often somewhat mysterious) between the Higgs and Coulomb branches of certain $3d$ $\mathcal{N}=4$ supersymmetric gauge theories. In particular, Hikita's conjecture relates the cohomology ring of the Higgs branch with the coordinate ring of the fixed scheme of the Coulomb branch, with respect to a torus action on the latter. I will discuss joint work with Roman Bezrukavnikov, which proposes a categorical analogue of this conjecture for abelian gauge theories. It relates constructible sheaves on the loop space of the Higgs branch with coherent sheaves on the fixed scheme of the Coulomb branch. I will give a combinatorial model for for categories and sketch a proof.

8 Categorical Hikita duality

Symplectic duality predicts various relationships (often somewhat mysterious) between the Higgs and Coulomb branches of certain $3d$ $\mathcal{N}=4$ supersymmetric gauge theories. In particular, Hikita's conjecture relates the cohomology ring of the Higgs branch with the coordinate ring of the fixed scheme of the Coulomb branch, with respect to a torus action on the latter. I will discuss joint work with Roman Bezrukavnikov, which proposes a categorical analogue of this conjecture for abelian gauge theories. It relates constructible sheaves on the loop space of the Higgs branch with coherent sheaves on the fixed scheme of the Coulomb branch. I will give a combinatorial model for for categories and sketch a proof.

Speaker: Michael McBreen

Wednesday 6 February

Takuya Okuda: SUSY localization for Coulom branch operators in 3 and 4 dimensions

We calculate, via SUSY localization, the correlators of the operators whose vevs parametrize the Coulomb branches. In $4d$, we review the computation of the correlators of Wilson-'t Hooft line operators in $N=2$ gauge theories on $S^1 \times \mathbb{R}^3$. The results involve $Z_{\text{mono}}$, the monopole analog of the Nekrasov instanton partition function. For a class $\mathcal{S}$ theory, the correlators describe deformation quantization of the Hitchin moduli space in terms of Fenchel-Nielsen coordinates. In $3d$, we compute correlators of dressed monopole operators in $N=4$ gauge theories on $\mathbb{R}^3$ with omega deformation and develop similar stories. We compare our results with those obtained in other approaches. Based on arXiv:1111.4221 with Ito and Taki, as well as on a work in progress with Y. Yoshida.

9 SUSY localization for Coulomb branch operators in 3 and 4 dimensions

We calculate, via SUSY localization, the correlators of the operators whose vevs parametrize the Coulomb branches. In $4d$, we review the computation of the correlators of Wilson-'t Hooft line operators in $N=2$ gauge theories on $S^1 \times \mathbb{R}^3$. The results involve $Z_{\text{mono}}$, the monopole analog of the Nekrasov instanton partition function. For a class $\mathcal{S}$ theory, the correlators describe deformation quantization of the Hitchin moduli space in terms of Fenchel-Nielsen coordinates. In $3d$, we compute correlators of dressed monopole operators in $N=4$ gauge theories on $\mathbb{R}^3$ with omega deformation and develop similar stories. We compare our results with those obtained in other approaches. Based on arXiv:1111.4221 with Ito and Taki, as well as on a work in progress with Y. Yoshida.

Speaker: Takuya Okuda

10:30 Coffee and Tea Break

Syu Kato: Frobenius splitting of semi-infinite flag manifolds

We explain that extremal weight modules of quantum loop algebras give rise to the projective coordinate ring of the formal model of the semi-infinite flag manifolds over the ring of integers with two inverted. Then, we exhibit how this gives rise to the Frobenius splitting of such an (ind-)scheme. This particularly implies that the Schubert varieties of the quasi-map spaces from a projective line to a (partial) flag manifold admits a Frobenius splitting compatible with the boundaries, and consequently such varieties are normal and has rational singularity in characteristic zero. This extends the case of the genuine quasi-map spaces by Braverman-Finkelberg and the asymptotic case by myself.

If time allows, we explain how to use such results to understand the structure of equivariant small quantum $K$-theory of a (partial) flag manifold.

10 Frobenius splitting of semi-infinite flag manifolds

We explain that extremal weight modules of quantum loop algebras give rise to the projective coordinate ring of the formal model of the semi-infinite flag manifolds over the ring of integers with two inverted. Then, we exhibit how this gives rise to the Frobenius splitting of such an (ind-)scheme. This particularly implies that the Schubert varieties of the quasi-map spaces from a projective line to a (partial) flag manifold admits a Frobenius splitting compatible with the boundaries, and consequently such varieties are normal and has rational singularity in characteristic zero. This extends the case of the genuine quasi-map spaces by Braverman-Finkelberg and the asymptotic case by myself.

If time allows, we explain how to use such results to understand the structure of equivariant small quantum $K$-theory of a (partial) flag manifold.

Speaker: Syu Kato

12:00 Lunch break

Mikhail Bershtein: Deatonomization of cluster integrable systems

To any Newton polygon one can assign the cluster integrable system. The group $G$ of discrete flows acts on the phase space, preserving the integrals of motion of the cluster integrable system. After deautonomization the action $G$ leads to $q$-difference equations, which are equations of isomonodromic deformations of linear $q$-difference equations. Finally, these equations can be explicitly solved using Nekrasov functions of $5d$ supersymmetric gauge theory or partition functions of topological strings. The Seiberg-Witten curve for corresponding supersymmetric gauge theory and toric Calabi-Yau are constructed from the initial Newton polygon.

Based on joint works with A. Marshakov and P. Gavrylenko.

11 Deatonomization of cluster integrable systems

To any Newton polygon one can assign the cluster integrable system. The group $G$ of discrete flows acts on the phase space, preserving the integrals of motion of the cluster integrable system. After deautonomization the action $G$ leads to $q$-difference equations, which are equations of isomonodromic deformations of linear $q$-difference equations. Finally, these equations can be explicitly solved using Nekrasov functions of $5d$ supersymmetric gauge theory or partition functions of topological strings. The Seiberg-Witten curve for corresponding supersymmetric gauge theory and toric Calabi-Yau are constructed from the initial Newton polygon.

Based on joint works with A. Marshakov and P. Gavrylenko.

Speaker: Mikhail Bershtein

15:00 Coffee and Tea Break

Shintaro Yanagida: Hall algebra of sheaves on abelian surfaces

I will give some explicit calculations on the Hall algebra of sheaves on abelian surfaces, and will explain their relationship to the elliptic integrable system of Macdonald-Ruijsenaars operators.

12 Hall algebra of sheaves on abelian surfaces

I will give some explicit calculations on the Hall algebra of sheaves on abelian surfaces, and will explain their relationship to the elliptic integrable system of Macdonald-Ruijsenaars operators.

Speaker: Shintaro Yanagida

Thursday 7 February

Dinakar Muthiah: Toward double affine flag varieties and Grassmannians

Recently there has been a growing interest in double affine Grassmannians, especially because of their relationship with Coulomb branches of quiver gauge theories. However, not much has been said about double affine flag varieties. I will discuss some results toward understanding double affine flag varieties and Grassmannians (and their Schubert subvarieties) from the point of view of $p$-adic Kac-Moody groups. I will discuss Hecke algebras, Bruhat order, and Kazhdan-Lusztig polynomials in this setting. This includes work joint with Daniel Orr and joint with Manish Patnaik.

13 Toward double affine flag varieties and Grassmannians

Recently there has been a growing interest in double affine Grassmannians, especially because of their relationship with Coulomb branches of quiver gauge theories. However, not much has been said about double affine flag varieties. I will discuss some results toward understanding double affine flag varieties and Grassmannians (and their Schubert subvarieties) from the point of view of $p$-adic Kac-Moody groups. I will discuss Hecke algebras, Bruhat order, and Kazhdan-Lusztig polynomials in this setting. This includes work joint with Daniel Orr and joint with Manish Patnaik.

Speaker: Dinakar Muthiah

10:30 Coffee and Tea Break

Taro Kimura: Geometry of quiver W-algebra

Quiver W-algebra is the gauge theoretical construction of q-deformed W-algebra. The generating current of the algebra is given as the operator analog of the qq-character associated with the representation on the quiver. I'd like to show that a master formula for such a generating current is obtained through the geometric construction of the qq-character by Nekrasov.

14 Geometry of quiver W-algebra

Quiver W-algebra is the gauge theoretical construction of q-deformed W-algebra. The generating current of the algebra is given as the operator analog of the qq-character associated with the representation on the quiver. I'd like to show that a master formula for such a generating current is obtained through the geometric construction of the qq-character by Nekrasov.

Speaker: Taro Kimura

Friday 8 February

Changjian Su: Homology of affine Grassmannian and quantum cohomology

Let $G$ be a complex reductive group, and $X$ be a smooth projective $G$-variety. We will construct an algebra homomorphism from the homology of the affine Grassmannian $Gr_G$ to the G-equivariant quantum cohomology of $X$. The construction uses shift operators in quantum cohomolgies. Joint work with Alexander Braverman.

15 Homology of affine Grassmannian and quantum cohomology

Let $G$ be a complex reductive group, and $X$ be a smooth projective $G$-variety. We will construct an algebra homomorphism from the homology of the affine Grassmannian $Gr_G$ to the G-equivariant quantum cohomology of $X$. The construction uses shift operators in quantum cohomolgies. Joint work with Alexander Braverman.

Speaker: Changjian Su

10:30 Coffee and Tea Break

Justin Hilburn: Symplectic duality and Langlands duality

16 Symplectic duality and Langlands duality

In this talk I would like to sketch how one can use the tools of derived symplectic geometry and holomorphically twisted gauge theories to derive a relationship between symplectic duality and local Langlands. Our starting point will be an observation due to Gaiotto-Witten that a $3d$ $\mathcal{N}=4$ theory with a $G$ flavor symmetry is a boundary condition for $4d$ $\mathcal{N}=4$ SYM with gauge group $G$. By examining the relationship between boundary observables and bulk lines we will be able to derive constructions originally due to Braverman, Finkelberg, Nakajima. By examine the relationship between boundary lines and bulk surface operators one can derive new connections to local geometric Langlands.

This is based on joint work with Philsang Yoo, Tudor Dimofte, and Davide Gaiotto.

Speaker: Justin Hilburn

12:00 Lunch break

Andrea Appel: Cohomological construction of quantum symmetric pairs

Braided module categories provide a conceptual framework for universal solutions of the (twisted) reflection equation, in analogy of what braided monoidal categories are for the quantum Yang-Baxter equation. In the theory of quantum groups, natural examples of braided module categories arise from the category of representations of a quantum symmetric pair coideal subalgebra as recently proved by M. Balagovic and S. Kolb. In this talk, I will describe the semi-classical interpretation of their construction and how this leads to a cohomological construction of quantized symmetric pairs in the context of deformation theory.

17 Cohomological construction of quantum symmetric pairs

Braided module categories provide a conceptual framework for universal solutions of the (twisted) reflection equation, in analogy of what braided monoidal categories are for the quantum Yang-Baxter equation. In the theory of quantum groups, natural examples of braided module categories arise from the category of representations of a quantum symmetric pair coideal subalgebra as recently proved by M. Balagovic and S. Kolb. In this talk, I will describe the semi-classical interpretation of their construction and how this leads to a cohomological construction of quantized symmetric pairs in the context of deformation theory.

Speaker: Andrea Appel

Sergei Gukov: New TQFTs from DAHA

TBA

18 New TQFTs from DAHA

Speaker: Sergei Gukov