## Infinite Analysis Seminar Tokyo

Seminar information archive ～04/18｜Next seminar｜Future seminars 04/19～

Date, time & place | Saturday 13:30 - 16:00 117Room #117 (Graduate School of Math. Sci. Bldg.) |
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**Seminar information archive**

### 2023/12/15

13:00-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Bi-Hamiltonian structures of integrable many-body models from Poisson reduction (ENGLISH)

**Laszlo Feher**(University of Szeged, Hungary)Bi-Hamiltonian structures of integrable many-body models from Poisson reduction (ENGLISH)

[ Abstract ]

We review our results on bi-Hamiltonian structures of trigonometric spin Sutherland models

built on collective spin variables.

Our basic observation was that the cotangent bundle $T^*\mathrm{U}(n)$ and its holomorphic analogue $T^* \mathrm{GL}(n,{\mathbb C})$,

as well as $T^*\mathrm{GL}(n,{\mathbb C})_{\mathbb R}$, carry a natural quadratic Poisson bracket,

which is compatible with the canonical linear one. The quadratic bracket arises by change of variables and analytic continuation

from an associated Heisenberg double.

Then, the reductions of $T^*{\mathrm{U}}(n)$ and $T^*{\mathrm{GL}}(n,{\mathbb C})$ by the conjugation actions of the

corresponding groups lead to the real and holomorphic spin Sutherland models, respectively, equipped

with a bi-Hamiltonian structure. The reduction of $T^*{\mathrm{GL}}(n,{\mathbb C})_{\mathbb R}$ by the group $\mathrm{U}(n) \times \mathrm{U}(n)$ gives

a generalized Sutherland model coupled to two ${\mathfrak u}(n)^*$-valued spins.

We also show that

a bi-Hamiltonian structure on the associative algebra ${\mathfrak{gl}}(n,{\mathbb R})$ that appeared in the context

of Toda models can be interpreted as the quotient of compatible Poisson brackets on $T^*{\mathrm{GL}}(n,{\mathbb R})$.

Before our work, all these reductions were studied using the canonical Poisson structures of the cotangent bundles,

without realizing the bi-Hamiltonian aspect.

Finally, if time permits, the degenerate integrability of some of the reduced systems

will be explained as well.

[1] L. Feher, Reduction of a bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$

to spin Ruijsenaars--Sutherland models, Lett. Math. Phys. 110, 1057-1079 (2020).

[2] L. Feher, Bi-Hamiltonian structure of spin Sutherland models: the holomorphic case, Ann. Henri Poincar\'e 22, 4063-4085 (2021).

[3] L. Feher, Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n)^*$-valued spins from Poisson reduction,

Nonlinearity 35, 2971-3003 (2022).

[4] L. Feher and B. Juhasz,

A note on quadratic Poisson brackets on $\mathfrak{gl}(n,\mathbb{R})$ related to Toda lattices,

Lett. Math. Phys. 112:45 (2022).

[5] L. Feher,

Notes on the degenerate integrability of reduced systems obtained from the master systems of free motion on cotangent bundles of

compact Lie groups, arXiv:2309.16245

We review our results on bi-Hamiltonian structures of trigonometric spin Sutherland models

built on collective spin variables.

Our basic observation was that the cotangent bundle $T^*\mathrm{U}(n)$ and its holomorphic analogue $T^* \mathrm{GL}(n,{\mathbb C})$,

as well as $T^*\mathrm{GL}(n,{\mathbb C})_{\mathbb R}$, carry a natural quadratic Poisson bracket,

which is compatible with the canonical linear one. The quadratic bracket arises by change of variables and analytic continuation

from an associated Heisenberg double.

Then, the reductions of $T^*{\mathrm{U}}(n)$ and $T^*{\mathrm{GL}}(n,{\mathbb C})$ by the conjugation actions of the

corresponding groups lead to the real and holomorphic spin Sutherland models, respectively, equipped

with a bi-Hamiltonian structure. The reduction of $T^*{\mathrm{GL}}(n,{\mathbb C})_{\mathbb R}$ by the group $\mathrm{U}(n) \times \mathrm{U}(n)$ gives

a generalized Sutherland model coupled to two ${\mathfrak u}(n)^*$-valued spins.

We also show that

a bi-Hamiltonian structure on the associative algebra ${\mathfrak{gl}}(n,{\mathbb R})$ that appeared in the context

of Toda models can be interpreted as the quotient of compatible Poisson brackets on $T^*{\mathrm{GL}}(n,{\mathbb R})$.

Before our work, all these reductions were studied using the canonical Poisson structures of the cotangent bundles,

without realizing the bi-Hamiltonian aspect.

Finally, if time permits, the degenerate integrability of some of the reduced systems

will be explained as well.

[1] L. Feher, Reduction of a bi-Hamiltonian hierarchy on $T^*\mathrm{U}(n)$

to spin Ruijsenaars--Sutherland models, Lett. Math. Phys. 110, 1057-1079 (2020).

[2] L. Feher, Bi-Hamiltonian structure of spin Sutherland models: the holomorphic case, Ann. Henri Poincar\'e 22, 4063-4085 (2021).

[3] L. Feher, Bi-Hamiltonian structure of Sutherland models coupled to two $\mathfrak{u}(n)^*$-valued spins from Poisson reduction,

Nonlinearity 35, 2971-3003 (2022).

[4] L. Feher and B. Juhasz,

A note on quadratic Poisson brackets on $\mathfrak{gl}(n,\mathbb{R})$ related to Toda lattices,

Lett. Math. Phys. 112:45 (2022).

[5] L. Feher,

Notes on the degenerate integrability of reduced systems obtained from the master systems of free motion on cotangent bundles of

compact Lie groups, arXiv:2309.16245

### 2023/12/06

13:00-14:30 Room #056 (Graduate School of Math. Sci. Bldg.)

This seminar has been cancelled.

Flat coordinates of algebraic Frobenius manifolds (ENGLISH)

**Misha Feigin**(University of Glasgow)This seminar has been cancelled.

Flat coordinates of algebraic Frobenius manifolds (ENGLISH)

[ Abstract ]

This seminar has been cancelled.

Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide Frobenius manifolds with polynomial prepotentials. Flat coordinates of the corresponding flat metric, known as Saito metric, are distinguished basic invariants of the Coxeter group. They have applications in representations of Cherednik algebras. Frobenius manifolds with algebraic prepotentials remain not classified and they are typically related to quasi-Coxeter conjugacy classes in finite Coxeter groups. We obtain flat coordinates for the majority of known examples of algebraic Frobenius manifolds in dimensions up to 4. In all the cases, flat coordinates appear to be some algebraic functions on the orbit space of the Coxeter group. This is a joint work with Daniele Valeri and Johan Wright.

This seminar has been cancelled.

Orbit spaces of the reflection representation of finite irreducible Coxeter groups provide Frobenius manifolds with polynomial prepotentials. Flat coordinates of the corresponding flat metric, known as Saito metric, are distinguished basic invariants of the Coxeter group. They have applications in representations of Cherednik algebras. Frobenius manifolds with algebraic prepotentials remain not classified and they are typically related to quasi-Coxeter conjugacy classes in finite Coxeter groups. We obtain flat coordinates for the majority of known examples of algebraic Frobenius manifolds in dimensions up to 4. In all the cases, flat coordinates appear to be some algebraic functions on the orbit space of the Coxeter group. This is a joint work with Daniele Valeri and Johan Wright.

### 2023/04/24

16:00-17:30 Room #123 (Graduate School of Math. Sci. Bldg.)

Euler type integral formulas and hypergeometric solutions for

variants of the $q$ hypergeometric equations.

(Japanese)

**Takahiko Nobukawa**(Kobe University )Euler type integral formulas and hypergeometric solutions for

variants of the $q$ hypergeometric equations.

(Japanese)

[ Abstract ]

We know that Papperitz's differential equation is essentially obtained from

Gauss' hypergeometric equation by applying a Moebius transformation,

implying that we have Euler type integral formulas or hypergeometric solutions.

The variants of the $q$ hypergeometric equations, introduced by

Hatano-Matsunawa-Sato-Takemura (Funkcial. Ekvac.,2022), are second order

$q$-difference systems which can be regarded as $q$ analoges of Papperitz's equation.

This motivates us for deriving Euler type integral formulas and hypergeometric solutions

for the pertinent $q$-difference systems. If time admits, I explain

the relation with $q$-analogues of Kummer's 24 solutions,

or the variants of multivariate $q$-hypergeometric functions.

This talk is based on the collaboration with Taikei Fujii.

We know that Papperitz's differential equation is essentially obtained from

Gauss' hypergeometric equation by applying a Moebius transformation,

implying that we have Euler type integral formulas or hypergeometric solutions.

The variants of the $q$ hypergeometric equations, introduced by

Hatano-Matsunawa-Sato-Takemura (Funkcial. Ekvac.,2022), are second order

$q$-difference systems which can be regarded as $q$ analoges of Papperitz's equation.

This motivates us for deriving Euler type integral formulas and hypergeometric solutions

for the pertinent $q$-difference systems. If time admits, I explain

the relation with $q$-analogues of Kummer's 24 solutions,

or the variants of multivariate $q$-hypergeometric functions.

This talk is based on the collaboration with Taikei Fujii.

### 2020/02/18

15:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Hydrodynamics of a one dimensional lattice gas.

(ENGLISH)

**Vincent Pasquier**(IPhT Saclay)Hydrodynamics of a one dimensional lattice gas.

(ENGLISH)

[ Abstract ]

The simplest boxball model is a one dimensional lattice gas obtained as

a certain (cristal) limit of the six vertex model where the evolution

determined by the transfer matrix becomes deterministic. One can

study its thermodynamics in and out of equilibrium and we shall present

preliminary results in this direction.

Collaboration with Atsuo Kuniba and Grégoire Misguich.

The simplest boxball model is a one dimensional lattice gas obtained as

a certain (cristal) limit of the six vertex model where the evolution

determined by the transfer matrix becomes deterministic. One can

study its thermodynamics in and out of equilibrium and we shall present

preliminary results in this direction.

Collaboration with Atsuo Kuniba and Grégoire Misguich.

### 2019/12/17

15:00-16:00 Room #駒場国際教育研究棟（旧６号館）108 (Graduate School of Math. Sci. Bldg.)

(-2) blow-up formula (JAPANESE)

**Ryo Ohkawa**(Waseda University)(-2) blow-up formula (JAPANESE)

[ Abstract ]

In this talk, we will consider the moduli of ADHM data

corresponding to the affine A_1 Dynkin diagram.

It is a moduli of framed sheaves on the (-2) curve or the projective

plane with a group action.

Each of these two types of moduli integrals has a combinatorial

description. In particular, the Hirota derivative of the Nekrasov

function can be obtained on the (-2) curve.

We introduce equalities among these two integrals and the

corresponding functional equations in some cases.

This is similar to the blow-up formula by Nakajima-Yoshioka.

I would also like to talk about relationships with the study of the

Painleve tau function by Bershtein-Shchechkin.

In this talk, we will consider the moduli of ADHM data

corresponding to the affine A_1 Dynkin diagram.

It is a moduli of framed sheaves on the (-2) curve or the projective

plane with a group action.

Each of these two types of moduli integrals has a combinatorial

description. In particular, the Hirota derivative of the Nekrasov

function can be obtained on the (-2) curve.

We introduce equalities among these two integrals and the

corresponding functional equations in some cases.

This is similar to the blow-up formula by Nakajima-Yoshioka.

I would also like to talk about relationships with the study of the

Painleve tau function by Bershtein-Shchechkin.

### 2018/12/25

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Q-operators for generalised eight vertex models associated

to the higher spin representations of the Sklyanin algebra. (ENGLISH)

**Takashi Takebe**(National Research University Higher School of Economics (Moscow))Q-operators for generalised eight vertex models associated

to the higher spin representations of the Sklyanin algebra. (ENGLISH)

[ Abstract ]

The Q-operator was first introduced by Baxter in 1972 as a

tool to solve the eight vertex model and recently attracts

attention from the representation theoretical viewpoint. In

this talk, we show that Baxter's apparently quite ad hoc and

technical construction can be generalised to the model

associated to the higher spin representations of the

Sklyanin algebra. If everybody in the audience understands Japanese, the talk

will be in Japanese.

The Q-operator was first introduced by Baxter in 1972 as a

tool to solve the eight vertex model and recently attracts

attention from the representation theoretical viewpoint. In

this talk, we show that Baxter's apparently quite ad hoc and

technical construction can be generalised to the model

associated to the higher spin representations of the

Sklyanin algebra. If everybody in the audience understands Japanese, the talk

will be in Japanese.

### 2018/12/06

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Integrability and TBA in non-equilibrium emergent hydrodynamics (ENGLISH)

**Francesco Ravanini**(University of Bologna)Integrability and TBA in non-equilibrium emergent hydrodynamics (ENGLISH)

[ Abstract ]

The paradigm of investigating non-equilibrium phenomena by considering stationary states of emergent hydrodynamics has attracted a lot of attention in the last years. Recent proposals of an exact approach in integrable cases, making use of TBA techniques, are presented and discussed.

The paradigm of investigating non-equilibrium phenomena by considering stationary states of emergent hydrodynamics has attracted a lot of attention in the last years. Recent proposals of an exact approach in integrable cases, making use of TBA techniques, are presented and discussed.

### 2018/10/04

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Integrable quad equations derived from the quantum Yang-Baxter

equation. (ENGLISH)

**Andrew Kels**(Graduate School of Arts and Sciences, University of Tokyo)Integrable quad equations derived from the quantum Yang-Baxter

equation. (ENGLISH)

[ Abstract ]

I will give an overview of an explicit correspondence that exists between

two different types of integrable equations; 1) the quantum Yang-Baxter

equation in its star-triangle relation (STR) form, and 2) the classical

3D-consistent quad equations in the Adler-Bobenko-Suris (ABS)

classification. The fundamental aspect of this correspondence is that the

equation of the critical point of a STR is equivalent to an ABS quad

equation. The STR's considered here are in fact equivalent to

hypergeometric integral transformation formulas. For example, a STR for

$H1_{(\varepsilon=0)}$ corresponds to the Euler Beta function, a STR for

$Q1_{(\delta=0)}$ corresponds to the $n=1$ Selberg integral, and STR's for

$H2_{\varepsilon=0,1}$, $H1_{(\varepsilon=1)}$, correspond to different

hypergeometric integral formulas of Barnes. I will discuss some of these

examples and some directions for future research.

I will give an overview of an explicit correspondence that exists between

two different types of integrable equations; 1) the quantum Yang-Baxter

equation in its star-triangle relation (STR) form, and 2) the classical

3D-consistent quad equations in the Adler-Bobenko-Suris (ABS)

classification. The fundamental aspect of this correspondence is that the

equation of the critical point of a STR is equivalent to an ABS quad

equation. The STR's considered here are in fact equivalent to

hypergeometric integral transformation formulas. For example, a STR for

$H1_{(\varepsilon=0)}$ corresponds to the Euler Beta function, a STR for

$Q1_{(\delta=0)}$ corresponds to the $n=1$ Selberg integral, and STR's for

$H2_{\varepsilon=0,1}$, $H1_{(\varepsilon=1)}$, correspond to different

hypergeometric integral formulas of Barnes. I will discuss some of these

examples and some directions for future research.

### 2018/09/25

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Classification of quad-equations on a cuboctahedron (JAPANESE)

**Nobutaka Nakazono**(Aoyama Gakuin University Department of Physics and Mathematics)Classification of quad-equations on a cuboctahedron (JAPANESE)

[ Abstract ]

Adelr-Bobenko-Suris (2003, 2009) and Boll (2011) classified quad-equations on a cube using a consistency around a cube. By use of this consistency, we can define integrable two-dimensional partial difference equations called ABS equations. A major example of ABS equation is the lattice modified KdV equation, which is a discrete analogue of the modified KdV equation. It is known that Lax representations and Backlund transformations of ABS equations can be constructed by using the consistency around a cube, and ABS equations can be reduced to differential and difference Painlevé equations via periodically reductions.

In this talk, we show a classification of quad-equations on a cuboctahedron using a consistency around a cuboctahedron and the relation between a resulting partial difference equation and a discrete Painlevé equation.

This work has been done in collaboration with Prof Nalini Joshi (The University of Sydney).

Adelr-Bobenko-Suris (2003, 2009) and Boll (2011) classified quad-equations on a cube using a consistency around a cube. By use of this consistency, we can define integrable two-dimensional partial difference equations called ABS equations. A major example of ABS equation is the lattice modified KdV equation, which is a discrete analogue of the modified KdV equation. It is known that Lax representations and Backlund transformations of ABS equations can be constructed by using the consistency around a cube, and ABS equations can be reduced to differential and difference Painlevé equations via periodically reductions.

In this talk, we show a classification of quad-equations on a cuboctahedron using a consistency around a cuboctahedron and the relation between a resulting partial difference equation and a discrete Painlevé equation.

This work has been done in collaboration with Prof Nalini Joshi (The University of Sydney).

### 2018/07/17

16:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Operator algebra for statistical model of square ladder (ENGLISH)

**Valerii Sopin:**(Higher School of Economics (Moscow))Operator algebra for statistical model of square ladder (ENGLISH)

[ Abstract ]

In this talk we will define operator algebra for square ladder on the basis

of semi-infinite forms.

Keywords: hard-square model, square ladder, operator algebra, semi-infinite

forms, fermions, quadratic algebra, cohomology, Demazure modules,

Heisenberg algebra.

In this talk we will define operator algebra for square ladder on the basis

of semi-infinite forms.

Keywords: hard-square model, square ladder, operator algebra, semi-infinite

forms, fermions, quadratic algebra, cohomology, Demazure modules,

Heisenberg algebra.

### 2018/04/03

15:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Screening Operators and Parabolic inductions for W-algebras

(ENGLISH)

**Naoki Genra**(RIMS, Kyoto U.)Screening Operators and Parabolic inductions for W-algebras

(ENGLISH)

[ Abstract ]

(Affine) W-algebras are the family of vertex algebras defined by

Drinfeld-Sokolov reductions. We introduce the free field realizations of

W-algebras by the Wakimoto representations of affine Lie algebras, which

we call the Wakimoto representations of W-algebras. Then W-algebras may be

described as the intersections of the kernels of the screening operators.

As applications, the parabolic inductions for W-algebras are obtained.

This is motivated by results of Premet and Losev on finite W-algebras. In

A-types, this becomes a chiralization of coproducts by Brundan-Kleshchev.

In BCD-types, we also have analogue results in special cases.

(Affine) W-algebras are the family of vertex algebras defined by

Drinfeld-Sokolov reductions. We introduce the free field realizations of

W-algebras by the Wakimoto representations of affine Lie algebras, which

we call the Wakimoto representations of W-algebras. Then W-algebras may be

described as the intersections of the kernels of the screening operators.

As applications, the parabolic inductions for W-algebras are obtained.

This is motivated by results of Premet and Losev on finite W-algebras. In

A-types, this becomes a chiralization of coproducts by Brundan-Kleshchev.

In BCD-types, we also have analogue results in special cases.

### 2018/03/27

15:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Schramm-Loewner evolutions and Liouville field theory (JAPANESE)

**Yoshiki Fukusumi**(The University of Tokyo, The Institute for Solid State Physics)Schramm-Loewner evolutions and Liouville field theory (JAPANESE)

[ Abstract ]

Schramm-Loewner evolutions (SLEs) are stochastic processes driven by Brownian motions which preserves conformal invariance. They describe the cluster boundaries associated with the minimal models of the conformal field theory, including the Ising model and the percolation as typical examples. The correlation functions of such models remarkably satisfy the martingale condition. We briefly review some known results. Then we analyse the time reversing procedure of Schramm Loewner evolutions and its relation to Liouville field theory or 2d pure gravity. We can get martingale observables by the calculation of the correlation functions of Liouville field theory without matter.

Schramm-Loewner evolutions (SLEs) are stochastic processes driven by Brownian motions which preserves conformal invariance. They describe the cluster boundaries associated with the minimal models of the conformal field theory, including the Ising model and the percolation as typical examples. The correlation functions of such models remarkably satisfy the martingale condition. We briefly review some known results. Then we analyse the time reversing procedure of Schramm Loewner evolutions and its relation to Liouville field theory or 2d pure gravity. We can get martingale observables by the calculation of the correlation functions of Liouville field theory without matter.

### 2018/02/06

15:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Sine-square deformation of one-dimensional critical systems (ENGLISH)

Modular invariant representations of the $N=2$ vertex operator superalgebra (ENGLISH)

**Hosho Katsura**(Department of Physics, Graduate School of Science, The Univeristy of Tokyo ) 15:00-16:00Sine-square deformation of one-dimensional critical systems (ENGLISH)

[ Abstract ]

Sine-square deformation (SSD) is one example of smooth boundary conditions that have significantly smaller finite-size effects than open boundary conditions. In a one-dimensional system with SSD, the interaction strength varies smoothly from the center to the edges according to the sine-square function. This means that the Hamiltonian of the system is inhomogeneous, as it lacks translational symmetry. Nevertheless, previous studies have revealed that the SSD leaves the ground state of the uniform chain with periodic boundary conditions (PBC) almost unchanged for critical systems. In particular, I showed in [1,2,3] that the correspondence is exact for critical XY and quantum Ising chains. The same correspondence between SSD and PBC holds for Dirac fermions in 1+1 dimension and a family of more general conformal field theories. If time permits, I will also introduce more recent results [4,5] and discuss the excited states of the SSD systems.

[1] H. Katsura, J. Phys. A: Math. Theor. 44, 252001 (2011).

[2] H. Katsura, J. Phys. A: Math. Theor. 45, 115003 (2012).

[3] I. Maruyama, H. Katsura, T. Hikihara, Phys. Rev. B 84, 165132 (2011).

[4] K. Okunishi and H. Katsura, J. Phys. A: Math. Theor. 48, 445208 (2015).

[5] S. Tamura and H. Katsura, Prog. Theor. Exp. Phys 2017, 113A01 (2017).

Sine-square deformation (SSD) is one example of smooth boundary conditions that have significantly smaller finite-size effects than open boundary conditions. In a one-dimensional system with SSD, the interaction strength varies smoothly from the center to the edges according to the sine-square function. This means that the Hamiltonian of the system is inhomogeneous, as it lacks translational symmetry. Nevertheless, previous studies have revealed that the SSD leaves the ground state of the uniform chain with periodic boundary conditions (PBC) almost unchanged for critical systems. In particular, I showed in [1,2,3] that the correspondence is exact for critical XY and quantum Ising chains. The same correspondence between SSD and PBC holds for Dirac fermions in 1+1 dimension and a family of more general conformal field theories. If time permits, I will also introduce more recent results [4,5] and discuss the excited states of the SSD systems.

[1] H. Katsura, J. Phys. A: Math. Theor. 44, 252001 (2011).

[2] H. Katsura, J. Phys. A: Math. Theor. 45, 115003 (2012).

[3] I. Maruyama, H. Katsura, T. Hikihara, Phys. Rev. B 84, 165132 (2011).

[4] K. Okunishi and H. Katsura, J. Phys. A: Math. Theor. 48, 445208 (2015).

[5] S. Tamura and H. Katsura, Prog. Theor. Exp. Phys 2017, 113A01 (2017).

**Ryo Sato**(Graduate School of Mathematical Sciences, The University of Tokyo) 16:30-17:30Modular invariant representations of the $N=2$ vertex operator superalgebra (ENGLISH)

[ Abstract ]

One of the most remarkable features in representation theory of a (``good'') vertex operator superalgebra (VOSA) is the modular invariance property of the characters. As an application of the property, M. Wakimoto and D. Adamovic proved that all the fusion rules for the simple $N=2$ VOSA of central charge $c_{p,1}=3(1-2/p)$ are computed from the modular $S$-matrix by the so-called Verlinde formula. In this talk, we present a new ``modular invariant'' family of irreducible highest weight modules over the simple $N=2$ VOSA of central charge $c_{p,p'}:=3(1-2p'/p)$. Here $(p,p')$ is a pair of coprime integers such that $p,p'>1$. In addition, we will discuss some generalization of the Verlinde formula in the spirit of Creutzig--Ridout.

One of the most remarkable features in representation theory of a (``good'') vertex operator superalgebra (VOSA) is the modular invariance property of the characters. As an application of the property, M. Wakimoto and D. Adamovic proved that all the fusion rules for the simple $N=2$ VOSA of central charge $c_{p,1}=3(1-2/p)$ are computed from the modular $S$-matrix by the so-called Verlinde formula. In this talk, we present a new ``modular invariant'' family of irreducible highest weight modules over the simple $N=2$ VOSA of central charge $c_{p,p'}:=3(1-2p'/p)$. Here $(p,p')$ is a pair of coprime integers such that $p,p'>1$. In addition, we will discuss some generalization of the Verlinde formula in the spirit of Creutzig--Ridout.

### 2017/11/10

17:00-18:30 Room #122 (Graduate School of Math. Sci. Bldg.)

Chern-Simons, gravity and integrable systems. (ENGLISH)

http://www.iip.ufrn.br/eventslecturer?inf==0EVRpXTR1TP

**Fabio Novaes**(International Institute of Physics (UFRN))Chern-Simons, gravity and integrable systems. (ENGLISH)

[ Abstract ]

It is known since the 80's that pure three-dimensional gravity is classically equivalent to a Chern-Simons theory with gauge group SL(2,R) x SL(2,R). For a given set of boundary conditions, the asymptotic classical phase space has a central extension in terms of two copies of Virasoro algebra. In particular, this gives a conformal field theory representation of black hole solutions in 3d gravity, also known as BTZ black holes. The BTZ black hole entropy can then be recovered using CFT. In this talk, we review this story and discuss recent results on how to relax the BTZ boundary conditions to obtain the KdV hierarchy at the boundary. More generally, this shows that Chern-Simons theory can represent virtually any integrable system at the boundary, given some consistency conditions. We also briefly discuss how this formulation can be useful to describe non-relativistic systems.

[ Reference URL ]It is known since the 80's that pure three-dimensional gravity is classically equivalent to a Chern-Simons theory with gauge group SL(2,R) x SL(2,R). For a given set of boundary conditions, the asymptotic classical phase space has a central extension in terms of two copies of Virasoro algebra. In particular, this gives a conformal field theory representation of black hole solutions in 3d gravity, also known as BTZ black holes. The BTZ black hole entropy can then be recovered using CFT. In this talk, we review this story and discuss recent results on how to relax the BTZ boundary conditions to obtain the KdV hierarchy at the boundary. More generally, this shows that Chern-Simons theory can represent virtually any integrable system at the boundary, given some consistency conditions. We also briefly discuss how this formulation can be useful to describe non-relativistic systems.

http://www.iip.ufrn.br/eventslecturer?inf==0EVRpXTR1TP

### 2017/05/30

17:30-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

$Q$-functions associated to the root system of type $C$ (JAPANESE)

**Soichi Okada**(Graduate School of Mathematics, Nagoya University)$Q$-functions associated to the root system of type $C$ (JAPANESE)

[ Abstract ]

Schur $Q$-functions are a family of symmetric functions introduced

by Schur in his study of projective representations of symmetric

groups. They are obtained by putting $t=-1$ in the Hall-Littlewood

functions associated to the root system of type $A$. (Schur

functions are the $t=0$ specialization.) This talk concerns

symplectic $Q$-functions, which are obtained by putting $t=-1$

in the Hall-Littlewood functions associated to the root system

of type $C$. We discuss several Pfaffian identities as well

as a combinatorial description for them. Also we present some

positivity conjectures.

Schur $Q$-functions are a family of symmetric functions introduced

by Schur in his study of projective representations of symmetric

groups. They are obtained by putting $t=-1$ in the Hall-Littlewood

functions associated to the root system of type $A$. (Schur

functions are the $t=0$ specialization.) This talk concerns

symplectic $Q$-functions, which are obtained by putting $t=-1$

in the Hall-Littlewood functions associated to the root system

of type $C$. We discuss several Pfaffian identities as well

as a combinatorial description for them. Also we present some

positivity conjectures.

### 2016/12/22

14:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Homology cobordisms over a surface of genus one (JAPANESE)

Generalization of Schur partition theorem (JAPANESE)

**Yuta Nozaki**(Graduate School of Mathematical Sciences, the University of Tokyo) 14:00-15:30Homology cobordisms over a surface of genus one (JAPANESE)

[ Abstract ]

Morimoto showed that some lens spaces have no genus one fibered knot,

and Baker completely determined such lens spaces.

In this talk, we introduce our results for the corresponding problem

formulated in terms of homology cobordisms.

The Chebotarev density theorem and binary quadratic forms play a key

role in the proof.

Morimoto showed that some lens spaces have no genus one fibered knot,

and Baker completely determined such lens spaces.

In this talk, we introduce our results for the corresponding problem

formulated in terms of homology cobordisms.

The Chebotarev density theorem and binary quadratic forms play a key

role in the proof.

**Shunsuke Tsuchioka**(Graduate School of Mathematical Sciences, the University of Tokyo) 16:00-17:30Generalization of Schur partition theorem (JAPANESE)

[ Abstract ]

The celebrated Rogers-Ramanujan partition theorem (RRPT) claims that

the number of partitions of n whose parts are ¥pm1 modulo 5

is equinumerous to the number of partitions of n whose successive

differences are

at least 2. Schur found a mod 6 analog of RRPT in 1926.

We will report a generalization for odd $p¥geq 3$ via representation

theory of quantum groups.

At p=3, it is Schur's theorem. The statement for p=5 was conjectured by

Andrews in 1970s

in a course of his 3 parameter generalization of RRPT and proved in 1994

by Andrews-Bessenrodt-Olsson with an aid of computer.

This is a joint work with Masaki Watanabe (arXiv:1609.01905).

The celebrated Rogers-Ramanujan partition theorem (RRPT) claims that

the number of partitions of n whose parts are ¥pm1 modulo 5

is equinumerous to the number of partitions of n whose successive

differences are

at least 2. Schur found a mod 6 analog of RRPT in 1926.

We will report a generalization for odd $p¥geq 3$ via representation

theory of quantum groups.

At p=3, it is Schur's theorem. The statement for p=5 was conjectured by

Andrews in 1970s

in a course of his 3 parameter generalization of RRPT and proved in 1994

by Andrews-Bessenrodt-Olsson with an aid of computer.

This is a joint work with Masaki Watanabe (arXiv:1609.01905).

### 2016/11/10

15:00-17:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Superconducting phase in the BCS model with imaginary

magnetic field (JAPANESE)

**Yohei Kashima**(Graduate School of Mathematical Scineces, The University of Tokyo)Superconducting phase in the BCS model with imaginary

magnetic field (JAPANESE)

[ Abstract ]

We prove that in the BCS model with an imaginary magnetic field

at positive temperature a spontaneous symmetry breaking (SSB) and

an off-diagonal long range order (ODLRO) occur. Here the BCS model

is meant to be a self-adjoint operator on the Fermionic Fock space,

consisting of a free part describing the electrons' nearest neighbor

hopping and a quartic interacting part describing a long range

interaction between Cooper pairs. The interaction with the imaginary

magnetic field is given by the z-component of the spin operator

multiplied by a pure imaginary parameter. The SSB and the ODLRO are

shown in the infinite-volume limit of the thermal average over the

full Fermionic Fock space. The insertion of the imaginary magnetic

field changes the gap equation. Consequently the SSB and the ODLRO

are shown in high temperature, weak coupling regimes where these

phenomena do not take place in the conventional BCS model. The proof

is based on the method of Grassmann integration.

We prove that in the BCS model with an imaginary magnetic field

at positive temperature a spontaneous symmetry breaking (SSB) and

an off-diagonal long range order (ODLRO) occur. Here the BCS model

is meant to be a self-adjoint operator on the Fermionic Fock space,

consisting of a free part describing the electrons' nearest neighbor

hopping and a quartic interacting part describing a long range

interaction between Cooper pairs. The interaction with the imaginary

magnetic field is given by the z-component of the spin operator

multiplied by a pure imaginary parameter. The SSB and the ODLRO are

shown in the infinite-volume limit of the thermal average over the

full Fermionic Fock space. The insertion of the imaginary magnetic

field changes the gap equation. Consequently the SSB and the ODLRO

are shown in high temperature, weak coupling regimes where these

phenomena do not take place in the conventional BCS model. The proof

is based on the method of Grassmann integration.

### 2016/10/27

15:00-17:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Non-unitary highest-weight modules over the $N=2$ superconformal algebra (JAPANESE)

**Ryou Sato**(Graduate School of Mathematical Scineces, The University of Tokyo)Non-unitary highest-weight modules over the $N=2$ superconformal algebra (JAPANESE)

[ Abstract ]

The $N=2$ superconformal algebra is a generalization of the Virasoro algebra having the super symmetry.

The character formulas associated with the unitary highest weight representations

are expressed in terms of the classical theta functions, and have the remarkable

modular invariance. Based on the method of the $W$-algebras,

Kac and Wakimoto, on the other hand, showed that the

characters for a certain class of non-unitary highest weight representations

can be written in terms of the mock theta functions associated with the affine ${sl}_{2|1}$.

Then they found a way to identify these formulas with

real analytic modular forms by using the correction terms given by Zwegers.

In this seminar, we explain a way to construct the above mentioned

non-unitary representations from the representations of the algebra affine ${sl}_{2}$,

based on the Kazama-Suzuki coset construction, namely not from the $W$-algebra method.

We also investigate the relations between the mock theta functions and the ordinary

theta functions, appearing in this method.

The $N=2$ superconformal algebra is a generalization of the Virasoro algebra having the super symmetry.

The character formulas associated with the unitary highest weight representations

are expressed in terms of the classical theta functions, and have the remarkable

modular invariance. Based on the method of the $W$-algebras,

Kac and Wakimoto, on the other hand, showed that the

characters for a certain class of non-unitary highest weight representations

can be written in terms of the mock theta functions associated with the affine ${sl}_{2|1}$.

Then they found a way to identify these formulas with

real analytic modular forms by using the correction terms given by Zwegers.

In this seminar, we explain a way to construct the above mentioned

non-unitary representations from the representations of the algebra affine ${sl}_{2}$,

based on the Kazama-Suzuki coset construction, namely not from the $W$-algebra method.

We also investigate the relations between the mock theta functions and the ordinary

theta functions, appearing in this method.

### 2016/02/08

13:30-15:00 Room #002 (Graduate School of Math. Sci. Bldg.)

q-Bosons, Toda lattice and Baxter Q-Operator (ENGLISH)

**Vincent Pasquier**(IPhT Saclay)q-Bosons, Toda lattice and Baxter Q-Operator (ENGLISH)

[ Abstract ]

I will use the Pieri rules of the Hall Littlewood polynomials to construct some

lattice models, namely the q-Boson model and the Toda Lattice Q matrix.

I will identify the semi infinite chain transfer matrix of these models with well known

half vertex operators. Finally, I will explain how the Gaudin determinant appears in the evaluation

of the semi infine chain scalar products for an arbitrary spin and is related to the Macdonald polynomials.

I will use the Pieri rules of the Hall Littlewood polynomials to construct some

lattice models, namely the q-Boson model and the Toda Lattice Q matrix.

I will identify the semi infinite chain transfer matrix of these models with well known

half vertex operators. Finally, I will explain how the Gaudin determinant appears in the evaluation

of the semi infine chain scalar products for an arbitrary spin and is related to the Macdonald polynomials.

### 2015/09/17

14:00-15:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Classifying simple modules at admissible levels through

symmetric polynomials (ENGLISH)

**Simon Wood**(The Australian National University)Classifying simple modules at admissible levels through

symmetric polynomials (ENGLISH)

[ Abstract ]

From infinite dimensional Lie algebras such as the Virasoro

algebra or affine Lie (super)algebras one can construct universal

vertex operator algebras. These vertex operator algebras are simple at

generic central charges or levels and only contain proper ideals at so

called admissible levels. The simple quotient vertex operator algebras

at these admissible levels are called minimal model algebras. In this

talk I will present free field realisations of the universal vertex

operator algebras and show how they allow one to elegantly classify

the simple modules over the simple quotient vertex operator algebras

by using a deep connection to symmetric polynomials.

From infinite dimensional Lie algebras such as the Virasoro

algebra or affine Lie (super)algebras one can construct universal

vertex operator algebras. These vertex operator algebras are simple at

generic central charges or levels and only contain proper ideals at so

called admissible levels. The simple quotient vertex operator algebras

at these admissible levels are called minimal model algebras. In this

talk I will present free field realisations of the universal vertex

operator algebras and show how they allow one to elegantly classify

the simple modules over the simple quotient vertex operator algebras

by using a deep connection to symmetric polynomials.

### 2015/07/17

14:00-16:00 Room #002 (Graduate School of Math. Sci. Bldg.)

Classifying simple modules at admissible levels through symmetric polynomials (ENGLISH)

**Simon Wood**(The Australian National University)Classifying simple modules at admissible levels through symmetric polynomials (ENGLISH)

[ Abstract ]

From infinite dimensional Lie algebras such as the Virasoro

algebra or affine Lie (super)algebras one can construct universal

vertex operator algebras. These vertex operator algebras are simple at

generic central charges or levels and only contain proper ideals at so

called admissible levels. The simple quotient vertex operator algebras

at these admissible levels are called minimal model algebras. In this

talk I will present free field realisations of the universal vertex

operator algebras and show how they allow one to elegantly classify

the simple modules over the simple quotient vertex operator algebras

by using a deep connection to symmetric polynomials.

From infinite dimensional Lie algebras such as the Virasoro

algebra or affine Lie (super)algebras one can construct universal

vertex operator algebras. These vertex operator algebras are simple at

generic central charges or levels and only contain proper ideals at so

called admissible levels. The simple quotient vertex operator algebras

at these admissible levels are called minimal model algebras. In this

talk I will present free field realisations of the universal vertex

operator algebras and show how they allow one to elegantly classify

the simple modules over the simple quotient vertex operator algebras

by using a deep connection to symmetric polynomials.

### 2015/07/09

15:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

An extension of the LMO functor and formal Gaussian integrals (JAPANESE)

On the relative number of ends of higher dimensional Thompson groups (JAPANESE)

**Yuta Nozaki**(Graduate School of Mathematical Sciences, the University of Tokyo) 15:00-16:30An extension of the LMO functor and formal Gaussian integrals (JAPANESE)

[ Abstract ]

Cheptea, Habiro and Massuyeau introduced the LMO functor as an

extension of the LMO invariant of closed 3-manifolds.

The LMO functor is “the monoidal category of Lagrangian cobordisms

between surfaces with at most one boundary component” to “the monoidal

category of certain Jacobi diagrams”.

In this talk, we extend the LMO functor to the case of any number of

boundary components.

In particular, we focus on a formal Gaussian integral, that is an

essential tool to construct the LMO functor.

Cheptea, Habiro and Massuyeau introduced the LMO functor as an

extension of the LMO invariant of closed 3-manifolds.

The LMO functor is “the monoidal category of Lagrangian cobordisms

between surfaces with at most one boundary component” to “the monoidal

category of certain Jacobi diagrams”.

In this talk, we extend the LMO functor to the case of any number of

boundary components.

In particular, we focus on a formal Gaussian integral, that is an

essential tool to construct the LMO functor.

**Motoko Kato**(Graduate School of Mathematical Sciences, the University of Tokyo) 17:00-18:30On the relative number of ends of higher dimensional Thompson groups (JAPANESE)

[ Abstract ]

In 2004, Brin defined n−dimensional Thompson group nV for every natural number n ≥ 1. nV is a generalization of the Thompson group V . The Thompson group V can be described as a subgroup of the homeomorphism group of the Cantor set C. In this point of view, nV is a subgroup of the homeomorphism group of Cn. We prove that the number of ends of nV is equal to 1 and there is a subgroup of nV such that the relative number of ends is ∞. As a corollary of the second result, for each n, nV has Haagerup property and it can not be the fundamental group of a compact K ̈ahler manifold. These results are the generalizations of the corresponding results of Farley, who studied the Thompson group V .

In 2004, Brin defined n−dimensional Thompson group nV for every natural number n ≥ 1. nV is a generalization of the Thompson group V . The Thompson group V can be described as a subgroup of the homeomorphism group of the Cantor set C. In this point of view, nV is a subgroup of the homeomorphism group of Cn. We prove that the number of ends of nV is equal to 1 and there is a subgroup of nV such that the relative number of ends is ∞. As a corollary of the second result, for each n, nV has Haagerup property and it can not be the fundamental group of a compact K ̈ahler manifold. These results are the generalizations of the corresponding results of Farley, who studied the Thompson group V .

### 2015/06/25

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Autonomous limit of the 4-dimensional Painlev¥.FN"e-type equations and degeneration of curves of genus two (JAPANESE)

**Akane Nakamura**(Tokyo University, Graduate School of Mathematical Sciences)Autonomous limit of the 4-dimensional Painlev¥.FN"e-type equations and degeneration of curves of genus two (JAPANESE)

[ Abstract ]

The Painlev¥.FN"e equations have been generalized from various aspects. Recently, the 4-dimensional Painlev¥N"e-type equations were classified by corresponding linear equations(Sakai, Kawakami-N.-Sakai, Kawakami). In this talk, I explain an attempt to characterize the 40 types of integrable systems obtained as the autonomous limit of the 4-dimensional Painlev¥N"e-type equations, by inspecting the degenerations of their spectral curves, which are curves of genus two.

The Painlev¥.FN"e equations have been generalized from various aspects. Recently, the 4-dimensional Painlev¥N"e-type equations were classified by corresponding linear equations(Sakai, Kawakami-N.-Sakai, Kawakami). In this talk, I explain an attempt to characterize the 40 types of integrable systems obtained as the autonomous limit of the 4-dimensional Painlev¥N"e-type equations, by inspecting the degenerations of their spectral curves, which are curves of genus two.

### 2015/05/28

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Unitary spherical representations of Drinfeld doubles (JAPANESE)

**Yuki Arano**(Graduate School of Mathematical Sciences, the University of Tokyo)Unitary spherical representations of Drinfeld doubles (JAPANESE)

[ Abstract ]

It is known that the Drinfeld double of the quantized

enveloping algebra of a semisimple Lie algebra looks similar to the

quantized enveloping algebra of the complexification of the Lie algebra.

In this talk, we investigate the unitary representation theory of such

Drinfeld double via its analogy to that of the complex Lie group.

We also talk on an application to operator algebras.

It is known that the Drinfeld double of the quantized

enveloping algebra of a semisimple Lie algebra looks similar to the

quantized enveloping algebra of the complexification of the Lie algebra.

In this talk, we investigate the unitary representation theory of such

Drinfeld double via its analogy to that of the complex Lie group.

We also talk on an application to operator algebras.

### 2015/04/23

17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)

Parabolic analogue of periodic Kazhdan-Lusztig polynomials (JAPANESE)

**Hideya Watanabe**(Department of Mathematics, Tokyo Institute of Technology, Graduate school of science and Engineering)Parabolic analogue of periodic Kazhdan-Lusztig polynomials (JAPANESE)

[ Abstract ]

We construct a parabolic analogue of so-called periodic modules, which are modules over the Hecke algebra

associated with an affine Weyl group.

These modules have a basis similar to Kazhdan-Lusztig basis.

Our construction enables us to see the relation between (ordinary)periodic KL-polynomials and parabolic ones.

We construct a parabolic analogue of so-called periodic modules, which are modules over the Hecke algebra

associated with an affine Weyl group.

These modules have a basis similar to Kazhdan-Lusztig basis.

Our construction enables us to see the relation between (ordinary)periodic KL-polynomials and parabolic ones.