## Kavli IPMU Komaba Seminar

過去の記録 ～10/03｜次回の予定｜今後の予定 10/04～

開催情報 | 月曜日 16:30～18:00 数理科学研究科棟(駒場) 002号室 |
---|---|

担当者 | 河野 俊丈 |

**過去の記録**

### 2017年11月09日(木)

13:30-14:30 数理科学研究科棟(駒場) 056号室

Various applications of supersymmetry in statistical physics (English)

**Edouard Brezin 氏**(lpt ens, Paris)Various applications of supersymmetry in statistical physics (English)

[ 講演概要 ]

Supersymmetry is a fundamental concept in particle physics (although it has not been seen experimentally so far). But it is although a powerful tool in a number of problems arising in quantum mechanics and statistical physics. It has been widely used in the theory of disordered systems (Efetov et al.), it led to dimensional reduction for branched polymers (Parisi-Sourlas), for the susy classical gas (Brydges and Imbrie), for Landau levels with impurities. If has also many powerful applications in the theory of random matrices. I will briefly review some of these topics.

Supersymmetry is a fundamental concept in particle physics (although it has not been seen experimentally so far). But it is although a powerful tool in a number of problems arising in quantum mechanics and statistical physics. It has been widely used in the theory of disordered systems (Efetov et al.), it led to dimensional reduction for branched polymers (Parisi-Sourlas), for the susy classical gas (Brydges and Imbrie), for Landau levels with impurities. If has also many powerful applications in the theory of random matrices. I will briefly review some of these topics.

### 2014年11月25日(火)

10:30-11:30 数理科学研究科棟(駒場) 128号室

Donaldson-Thomas theory for Calabi-Yau fourfolds.

(ENGLISH)

**Naichung Conan Leung 氏**(The Chinese University of Hong Kong)Donaldson-Thomas theory for Calabi-Yau fourfolds.

(ENGLISH)

[ 講演概要 ]

Donaldson-Thomas theory for Calabi-Yau threefolds is a

complexification of Chern-Simons theory. In this talk I will discuss

my joint work with Cao on the complexification of Donaldson theory.

This work is supported by a RGC grant of HK Government.

Donaldson-Thomas theory for Calabi-Yau threefolds is a

complexification of Chern-Simons theory. In this talk I will discuss

my joint work with Cao on the complexification of Donaldson theory.

This work is supported by a RGC grant of HK Government.

### 2014年06月30日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

On some quadratic algebras with applications to Topology,

Algebra, Combinatorics, Schubert Calculus and Integrable Systems. (ENGLISH)

**Anatol Kirillov 氏**(RIMS, Kyoto University)On some quadratic algebras with applications to Topology,

Algebra, Combinatorics, Schubert Calculus and Integrable Systems. (ENGLISH)

[ 講演概要 ]

The main purpose of my talk is to draw attention of the

participants of the seminar to a certain family of quadratic algebras

which has a wide range of applications to the subject mentioned in the

title of my talk.

The main purpose of my talk is to draw attention of the

participants of the seminar to a certain family of quadratic algebras

which has a wide range of applications to the subject mentioned in the

title of my talk.

### 2014年06月16日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Universal formulae for Lie groups and Chern-Simons theory (ENGLISH)

**A.P. Veselov 氏**(Loughborough, UK and Tokyo)Universal formulae for Lie groups and Chern-Simons theory (ENGLISH)

[ 講演概要 ]

In 1990s Vogel introduced an interesting parametrization of simple

Lie algebras by 3 parameters defined up to a common multiple and

permutations. Numerical characteristic is called universal if it can be

expressed in terms of Vogel's parameters (example - the dimension of Lie

algebra). I will discuss some universal formulae for Lie groups

and Chern-Simons theory on 3D sphere.

The talk is based on joint work with R.L. Mkrtchyan and A.N. Sergeev.

In 1990s Vogel introduced an interesting parametrization of simple

Lie algebras by 3 parameters defined up to a common multiple and

permutations. Numerical characteristic is called universal if it can be

expressed in terms of Vogel's parameters (example - the dimension of Lie

algebra). I will discuss some universal formulae for Lie groups

and Chern-Simons theory on 3D sphere.

The talk is based on joint work with R.L. Mkrtchyan and A.N. Sergeev.

### 2014年01月30日(木)

17:00-18:30 数理科学研究科棟(駒場) 128号室

Characteristic classes from 2d renormalized sigma-models (ENGLISH)

**Hans Jockers 氏**(The University of Bonn)Characteristic classes from 2d renormalized sigma-models (ENGLISH)

[ 講演概要 ]

The Hirzebruch-Riemann-Roch formula relates the holomorphic Euler characteristic

of holomorphic vector bundles to topological invariants of compact complex manifold.

I will explain a generalization of the Mukai's modified first Chern character map, which

introduces certain characteristic classes that have not been considered in this form by

Hirzebruch. This naturally leads to the characteristic Gamma class based on the Gamma

function. The characteristic Gamma class has a surprising relation to the quantum theory

of certain 2d sigma-models with compact complex manifolds as their target spaces. I will

argue that the Gamma class describes perturbative quantum corrections to the classical

theory of those sigma models.

The Hirzebruch-Riemann-Roch formula relates the holomorphic Euler characteristic

of holomorphic vector bundles to topological invariants of compact complex manifold.

I will explain a generalization of the Mukai's modified first Chern character map, which

introduces certain characteristic classes that have not been considered in this form by

Hirzebruch. This naturally leads to the characteristic Gamma class based on the Gamma

function. The characteristic Gamma class has a surprising relation to the quantum theory

of certain 2d sigma-models with compact complex manifolds as their target spaces. I will

argue that the Gamma class describes perturbative quantum corrections to the classical

theory of those sigma models.

### 2013年11月18日(月)

17:00-18:30 数理科学研究科棟(駒場) 002号室

Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

(ENGLISH)

**Mauricio Romo 氏**(Kavli IPMU)Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary

(ENGLISH)

[ 講演概要 ]

I will talk about the recent computation, done in joint work with Prof. K. Hori, of the partition function on the hemisphere of a class of two-dimensional (2,2) supersymmetric field theories including gauged linear sigma models (GLSM). The result provides a general exact formula for the central charge of the D-branes placed at the boundary. From the mathematical point of view, for the case of GLSMs that admit a geometrical interpretation, this formula provides an expression for the central charge of objects in the derived category at any point of the stringy Kahler moduli space. I will describe how this formula arises from physics and give simple, yet important, examples that supports its validity. If time allows, I will also explain some of its consequences such as how it can be used to obtain the grade restriction rule for branes near phase boundaries.

I will talk about the recent computation, done in joint work with Prof. K. Hori, of the partition function on the hemisphere of a class of two-dimensional (2,2) supersymmetric field theories including gauged linear sigma models (GLSM). The result provides a general exact formula for the central charge of the D-branes placed at the boundary. From the mathematical point of view, for the case of GLSMs that admit a geometrical interpretation, this formula provides an expression for the central charge of objects in the derived category at any point of the stringy Kahler moduli space. I will describe how this formula arises from physics and give simple, yet important, examples that supports its validity. If time allows, I will also explain some of its consequences such as how it can be used to obtain the grade restriction rule for branes near phase boundaries.

### 2013年07月17日(水)

17:00-18:30 数理科学研究科棟(駒場) 002号室

Homological Mirror Symmetry for toric Calabi-Yau varieties (ENGLISH)

**Daniel Pomerleano 氏**(Kavli IPMU)Homological Mirror Symmetry for toric Calabi-Yau varieties (ENGLISH)

[ 講演概要 ]

I will discuss some recent developments in Homological Mirror

Symmetry for toric Calabi-Yau varieties.

I will discuss some recent developments in Homological Mirror

Symmetry for toric Calabi-Yau varieties.

### 2013年07月08日(月)

17:00-18:30 数理科学研究科棟(駒場) 122号室

Elliptic genera and two dimensional gauge theories (ENGLISH)

**Richard Eager 氏**(Kavli IPMU)Elliptic genera and two dimensional gauge theories (ENGLISH)

[ 講演概要 ]

The elliptic genus is an important invariant of two dimensional conformal field theories that generalizes the Witten index. In this talk, I will first review the geometric meaning of the elliptic genus and Witten's GLSM construction. Then I will explain how the elliptic genus can be computed directly from a two dimensional gauge theory using localization. The central example of this talk will be the quintic threefold. The GLSM description of the quintic threefold has both a large-volume sigma model description and a Landau-Ginzburg description. I will explain how the GLSM calculation of the index reproduces the old results in these two phases. Time permitting, further applications and generalizations will be discussed.

The elliptic genus is an important invariant of two dimensional conformal field theories that generalizes the Witten index. In this talk, I will first review the geometric meaning of the elliptic genus and Witten's GLSM construction. Then I will explain how the elliptic genus can be computed directly from a two dimensional gauge theory using localization. The central example of this talk will be the quintic threefold. The GLSM description of the quintic threefold has both a large-volume sigma model description and a Landau-Ginzburg description. I will explain how the GLSM calculation of the index reproduces the old results in these two phases. Time permitting, further applications and generalizations will be discussed.

### 2013年04月24日(水)

17:00-18:30 数理科学研究科棟(駒場) 002号室

Calabi-Yau threefolds of Type K (ENGLISH)

**金沢 篤 氏**(University of British Columbia)Calabi-Yau threefolds of Type K (ENGLISH)

[ 講演概要 ]

We will provide a full classification of Calabi-Yau threefolds of Type

K studied by Oguiso and Sakurai. Our study completes the

classification of Calabi-Yau threefolds with infinite fundamental

group. I will then discuss special Lagrangian T3-fibrations of

Calabi-Yau threefolds of type K. This talk is based on a joint work

with Kenji Hashimoto.

We will provide a full classification of Calabi-Yau threefolds of Type

K studied by Oguiso and Sakurai. Our study completes the

classification of Calabi-Yau threefolds with infinite fundamental

group. I will then discuss special Lagrangian T3-fibrations of

Calabi-Yau threefolds of type K. This talk is based on a joint work

with Kenji Hashimoto.

### 2012年06月11日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Quantum cohomology of flag varieties (ENGLISH)

**Changzheng Li 氏**(Kavli IPMU)Quantum cohomology of flag varieties (ENGLISH)

[ 講演概要 ]

In this talk, I will give a brief introduction to the quantum cohomology of flag varieties first. Then I will introduce a Z^2-filtration on the quantum cohomology of complete flag varieties. In the end, we will study the quantum Pieri rules for complex/symplectic Grassmannians, as applications of the Z^2-filtration.

In this talk, I will give a brief introduction to the quantum cohomology of flag varieties first. Then I will introduce a Z^2-filtration on the quantum cohomology of complete flag varieties. In the end, we will study the quantum Pieri rules for complex/symplectic Grassmannians, as applications of the Z^2-filtration.

### 2012年06月08日(金)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Period Integrals and Tautological Systems (ENGLISH)

**Bong Lian 氏**(Brandeis University)Period Integrals and Tautological Systems (ENGLISH)

[ 講演概要 ]

We develop a global Poincar\\'e residue formula to study

period integrals of families of complex manifolds. For any compact

complex manifold $X$ equipped with a linear system $V^*$ of

generically smooth CY hypersurfaces, the formula expresses period

integrals in terms of a canonical global meromorphic top form on $X$.

Two important ingredients of this construction are the notion of a CY

principal bundle, and a classification of such rank one bundles.

We also generalize the construction to CY and general type complete

intersections. When $X$ is an algebraic manifold having a sufficiently

large automorphism group $G$ and $V^*$ is a linear representation of

$G$, we construct a holonomic D-module that governs the period

integrals. The construction is based in part on the theory of

tautological systems we have developed earlier. The approach allows us

to explicitly describe a Picard-Fuchs type system for complete

intersection varieties of general types, as well as CY, in any Fano

variety, and in a homogeneous space in particular. In addition, the

approach provides a new perspective of old examples such as CY

complete intersections in a toric variety or partial flag variety. The

talk is based on recent joint work with R. Song and S.T. Yau.

We develop a global Poincar\\'e residue formula to study

period integrals of families of complex manifolds. For any compact

complex manifold $X$ equipped with a linear system $V^*$ of

generically smooth CY hypersurfaces, the formula expresses period

integrals in terms of a canonical global meromorphic top form on $X$.

Two important ingredients of this construction are the notion of a CY

principal bundle, and a classification of such rank one bundles.

We also generalize the construction to CY and general type complete

intersections. When $X$ is an algebraic manifold having a sufficiently

large automorphism group $G$ and $V^*$ is a linear representation of

$G$, we construct a holonomic D-module that governs the period

integrals. The construction is based in part on the theory of

tautological systems we have developed earlier. The approach allows us

to explicitly describe a Picard-Fuchs type system for complete

intersection varieties of general types, as well as CY, in any Fano

variety, and in a homogeneous space in particular. In addition, the

approach provides a new perspective of old examples such as CY

complete intersections in a toric variety or partial flag variety. The

talk is based on recent joint work with R. Song and S.T. Yau.

### 2012年05月21日(月)

17:00-18:30 数理科学研究科棟(駒場) 002号室

Topological Strings on Elliptic Fibrations (ENGLISH)

**Emanuel Scheidegger 氏**(The University of Freiburg)Topological Strings on Elliptic Fibrations (ENGLISH)

[ 講演概要 ]

We will explain a conjecture that expresses the BPS invariants

(Gopakumar-Vafa invariants) for elliptically fibered Calabi-Yau

threefolds in terms of modular forms. In particular, there is a

recursion relation which governs these modular forms. Evidence comes

from the polynomial formulation of the higher genus topological string

amplitudes with insertions.

We will explain a conjecture that expresses the BPS invariants

(Gopakumar-Vafa invariants) for elliptically fibered Calabi-Yau

threefolds in terms of modular forms. In particular, there is a

recursion relation which governs these modular forms. Evidence comes

from the polynomial formulation of the higher genus topological string

amplitudes with insertions.

### 2012年01月20日(金)

14:45-16:15 数理科学研究科棟(駒場) 056号室

Refined holomorphic anomaly equations (ENGLISH)

**Albrecht Klemm 氏**(The University of Bonn)Refined holomorphic anomaly equations (ENGLISH)

[ 講演概要 ]

We propose a derivation of refined holomophic

anomaly equation from the word-sheet point of

view and discuss the interpretation of the

refined BPS invariants for local Calabi-Yau

spaces.

We propose a derivation of refined holomophic

anomaly equation from the word-sheet point of

view and discuss the interpretation of the

refined BPS invariants for local Calabi-Yau

spaces.

### 2011年11月21日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Enuemerative meaning of mirror maps for toric Calabi-Yau manifolds (ENGLISH)

**Siu-Cheong Lau 氏**(IPMU)Enuemerative meaning of mirror maps for toric Calabi-Yau manifolds (ENGLISH)

[ 講演概要 ]

For a mirror pair of smooth manifolds X and Y, mirror symmetry associates a complex structure on Y to each Kaehler structure on X, and this association is called the mirror map. Traditionally mirror maps are defined by solving Picard-Fuchs equations and its geometric meaning was unclear. In this talk I explain a recent joint work with K.W. Chan, N.C. Leung and H.H. Tseng which proves that mirror maps can be obtained by taking torus duality (the SYZ approach) and disk-counting for a class of toric Calabi-Yau manifolds in any dimensions. As a consequence we can compute disk-counting invariants by solving Picard-Fuchs equations.

For a mirror pair of smooth manifolds X and Y, mirror symmetry associates a complex structure on Y to each Kaehler structure on X, and this association is called the mirror map. Traditionally mirror maps are defined by solving Picard-Fuchs equations and its geometric meaning was unclear. In this talk I explain a recent joint work with K.W. Chan, N.C. Leung and H.H. Tseng which proves that mirror maps can be obtained by taking torus duality (the SYZ approach) and disk-counting for a class of toric Calabi-Yau manifolds in any dimensions. As a consequence we can compute disk-counting invariants by solving Picard-Fuchs equations.

### 2011年01月31日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Mirror symmetry for toric Calabi-Yau manifolds from the SYZ viewpoint (ENGLISH)

**Kwok-Wai Chan 氏**(IPMU, the University of Tokyo)Mirror symmetry for toric Calabi-Yau manifolds from the SYZ viewpoint (ENGLISH)

[ 講演概要 ]

In this talk, I will discuss mirror symmetry for toric

Calabi-Yau (CY) manifolds from the viewpoint of the SYZ program. I will

start with a special Lagrangian torus fibration on a toric CY manifold,

and then construct its instanton-corrected mirror by a T-duality modified

by quantum corrections. A remarkable feature of this construction is that

the mirror family is inherently written in canonical flat coordinates. As

a consequence, we get a conjectural enumerative meaning for the inverse

mirror maps. If time permits, I will explain the verification of this

conjecture in several examples via a formula which computes open

Gromov-Witten invariants for toric manifolds.

In this talk, I will discuss mirror symmetry for toric

Calabi-Yau (CY) manifolds from the viewpoint of the SYZ program. I will

start with a special Lagrangian torus fibration on a toric CY manifold,

and then construct its instanton-corrected mirror by a T-duality modified

by quantum corrections. A remarkable feature of this construction is that

the mirror family is inherently written in canonical flat coordinates. As

a consequence, we get a conjectural enumerative meaning for the inverse

mirror maps. If time permits, I will explain the verification of this

conjecture in several examples via a formula which computes open

Gromov-Witten invariants for toric manifolds.

### 2010年11月29日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Borcherds products in monstrous moonshine. (ENGLISH)

**Scott Carnahan 氏**(IPMU)Borcherds products in monstrous moonshine. (ENGLISH)

[ 講演概要 ]

During the 1980s, Koike, Norton, and Zagier independently found an

infinite product expansion for the difference of two modular j-functions

on a product of half planes. Borcherds showed that this product identity

is the Weyl denominator formula for an infinite dimensional Lie algebra

that has an action of the monster simple group by automorphisms, and used

this action to prove the monstrous moonshine conjectures.

I will describe a more general construction that yields an infinite

product identity and an infinite dimensional Lie algebra for each element

of the monster group. The above objects then arise as the special cases

assigned to the identity element. Time permitting, I will attempt to

describe a connection to conformal field theory.

During the 1980s, Koike, Norton, and Zagier independently found an

infinite product expansion for the difference of two modular j-functions

on a product of half planes. Borcherds showed that this product identity

is the Weyl denominator formula for an infinite dimensional Lie algebra

that has an action of the monster simple group by automorphisms, and used

this action to prove the monstrous moonshine conjectures.

I will describe a more general construction that yields an infinite

product identity and an infinite dimensional Lie algebra for each element

of the monster group. The above objects then arise as the special cases

assigned to the identity element. Time permitting, I will attempt to

describe a connection to conformal field theory.

### 2010年11月26日(金)

14:40-16:10 数理科学研究科棟(駒場) 002号室

Hamiltonian torus actions on orbifolds and orbifold-GKM theorem (joint

work with T. Holm) (JAPANESE)

**松村 朝雄 氏**(Cornell University)Hamiltonian torus actions on orbifolds and orbifold-GKM theorem (joint

work with T. Holm) (JAPANESE)

[ 講演概要 ]

When a symplectic manifold M carries a Hamiltonian torus R action, the

injectivity theorem states that the R-equivariant cohomology of M is a

subring of the one of the fixed points and the GKM theorem allows us

to compute this subring by only using the data of 1-dimensional

orbits. The results in the first part of this talk are a

generalization of this technique to Hamiltonian R actions on orbifolds

and an application to the computation of the equivariant cohomology of

toric orbifolds. In the second part, we will introduce the equivariant

Chen-Ruan cohomology ring which is a symplectic invariant of the

action on the orbifold and explain the injectivity/GKM theorem for this ring.

When a symplectic manifold M carries a Hamiltonian torus R action, the

injectivity theorem states that the R-equivariant cohomology of M is a

subring of the one of the fixed points and the GKM theorem allows us

to compute this subring by only using the data of 1-dimensional

orbits. The results in the first part of this talk are a

generalization of this technique to Hamiltonian R actions on orbifolds

and an application to the computation of the equivariant cohomology of

toric orbifolds. In the second part, we will introduce the equivariant

Chen-Ruan cohomology ring which is a symplectic invariant of the

action on the orbifold and explain the injectivity/GKM theorem for this ring.

### 2010年10月18日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Quasi-modular forms and Gromov--Witten theory of elliptic orbifold $\\mathbb{P}^1$ (ENGLISH)

**Todor Milanov 氏**(IPMU)Quasi-modular forms and Gromov--Witten theory of elliptic orbifold $\\mathbb{P}^1$ (ENGLISH)

[ 講演概要 ]

This talk is based on my current work with Y. Ruan. Our project is part of the so called Landau--Ginzburg/Calabi-Yau correspondence. The latter is a conjecture, due to Ruan, that describes the relation between the $W$-spin invariants of a Landau-Ginzburg potential $W$ and the Gromov--Witten invariants of a certain Calabi--Yau orbifold. I am planning first to explain the higher-genus reconstruction formalism of Givental. This formalism together with the work of M. Krawitz and Y. Shen allows us to express the Gromov--Witten invariants of the orbifold $\\mathbb{P}^1$'s with weights $(3,3,3)$, $(2,4,4)$, and $(2,3,6)$ in terms of Saito's Frobenius structure associated with the simple elliptic singularities $P_8$, $X_9$, and $J_{10}$ respectively. After explaining Givental's formalism, my goal would be to discuss the Saito's flat structure, and to explain how its modular behavior fits in the Givental's formalism. This allows us to prove that the Gromov--Witten invariants are quasi-modular forms on an appropriate modular group.

This talk is based on my current work with Y. Ruan. Our project is part of the so called Landau--Ginzburg/Calabi-Yau correspondence. The latter is a conjecture, due to Ruan, that describes the relation between the $W$-spin invariants of a Landau-Ginzburg potential $W$ and the Gromov--Witten invariants of a certain Calabi--Yau orbifold. I am planning first to explain the higher-genus reconstruction formalism of Givental. This formalism together with the work of M. Krawitz and Y. Shen allows us to express the Gromov--Witten invariants of the orbifold $\\mathbb{P}^1$'s with weights $(3,3,3)$, $(2,4,4)$, and $(2,3,6)$ in terms of Saito's Frobenius structure associated with the simple elliptic singularities $P_8$, $X_9$, and $J_{10}$ respectively. After explaining Givental's formalism, my goal would be to discuss the Saito's flat structure, and to explain how its modular behavior fits in the Givental's formalism. This allows us to prove that the Gromov--Witten invariants are quasi-modular forms on an appropriate modular group.

### 2010年04月26日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

The correspondence between Frobenius algebra of Hurwitz numbers

and matrix models (JAPANESE)

**池田 暁志 氏**(東京大学大学院数理科学研究科)The correspondence between Frobenius algebra of Hurwitz numbers

and matrix models (JAPANESE)

[ 講演概要 ]

The number of branched coverings of closed surfaces are called Hurwitz

numbers. They constitute a Frobenius algebra structure, or

two dimensional topological field theory. On the other hand, correlation

functions of matrix models are expressed in term of ribbon graphs

(graphs embedded in closed surfaces).

In this talk, I explain how the Frobenius algebra structure of Hurwitz

numbers are described in terms of matrix models. We use the

correspondence between ribbon graphs and covering of S^2 ramified at

three points, both of which have natural symmetric group actions.

As an application I use Frobenius algebra structure to compute Hermitian

matrix models, multi-variable matrix models, and their large N

expansions. The generating function of Hurwitz numbers is also expressed

in terms of matrix models. The relation to integrable hierarchies and

random partitions is briefly discussed.

The number of branched coverings of closed surfaces are called Hurwitz

numbers. They constitute a Frobenius algebra structure, or

two dimensional topological field theory. On the other hand, correlation

functions of matrix models are expressed in term of ribbon graphs

(graphs embedded in closed surfaces).

In this talk, I explain how the Frobenius algebra structure of Hurwitz

numbers are described in terms of matrix models. We use the

correspondence between ribbon graphs and covering of S^2 ramified at

three points, both of which have natural symmetric group actions.

As an application I use Frobenius algebra structure to compute Hermitian

matrix models, multi-variable matrix models, and their large N

expansions. The generating function of Hurwitz numbers is also expressed

in terms of matrix models. The relation to integrable hierarchies and

random partitions is briefly discussed.

### 2010年02月01日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Bases in the solution space of the Mellin system

**Timur Sadykov 氏**(Siberian Federal University)Bases in the solution space of the Mellin system

[ 講演概要 ]

I will present a joint work with Alicia Dickenstein.

We consider algebraic functions $z$ satisfying equations of the

form

\\begin{equation}

a_0 z^m + a_1z^{m_1} + a_2 z^{m_2} + \\ldots + a_n z^{m_n} +

a_{n+1} =0.

\\end{equation}

Here $m > m_1 > \\ldots > m_n>0,$ $m,m_i \\in \\N,$ and

$z=z(a_0,\\ldots,a_{n+1})$ is a function of the complex variables

$a_0, \\ldots, a_{n+1}.$ Solutions to such equations are

classically known to satisfy holonomic systems of linear partial

differential equations with polynomial coefficients. In the talk

I will investigate one of such systems of differential equations which

was introduced by Mellin. We compute the holonomic rank of the

Mellin system as well as the dimension of the space of its

algebraic solutions. Moreover, we construct explicit bases of

solutions in terms of the roots of initial algebraic equation and their

logarithms. We show that the monodromy of the Mellin system is

always reducible and give some factorization results in the

univariate case.

I will present a joint work with Alicia Dickenstein.

We consider algebraic functions $z$ satisfying equations of the

form

\\begin{equation}

a_0 z^m + a_1z^{m_1} + a_2 z^{m_2} + \\ldots + a_n z^{m_n} +

a_{n+1} =0.

\\end{equation}

Here $m > m_1 > \\ldots > m_n>0,$ $m,m_i \\in \\N,$ and

$z=z(a_0,\\ldots,a_{n+1})$ is a function of the complex variables

$a_0, \\ldots, a_{n+1}.$ Solutions to such equations are

classically known to satisfy holonomic systems of linear partial

differential equations with polynomial coefficients. In the talk

I will investigate one of such systems of differential equations which

was introduced by Mellin. We compute the holonomic rank of the

Mellin system as well as the dimension of the space of its

algebraic solutions. Moreover, we construct explicit bases of

solutions in terms of the roots of initial algebraic equation and their

logarithms. We show that the monodromy of the Mellin system is

always reducible and give some factorization results in the

univariate case.

### 2009年12月07日(月)

17:30-19:00 数理科学研究科棟(駒場) 002号室

Geometric quantization on noncompact manifolds

**Weiping Zhang 氏**(Chern Institute of Mathematics, Nankai University)Geometric quantization on noncompact manifolds

[ 講演概要 ]

We will describe our analytic approach with Youlinag Tian to the Guillemin-Sternberg geometric quantization conjecture which can be summarized as "quantization commutes with reduction". We will aslo describe a recent extension to the case of noncompact symplectic manifolds. This is a joint work with Xiaonan Ma in which we solve a conjecture of Vergne mentioned in her ICM2006 plenary lecture.

We will describe our analytic approach with Youlinag Tian to the Guillemin-Sternberg geometric quantization conjecture which can be summarized as "quantization commutes with reduction". We will aslo describe a recent extension to the case of noncompact symplectic manifolds. This is a joint work with Xiaonan Ma in which we solve a conjecture of Vergne mentioned in her ICM2006 plenary lecture.

### 2009年11月30日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Chiral Algebras of (0,2) Models: Beyond Perturbation Theory

**Junya Yagi 氏**(Rutgers University)Chiral Algebras of (0,2) Models: Beyond Perturbation Theory

[ 講演概要 ]

The chiral algebras of two-dimensional sigma models with (0,2)

supersymmetry are infinite-dimensional generalizations of the chiral

rings of (2,2) models. Perturbatively, they enjoy rich mathematical

structures described by sheaves of chiral differential operators.

Nonperturbatively, however, they vanish completely for certain (0,2)

models with no left-moving fermions. In this talk, I will explain how

this vanishing phenomenon takes places. The vanishing of the chiral

algebra of a (0, 2) model implies that supersymmetry is spontaneously

broken in the model, which in turn suggests that no harmonic spinors

exist on the loop space of the target space. In particular, the

elliptic genus of the model vanishes, thereby providing a physics

proof of a special case of the Hoelhn-Stolz conjecture.

The chiral algebras of two-dimensional sigma models with (0,2)

supersymmetry are infinite-dimensional generalizations of the chiral

rings of (2,2) models. Perturbatively, they enjoy rich mathematical

structures described by sheaves of chiral differential operators.

Nonperturbatively, however, they vanish completely for certain (0,2)

models with no left-moving fermions. In this talk, I will explain how

this vanishing phenomenon takes places. The vanishing of the chiral

algebra of a (0, 2) model implies that supersymmetry is spontaneously

broken in the model, which in turn suggests that no harmonic spinors

exist on the loop space of the target space. In particular, the

elliptic genus of the model vanishes, thereby providing a physics

proof of a special case of the Hoelhn-Stolz conjecture.

### 2009年11月09日(月)

16:30-18:00 数理科学研究科棟(駒場) 002号室

Differential Graded Categories and heterotic string theory

**Makoto Sakurai 氏**(東京大学大学院数理科学研究科)Differential Graded Categories and heterotic string theory

[ 講演概要 ]

The saying "category theory is an abstract nonsense" is even physically not true.

The schematic language of triangulated category presents a new stage of string theory.

To illuminate this idea, I will draw your attention to the blow-up minimal model

of complex algebraic surfaces. This is done under the hypothetical assumptions

of "generalized complex structure" of cotangent bundle due to Hitchin school.

The coordinate transformation Jacobian matrices of the measure of sigma model

with spin structures cause one part of the gravitational "anomaly cancellation"

of smooth Kahler manifold $X$ and Weyl anomaly of compact Riemann surface $\\Sigma$.

$Anom = c_1 (X) c_1 (\\Sigma) \\oplus ch_2 (X)$,

in terms of 1st and 2nd Chern characters. Note that when $\\Sigma$ is a puctured disk

with flat metric, the chiral algebra is nothing but the ordinary vertex algebra.

Note that I do not explain the complex differential geometry,

but essentially more recent works with the category of DGA (Diffenreial Graded Algebra),

which is behind the super conformal field theory of chiral algebras.

My result of "vanishing tachyon" (nil-radical part of vertex algebras)

and "causality resortation" in compactified non-critical heterotic sigma model

is physically a promising idea of new solution to unitary representation of operator algebras.

This idea is realized in the formalism of BRST cohomology and its generalization

in $\\mathcal{N} = (0,2)$ supersymmetry, that is, non-commutative geometry

with non-linear constraint condition of pure spinors for covariant quantization.

The saying "category theory is an abstract nonsense" is even physically not true.

The schematic language of triangulated category presents a new stage of string theory.

To illuminate this idea, I will draw your attention to the blow-up minimal model

of complex algebraic surfaces. This is done under the hypothetical assumptions

of "generalized complex structure" of cotangent bundle due to Hitchin school.

The coordinate transformation Jacobian matrices of the measure of sigma model

with spin structures cause one part of the gravitational "anomaly cancellation"

of smooth Kahler manifold $X$ and Weyl anomaly of compact Riemann surface $\\Sigma$.

$Anom = c_1 (X) c_1 (\\Sigma) \\oplus ch_2 (X)$,

in terms of 1st and 2nd Chern characters. Note that when $\\Sigma$ is a puctured disk

with flat metric, the chiral algebra is nothing but the ordinary vertex algebra.

Note that I do not explain the complex differential geometry,

but essentially more recent works with the category of DGA (Diffenreial Graded Algebra),

which is behind the super conformal field theory of chiral algebras.

My result of "vanishing tachyon" (nil-radical part of vertex algebras)

and "causality resortation" in compactified non-critical heterotic sigma model

is physically a promising idea of new solution to unitary representation of operator algebras.

This idea is realized in the formalism of BRST cohomology and its generalization

in $\\mathcal{N} = (0,2)$ supersymmetry, that is, non-commutative geometry

with non-linear constraint condition of pure spinors for covariant quantization.

### 2009年07月27日(月)

17:00-18:30 数理科学研究科棟(駒場) 002号室

Mapping class group for hyperkaehler manifolds

**Misha Verbitsky 氏**(ITEP Moscow/IPMU)Mapping class group for hyperkaehler manifolds

[ 講演概要 ]

A mapping class group is a group of orientation-preserving

diffeomorphisms up to isotopy. I explain how to compute a

mapping class group of a hyperkaehler manifold. It is

commensurable to an arithmetic lattice in a Lie group

$SO(n-3,3)$. This makes it possible to state and prove a

new version of Torelli theorem.

A mapping class group is a group of orientation-preserving

diffeomorphisms up to isotopy. I explain how to compute a

mapping class group of a hyperkaehler manifold. It is

commensurable to an arithmetic lattice in a Lie group

$SO(n-3,3)$. This makes it possible to state and prove a

new version of Torelli theorem.

### 2009年06月08日(月)

17:00-18:30 数理科学研究科棟(駒場) 002号室

Multiplication in differential cohomology and cohomology operation

**Kiyonori Gomi 氏**(Kyoto University)Multiplication in differential cohomology and cohomology operation

[ 講演概要 ]

The notion of differential cohomology refines generalized

cohomology theory so as to incorporate information of differential

forms. The differential version of the ordinary cohomology has been

known as the Cheeger-Simons cohomology or the smooth Deligne

cohomology, while the general case was introduced by Hopkins and

Singer around 2002.

The theme of my talk is the cohomology operation induced from the

squaring map in the differential ordinary cohomology and the

differential K-cohomology: I will relate these operations to the

Steenrod operation and the Adams operation. I will also explain the

roles that the squaring maps play in 5-dimensional Chern-Simons theory

for pairs of B-fields and Hamiltonian quantization of generalized

abelian gauge fields.

The notion of differential cohomology refines generalized

cohomology theory so as to incorporate information of differential

forms. The differential version of the ordinary cohomology has been

known as the Cheeger-Simons cohomology or the smooth Deligne

cohomology, while the general case was introduced by Hopkins and

Singer around 2002.

The theme of my talk is the cohomology operation induced from the

squaring map in the differential ordinary cohomology and the

differential K-cohomology: I will relate these operations to the

Steenrod operation and the Adams operation. I will also explain the

roles that the squaring maps play in 5-dimensional Chern-Simons theory

for pairs of B-fields and Hamiltonian quantization of generalized

abelian gauge fields.