Kavli IPMU Komaba Seminar

Seminar information archive ~07/24Next seminarFuture seminars 07/25~

Date, time & place Monday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.)

Seminar information archive


13:30-14:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Edouard Brezin (lpt ens, Paris)
Various applications of supersymmetry in statistical physics (English)
[ Abstract ]
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.


10:30-11:30   Room #128 (Graduate School of Math. Sci. Bldg.)
Naichung Conan Leung (The Chinese University of Hong Kong)
Donaldson-Thomas theory for Calabi-Yau fourfolds.
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Anatol Kirillov (RIMS, Kyoto University)
On some quadratic algebras with applications to Topology,
Algebra, Combinatorics, Schubert Calculus and Integrable Systems. (ENGLISH)
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
A.P. Veselov (Loughborough, UK and Tokyo)
Universal formulae for Lie groups and Chern-Simons theory (ENGLISH)
[ Abstract ]
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.


17:00-18:30   Room #128 (Graduate School of Math. Sci. Bldg.)
Hans Jockers (The University of Bonn)
Characteristic classes from 2d renormalized sigma-models (ENGLISH)
[ Abstract ]
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.


17:00-18:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Mauricio Romo (Kavli IPMU)
Exact Results In Two-Dimensional (2,2) Supersymmetric Gauge Theories With Boundary
[ Abstract ]
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.


17:00-18:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Daniel Pomerleano (Kavli IPMU)
Homological Mirror Symmetry for toric Calabi-Yau varieties (ENGLISH)
[ Abstract ]
I will discuss some recent developments in Homological Mirror
Symmetry for toric Calabi-Yau varieties.


17:00-18:30   Room #122 (Graduate School of Math. Sci. Bldg.)
Richard Eager (Kavli IPMU)
Elliptic genera and two dimensional gauge theories (ENGLISH)
[ Abstract ]
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.


17:00-18:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Atsushi Kanazawa (University of British Columbia)
Calabi-Yau threefolds of Type K (ENGLISH)
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Changzheng Li (Kavli IPMU)
Quantum cohomology of flag varieties (ENGLISH)
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Bong Lian (Brandeis University)
Period Integrals and Tautological Systems (ENGLISH)
[ Abstract ]
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.


17:00-18:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Emanuel Scheidegger (The University of Freiburg)
Topological Strings on Elliptic Fibrations (ENGLISH)
[ Abstract ]
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.


14:45-16:15   Room #056 (Graduate School of Math. Sci. Bldg.)
Albrecht Klemm (The University of Bonn)
Refined holomorphic anomaly equations (ENGLISH)
[ Abstract ]
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


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Siu-Cheong Lau (IPMU)
Enuemerative meaning of mirror maps for toric Calabi-Yau manifolds (ENGLISH)
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Kwok-Wai Chan (IPMU, the University of Tokyo)
Mirror symmetry for toric Calabi-Yau manifolds from the SYZ viewpoint (ENGLISH)
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Scott Carnahan (IPMU)
Borcherds products in monstrous moonshine. (ENGLISH)
[ Abstract ]
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.


14:40-16:10   Room #002 (Graduate School of Math. Sci. Bldg.)
Tomoo Matsumura (Cornell University)
Hamiltonian torus actions on orbifolds and orbifold-GKM theorem (joint
work with T. Holm) (JAPANESE)
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Todor Milanov (IPMU)
Quasi-modular forms and Gromov--Witten theory of elliptic orbifold $\\mathbb{P}^1$ (ENGLISH)
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Akishi Ikeda (The University of Tokyo)
The correspondence between Frobenius algebra of Hurwitz numbers
and matrix models (JAPANESE)
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Timur Sadykov (Siberian Federal University)
Bases in the solution space of the Mellin system
[ Abstract ]
I will present a joint work with Alicia Dickenstein.
We consider algebraic functions $z$ satisfying equations of the
a_0 z^m + a_1z^{m_1} + a_2 z^{m_2} + \\ldots + a_n z^{m_n} +
a_{n+1} =0.
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.


17:30-19:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Weiping Zhang (Chern Institute of Mathematics, Nankai University)
Geometric quantization on noncompact manifolds
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Junya Yagi (Rutgers University)
Chiral Algebras of (0,2) Models: Beyond Perturbation Theory
[ Abstract ]
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.


16:30-18:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Makoto Sakurai (東京大学大学院数理科学研究科)
Differential Graded Categories and heterotic string theory
[ Abstract ]
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.


17:00-18:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Misha Verbitsky (ITEP Moscow/IPMU)
Mapping class group for hyperkaehler manifolds
[ Abstract ]
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.


17:00-18:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Kiyonori Gomi (Kyoto University)
Multiplication in differential cohomology and cohomology operation
[ Abstract ]
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.

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