IPMU Komaba Seminar
Mathematical Sciences Building,
Komaba Campus, The University of Tokyo
This seminar is organized in collaboration with
Kavli Institute for the Physics and Mathematics
of the Universe (Kavli IPMU) and Graduate School of Mathematical Sciences,
The University of Tokyo.
Last updated May 10, 2012
Information : Akishi Kato, Toshitake Kohno
May 21, 2012 -- Room 002, 17:00 -- 18:30
Emanuel Scheidegger（The University of Freiburg）
Topological Strings on Elliptic Fibrations
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.
January 20, 2012 -- Room 056, 14:45 -- 16:15
Albrecht Klemm (The University of Bonn)
Refined holomorphic anomaly equations
Abstract: We propose a derivation of refined holomorphic
anomaly equation from the word-sheet point of
view and discuss the interpretation of the
refined BPS invariants for local Calabi-Yau
spaces.
November 21, 2011 -- Room 002, 16:30 -- 18:00
Siu-Cheong Lau (IPMU)
Enumerative meaning of mirror maps for toric Calabi-Yau manifolds
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.
January 31, 2011 -- Room 002, 16:30 -- 18:00
Kwok-Wai Chan (IPMU)
Mirror symmetry for toric Calabi-Yau manifolds from the SYZ viewpoint
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.
November 29, 2010 -- Room 002, 16:30 -- 18:00
Scott Carnahan (IPMU)
Borcherds products in monstrous moonshine.
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.
November 26, 2010 -- Room 002, 14:40 -- 16:10
Tomoo Matsumura (Cornell University)
Hamiltonian torus actions on orbifolds and orbifold-GKM theorem
(joint work with T. Holm)
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.
October 18, 2010 -- Room 002, 16:30 -- 18:00
Todor Milanov (IPMU)
Quasi-modular forms and Gromov--Witten theory of elliptic orbifold P1
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 P1'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.
April 26, 2010 -- Room 002, 16:30 -- 18:00
Akishi Ikeda (The University of Tokyo)
The correspondence between Frobenius algebra of Hurwitz numbers
and matrix models
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.
February 1, 2010 -- Room 002, 16:30 -- 18:00
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
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.
December 7, 2009 -- Room 002, 17:30 -- 19:00
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.
November 30, 2009 -- Room 002, 16:30 -- 18:00
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.
November 9, 2009 -- Room 002, 16:30 -- 18:00
Makoto Sakurai (The University of Tokyo)
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.
June 27, 2009 -- Room 002, 17:00 -- 18:30
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.
June 8, 2009 -- Room 002, 17:00 -- 18:30
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.
December 1, 2008 -- Room 002, 17:00 -- 18:30
Kentaro Hori (University of Toronto / IPMU)
A pair of non-birational but derived equivalent Calabi-Yau
manifolds from non-Abelian gauge theories
Abstract: We construct a family of (2,2) supersymmetric gauge theories
in 2-dimensions that flows to a family of (2,2) superconformal fields theories with \hat{c}=3. The family has two limits and three singular points. The two limits correspond to two Calabi-Yau manifolds which are not birationally equivalent. The two are, however, derived equivalent
by general principle of supersymmetric quantum field theory.
July 28, 2008 -- Room 002, 17:00 -- 18:30
Lin Weng (Kyushu University)
Symmetries and the Riemann Hypothesis
Abstract: Associated to each pair of a reductive group
and its maximal parabolic, we will introduce an abelian zeta function.
This zeta, defined using Weyl symmetries, is expected
to satisfy a standard functional equation and the Riemann Hypothesis.
Its relation with the so-called high rank zeta,
a very different but closely related non-abelian zeta,
defined using stable lattices and a new geo-arithmetical cohomology,
will be explained.
Examples for SL, SO, Sp and G_2 and confirmations of
(Lagarias and) Masatoshi Suzuki on the RH for zetas
associated to rank 1 and 2 groups will be presented
as well.
The manuscript is available at:
http://xxx.lanl.gov/abs/0803.1269
June 30, 2008-- Room 002, 17:00 -- 18:30
J.Manuel Garcia-Islas (National Autonomous University of Mexico)
Quantum topological invariants and black hole entropy
Abstract: A type of topological invariants of three manifolds were
introduced by Turaev and Viro. We will define an invariant of graphs
embedded in a three dimensional manifold in a Turaev-Viro spirit.
The relation of these invariants to mathematical physics is
a really nice one. We will show how entropy of a three dimensional
black hole known as BTZ can be described using our formulation.
June 2, 2008 -- Room 002, 17:00 -- 18:30
Shinobu Hikami (The University of Tokyo)
Intersection theory from duality and replica
Abstract: Kontsevich's work on Airy matrix integrals has led to explicit results for the
intersection numbers of the moduli space of curves. In this article we show that a duality between k-point functions on N by N matrices and N-point functions of k by k matrices, plus the replica method, familiar in the theory of disordered systems, allows one to recover Kontsevich's results on the intersection numbers, and to generalize them to other models. This provides an alternative and simple way to compute intersection numbers with one marked point, and leads also to some new results. This is a joint work with E. Brezin (Comm.Math. Phys. in press, arXiv:0708.2210).
Lectures by Jean-Michel Bismut (Univ. Paris-Sud, Orsay)
Lecture 1, May 12 , 2008-- Room 002, 17:00 -- 18:30
The hypoelliptic LaplacianA survey of Quillen metrics
Abstract: Let $X$ be a compact Riemannian manifold. The Laplace Beltrami
operator $-\Delta^{X}$, or more generally the Hodge Laplacian
$\square^{X}$, is an elliptic second order self adjoint operator on $X$.
We will explain the construction of a deformation of the elliptic
Laplacian to a family of hypoelliptic operators acting on the total
space of the cotangent bundle $\mathcal{X}$. These operators depend
on a parameter $b>0$, and interpolate between the Hodge Laplacian
(the limit as $b\to 0$) and the geodesic flow (the limit as $b\to +
\infty $).
Actually, the deformed Laplacian is associated with an exotic Hodge
theory on the total space of the cotangent bundle, in which the
standard $L_{2}$ scalar product on forms is replaced by a
symmetric bilinear form of signature $\left( \infty, \infty \right)$.
This deformation can be understood as a version of the Witten
deformation on the loop space associated with the energy functional.
From a probabilistic point of view, the deformed Laplacian
corresponds to a Langevin process.
The above considerations can also be used in complex geometry, in
which the Dolbeault cohomology is considered instead of the Rham cohomology.
Results obtained with Gilles Lebeau on the analysis of the
hypoelliptic Laplacian will also be presented, as well as
applications to analytic torsion.
Lecture 2, May 19, 2008 -- Room 002, 17:00 -- 18:30
A survey of Quillen metrics
Abstract: In this lecture, I will survey basic results
on Quillen metrics.
Indeed let $X$ be a complex K\"ahler manifold, and let $E$ be a
holomorphic Hermitian vector bundle on $X$. Let $\lambda$ be the complex line
which is the determinant of the cohomology of $E$. The Quillen metric
is a metric on the line $\lambda$, which one obtains using a spectral
invariant of the Hodge Laplacian, the Ray-Singer analytic torsion.
The Quillen metrics have a number of remarkable properties. Among them
the curvature theorem says that when one considers a family of such
$X$, the curvature of the holomorphic Hermitian connection on
$\lambda$ is given by a formula of Riemann-Roch-Grothendieck type.
I will explain some of the ideas which go into the proof of these
properties, which includes Quillen's superconnections.
April 24, 2008 -- Room 056, 17:00 -- 18:30
Motohico Mulase (University of California, Davis)
Recursion relations in intersection theory
on the moduli spaces of Riemann surfaces
Abstract: In this talk I will give a survey of
recent developments in the intersection theory
of tautological classes on the moduli spaces
of stable algebraic curves. The emphasis is
placed on explaining where the Virasoro
constraint conditions are originated. Recently
several authors have encountered the same
combinatorial recursion relation from completely
different contexts, that eventually leads to the
Virasoro constraint. This mysterious structure
of the theory will be surveyed.
IPMU Komaba Lectures, Tuesday, April -- July, 2008 -- Room 002, 10:30 -- 12:00
First Lecture : April 8
Akihiro Tsuchiya (IPMU)
Homotopy Theory (before 1970)
Recently the notion of homotopy theory has been widely used in many areas of
contemporary mathematics including mathematical physics.
The purpose of the lectures is to present an overview of the developments
of homotopy theory mainly from 1940's through 1960's, partly in view of
more recent progress in other areas.
(1) Prehistory of homotopy theory
-- Hurewicz theorem, Hopf theorem, Freudentahl suspension theorem
(2) Eilenberg-MacLane space and Postnikov system
(3) Steenrod algebras
(4) Serre's theorem on the homotopy groups of spheres
(5) Rational homotopy theory
(6) Stable homotopy category and Adams spectral sequence
(7) Vector bundles and characteristic classes
(8) Complex cobordism and Quillen's theorem
(9) Miscellaneous topics
Rereferences :
(1) J.P.May, A Concise Course in Algebraic Topology，
The University of Chicago Press
http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf
(2) Douglas Ravenel, Complex cobordism and stable homotopy groups of spheres
The second edition, AMS Chelsea Series
http://www.math.rochester.edu/u/faculty/doug/mu.html
(3) Mark Hovey, Model Category, AMS
(4) Gelfand and Manin, Homology Algebra
Special Lecture, March 10, 2008 -- Lecture Hall, 15:00 -- 16:00
Professor Shing-Tung Yau (Harvard University)
Geometric analysis and their applications in mathematics and theoretical physics
Over last ten years or so, we have seen many important developments in
mathematics and theoretical physics, which appeared from deep interactions
between them. As a special lecture of IPMU Komaba seminar, we have asked
Professor Shing-Tung Yau to survey recent developments in geometry and physics.
Professor Yau received the Fields Medal from the International Congress
of Mathematicians in 1982. He also received the Oswald Veblen Prize from the
American Mathematical Society (1981), the John J. Carty Award from the US
National Academy of Sciences (1981), the MacArthur Foundation Fellowship
Prize (1985), the Humboldt Research Award (1991), the Crafoord Prize (1994),
and the US National Medal of Science (1997).
February 12, 2008 -- Room 056, 17:00 -- 18:30
Katrin Wendland (University of Augsburg)
How to lift a construction by Hiroshi Inose to conformal field theory
The moduli space of Einstein metrics is well known to algebraic and differential geometers. Physicists have introduced the notion of conformal field theories (CFTs) associated to K3, and the moduli space of these objects is well understood as well. It can be interpreted as a generalisation of the moduli space of Einstein metrics on K3, which allows us to introduce this space without having to use background knowledge from conformal field theory. However, just as no smooth Einstein metrics on K3 are known explicitly, the explicit construction of CFTs associated to K3 in general remains an open problem. The only known constructions which allow to deal with families of CFTs give CFTs associated to K3 surfaces with orbifold singularities.
We use a classical construction by Hiroshi Inose to explicitly construct a family of CFTs which are associated to a family of smooth algebraic K3 surfaces. Although these CFTs were known before, it is remarkable that they allow a description in terms of a family of smooth surfaces whose complex structure is deformed while all other geometric data remain fixed.
We also discuss possible extensions of this result to higher dimensional Calabi-Yau threefolds.
Retrospectives