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Kavli IPMU Komaba Seminar
Seminar information archive ～04/24｜Next seminar｜Future seminars 04/25～
| Date, time & place | Monday 16:30 - 18:00 002Room #002 (Graduate School of Math. Sci. Bldg.) |
|---|
Seminar information archive
2008/12/01
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Kentaro Hori (University of Toronto / IPMU)
A pair of non-birational but derived equivalent Calabi-Yau
manifolds from non-Abelian gauge theories
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.
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.
2008/07/28
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Lin Weng (Kyushu University)
Symmetries and the Riemann Hypothesis
http://xxx.lanl.gov/abs/0803.1269
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.
[ Reference URL ]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.
http://xxx.lanl.gov/abs/0803.1269
2008/06/30
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
J.Manuel Garcia-Islas (National Autonomous University of Mexico)
Quantum topological invariants and black hole entropy
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.
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.
2008/06/02
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Shinobu Hikami (The University of Tokyo)
Intersection theory from duality and replica
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).
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).
2008/05/19
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Jean-Michel Bismut (Univ. Paris-Sud, Orsay)
A survey of Quillen metrics
Jean-Michel Bismut (Univ. Paris-Sud, Orsay)
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.
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.
2008/05/12
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Jean-Michel Bismut (Univ. Paris-Sud, Orsay)
The hypoelliptic Laplacian
Jean-Michel Bismut (Univ. Paris-Sud, Orsay)
The hypoelliptic Laplacian
[ 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.
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.
2008/04/24
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Motohico Mulase (University of California, Davis)
Recursion relations in intersection theory on the moduli spaces of Riemann surfaces
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.
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.
2008/04/08
10:30-12:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Akihiro Tsuchiya (IPMU, The University of Tokyo)
IPMU Komaba Lectures,Homotopy Theory (before 1970)
Akihiro Tsuchiya (IPMU, The University of Tokyo)
IPMU Komaba Lectures,Homotopy Theory (before 1970)
[ Abstract ]
Tuesday, April -- July, 2008
First Lecture Aprl 8
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
Tuesday, April -- July, 2008
First Lecture Aprl 8
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
2008/02/12
17:00-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)
Katrin Wendland (University of Augrburg)
How to lift a construction by Hiroshi Inose to conformal field theory
Katrin Wendland (University of Augrburg)
How to lift a construction by Hiroshi Inose to conformal field theory
[ Abstract ]
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.
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.
2007/12/17
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Ken-Ichi Yoshikawa (The University of Tokyo)
Analytic torsion for Calabi-Yau threefolds
Ken-Ichi Yoshikawa (The University of Tokyo)
Analytic torsion for Calabi-Yau threefolds
[ Abstract ]
In 1994, Bershadky-Cecotti-Ooguri-Vafa conjectured that analytic torsion
gives rise to a function on the moduli space of Calabi-Yau threefolds and
that it coincides with the quantity $F_{1}$ in string theory.
Since the holomorphic part of $F_{1}$ is conjecturally the generating function
of the counting problem of elliptic curves in the mirror Calabi-Yau threefold,
this implies the conjectural equivalence of analytic torsion and the counting
problem of elliptic curves for Calabi-Yau threefolds through mirror symmetry.
After Bershadsky-Cecotti-Ooguri-Vafa, we introduced an invariant of
Calabi-Yau threefolds, which we obtained using analytic torsion and
a Bott-Chern secondary class. In this talk, we will talk about the construction
and some explicit formulae of this analytic torsion invariant.
Some part of this talk is based on the joint work with H. Fang and Z. Lu.
In 1994, Bershadky-Cecotti-Ooguri-Vafa conjectured that analytic torsion
gives rise to a function on the moduli space of Calabi-Yau threefolds and
that it coincides with the quantity $F_{1}$ in string theory.
Since the holomorphic part of $F_{1}$ is conjecturally the generating function
of the counting problem of elliptic curves in the mirror Calabi-Yau threefold,
this implies the conjectural equivalence of analytic torsion and the counting
problem of elliptic curves for Calabi-Yau threefolds through mirror symmetry.
After Bershadsky-Cecotti-Ooguri-Vafa, we introduced an invariant of
Calabi-Yau threefolds, which we obtained using analytic torsion and
a Bott-Chern secondary class. In this talk, we will talk about the construction
and some explicit formulae of this analytic torsion invariant.
Some part of this talk is based on the joint work with H. Fang and Z. Lu.
2007/12/10
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Dmitry Kaledin (Steklov Institute and The University of Tokyo)
Deligne conjecture and the Drinfeld double.
Dmitry Kaledin (Steklov Institute and The University of Tokyo)
Deligne conjecture and the Drinfeld double.
[ Abstract ]
Deligne conjecture describes the structure which exists on
the Hochschild cohomology $HH(A)$ of an associative algebra
$A$. Several proofs exists, but they all combinatorial to a certain
extent. I will present another proof which is more categorical in
nature (in particular, the input data are not the algebra $A$, but
rather, the tensor category of $A$-bimodules). Combinatorics is
still there, but now it looks more natural -- in particular, the
action of the Gerstenhaber operad, which is know to consist of
homology of pure braid groups, is induced by the action of the braid
groups themselves on the so-called "Drinfeld double" of the category
$A$-bimod.
If time permits, I will also discuss what additional structures
appear in the Calabi-Yau case, and what one needs to impose to
insure Hodge-to-de Rham degeneration.
Deligne conjecture describes the structure which exists on
the Hochschild cohomology $HH(A)$ of an associative algebra
$A$. Several proofs exists, but they all combinatorial to a certain
extent. I will present another proof which is more categorical in
nature (in particular, the input data are not the algebra $A$, but
rather, the tensor category of $A$-bimodules). Combinatorics is
still there, but now it looks more natural -- in particular, the
action of the Gerstenhaber operad, which is know to consist of
homology of pure braid groups, is induced by the action of the braid
groups themselves on the so-called "Drinfeld double" of the category
$A$-bimod.
If time permits, I will also discuss what additional structures
appear in the Calabi-Yau case, and what one needs to impose to
insure Hodge-to-de Rham degeneration.
2007/11/26
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Mich\"ael Pevzner (Universit\'e de Reims and the University of Tokyo)
Kontsevich quantization of Poisson manifolds and Duflo isomorphism.
Mich\"ael Pevzner (Universit\'e de Reims and the University of Tokyo)
Kontsevich quantization of Poisson manifolds and Duflo isomorphism.
[ Abstract ]
Abstract: Since the fundamental results by Chevalley, Harish-Chandra and Dixmier one knows that the set of invariant polynomials on the dual of a Lie algebra of a particular type (solvable, simple or nilpotent) is isomorphic, as an algebra, to the center of the enveloping algebra. This fact was generalized to an arbitrary finite-dimensional real Lie algebra by M. Duflo in late 1970's. His proof was based on the Kirillov's orbits method that parametrizes infinitesimal characters of unitary irreducible representations of the corresponding Lie group in terms of co-adjoint orbits.
The Kontsevich' Formality theorem implies not only the existence of the Duflo map but shows that it is canonical. We shall describe this construction and indicate how does this construction extend to the whole Poisson cohomology of an arbitrary finite-dimensional real Lie algebra.
Abstract: Since the fundamental results by Chevalley, Harish-Chandra and Dixmier one knows that the set of invariant polynomials on the dual of a Lie algebra of a particular type (solvable, simple or nilpotent) is isomorphic, as an algebra, to the center of the enveloping algebra. This fact was generalized to an arbitrary finite-dimensional real Lie algebra by M. Duflo in late 1970's. His proof was based on the Kirillov's orbits method that parametrizes infinitesimal characters of unitary irreducible representations of the corresponding Lie group in terms of co-adjoint orbits.
The Kontsevich' Formality theorem implies not only the existence of the Duflo map but shows that it is canonical. We shall describe this construction and indicate how does this construction extend to the whole Poisson cohomology of an arbitrary finite-dimensional real Lie algebra.
2007/10/29
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Hiroshige Kajiura (RIMS, Kyoto University)
Some examples of triangulated and/or $A_\\infty$-categories
related to homological mirror symmetry
Hiroshige Kajiura (RIMS, Kyoto University)
Some examples of triangulated and/or $A_\\infty$-categories
related to homological mirror symmetry
[ Abstract ]
In this talk, I would like to discuss on some examples of
triangulated and/or $A_\\infty$-categories associated to
manifolds with additional structures
(symplectic structure, complex structure, ...)
which can appear in the homological mirror symmetry (HMS) set-up
proposed by Kontsevich'94.
The strongest form of the HMS may be to show the equivalence
of Fukaya category on a symplectic manifold with the category
of coherent sheaves on the mirror dual complex manifold
at the level of $A_\\infty$-categories.
On the other hand, for a given $A_\\infty$-category,
there is a canonical way (due to Bondal-Kapranov, Kontsevich)
to construct an enlarged $A_\\infty$-category
whose restriction to the zero-th cohomology forms a triangulated category.
I plan to talk about the triangulated structure of categories
associated to regular systems of weights
(joint work with Kyoji Saito and Atsushi Takahashi),
and also give a realization of higher $A_\\infty$-products in
Fukaya categories from the mirror dual complex manifold
via HMS in some easy examples.
In this talk, I would like to discuss on some examples of
triangulated and/or $A_\\infty$-categories associated to
manifolds with additional structures
(symplectic structure, complex structure, ...)
which can appear in the homological mirror symmetry (HMS) set-up
proposed by Kontsevich'94.
The strongest form of the HMS may be to show the equivalence
of Fukaya category on a symplectic manifold with the category
of coherent sheaves on the mirror dual complex manifold
at the level of $A_\\infty$-categories.
On the other hand, for a given $A_\\infty$-category,
there is a canonical way (due to Bondal-Kapranov, Kontsevich)
to construct an enlarged $A_\\infty$-category
whose restriction to the zero-th cohomology forms a triangulated category.
I plan to talk about the triangulated structure of categories
associated to regular systems of weights
(joint work with Kyoji Saito and Atsushi Takahashi),
and also give a realization of higher $A_\\infty$-products in
Fukaya categories from the mirror dual complex manifold
via HMS in some easy examples.
2007/10/15
17:00-18:30 Room #002 (Graduate School of Math. Sci. Bldg.)
Shinobu Hosono (The University of Tokyo)
Topics on string theory, mirror symmetry, and Gromov-Witten invariants
Shinobu Hosono (The University of Tokyo)
Topics on string theory, mirror symmetry, and Gromov-Witten invariants
[ Abstract ]
Recently, some technical developments in solving BCOV
(Bershadsky-Cecotti-Ooguri-Vafa) holomorphic anomaly equation has been
made and it has become possible to predict higher genus Gromov-Witten
invariants for some class of Calabi-Yau 3 folds.
With a brief introduction to BCOV equation, I will present some
predictions for Gromov-Witten invariants of certain Calabi-Yau 3 folds,
which are not birational but derived equivalent. (This is based on
a work with Y. Konishi which appeared in mathAG/0704.2928.)
Before coming to this specific topic, I will review some recent
topics of the homological mirror symmetry focusing on
its connection to the `classical' mirror symmetry, where the
variation theory of Hodge structures (VHS) plays a central role.
The BCOV equation and its open string generalization have their grounds
on the VHS.
Recently, some technical developments in solving BCOV
(Bershadsky-Cecotti-Ooguri-Vafa) holomorphic anomaly equation has been
made and it has become possible to predict higher genus Gromov-Witten
invariants for some class of Calabi-Yau 3 folds.
With a brief introduction to BCOV equation, I will present some
predictions for Gromov-Witten invariants of certain Calabi-Yau 3 folds,
which are not birational but derived equivalent. (This is based on
a work with Y. Konishi which appeared in mathAG/0704.2928.)
Before coming to this specific topic, I will review some recent
topics of the homological mirror symmetry focusing on
its connection to the `classical' mirror symmetry, where the
variation theory of Hodge structures (VHS) plays a central role.
The BCOV equation and its open string generalization have their grounds
on the VHS.
