
THE FIELDS
INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES 
Thematic
Program on CalabiYau Varieties: Arithmetic, Geometry
and Physics
September
1620, 2013
Workshop on Modular forms around string theory
Principal
Organizers:
Charles F. Doran, Matthias Schütt, NorikoYui
Fields Institute, Room 230



Preliminary Schedule
Time

September 16
Monday

September 17
Tuesday

September 18
Wednesday

Time

September 19
Thursday

September 20
Friday

9:0010:00




9:3010:30



10:0010:30

Coffee Break

10:3010:45

Coffee Break

10:30–11:30




10:45–11:45



11:30–12:30




11:45–12:45



12:3014:00

Lunch Break

12:4514:00

Lunch Break

14:00–15:00



Free

14:00–15:00



15:0015:30

Tea Break


Tea Break

15:30–16.30



Free

15:30–16.30


TBA

17:00–19:00


Reception





Speaker 
Title and Abstract 
Candelas, Philip
Oxford University 
Puzzles to do with the zetafunction for the
quintic threefold
The lines in the Dwork pencil of quintic threefold 
Clingher, Adrian
University of Missouri at St. Louis 
Modular forms associated to K3 surfaces endowed with lattice polarizations
of high Picard rank
I will discuss several cases of lattice polarizations of high
Picard rank on a K3 su rface. A classification for these objects will
be presented in terms of quartic normal forms. Modular forms of appropriate
group appear as coefficients of the normal forms.

Doran, Charles
University of Alberta 
Families of latticepolarized K3 surfaces with monodromy
We extend the notion of a latticepolarized K3 surface to
families, study the action of monodromy on the NeronSeveri group of
the general fiber, and use this to ”undo” the Kummer and ShiodaInose
structures in families.This technique sheds important light on the 14
families of CalabiYau threefolds with $h_{2,1}$ = 1 studied by Doran
Morgan.
This is joint work with Andrew Harder, Andrey Novoseltsev, and Alan
Thompson.

Peng Gao
Harvard and Simons Institute 
Extremal bundles on CalabiYau manifolds
Motivated by the goal to better understand the implications
of stability conditions on numerical invariants, we study explicit constructions
of (heterotic string) vector bundles on CalabiYau 3 folds. This includes
both the monad construction and spectral cover bundles over elliptically
fibered CY threefolds. We compare our results with the DRY (DouglasReinbacherYau)
conjecture about generalized BogomolovYau inequalities.
This is a joint work with Y.H. He and S.T. Yau.

Golyshev, Vasily
IITP Mosco 
Fano threefolds and mirror duality
We discuss recent joint work with Coates, Corti, Galkin and
Kasprzyk on a mirror link between Fano threefolds and a class of threefolds
obtained by generalizing certain modular threefolds.

Hosono, Shinobu
Tokyo University 
Mirror symmetry of determinantal quintics
I describe mirror symmetry of determinantal quintics defined
by generic 5 × 5 matrices with entries linear in coordinates of
$P^4$. A generic determinantal quintic is singular at 50 nodes, and
has a small resolution which is a CalabiYau threefold of $h^{1,1}$
= 2 and $h^{2,1}$ = 52. I will consider the mirror family of this quintic
by the orbifold construction starting from a special family of the determinantal
quintic. It turns out that the singularities of the special family are
similar to the BarthNieto quintic, although there are some complications
in our case. After making a crepant resolution, we obtain the mirror
family, namely we find that the orbifold group Gorg is trivial in this
case. I will also describe CalabiYau manifolds related to determinantal
quintics which admit free $Z_2$ quotients.
This is based on the collaborations with Hiromichi Takagi.

Kelly, Tyler
University of Pennsylvania

Berglund–Hübsch–Krawitz Mirrors via Shioda Maps
We will introduce the Shioda map into the BerglundHübschKrawitz
mirror duality proven by Chiodo and Ruan. In particular, we will find
a new proof of birationality of BHK mirrors to certain orbifold quotients
of hypersurfaces of weightedprojective nspace. We hope to talk about
workinprogress about generalized Shioda maps, BHK mirrors and PicardFuchs
equations.

Kudla, Steve
University of Toronto 
Another product formula for a Borcherds form
In his celebrated 1998 Inventiones paper, Borcherds constructed
meromorphic automorphic forms $\Psi(F)$ for arithmetic subgroups associated
to even integral lattices M of signature (n, 2). The input to his construction
is a vector valued weakly holomorphic modular form F of weight 1n/2,
and the resulting Borcherds form has an explicit divisor on the arithmetic
quotient X = $\Gamma_M \ D$. Most remarkably, in the neighborhood of
each cusp (= rational point boundary component), there is a beautiful
product formula for $\Psi(F)$, reminiscent of the classical product
formula for the Dedekind etafunction. In this lecture, we will describe
an analogous product formula for $\Psi(F)$ in the neighborhood of each
1dimensional rational boundary component. This formula, which, like
that of Borcherds, is obtained through the calculation of a regularized
theta integral, reveals the behavior of $\Psi(F)$ on a (partial) smooth
compactification of X.

Malmendier, Andreas
Colby College 
Heterotic/Ftheory duality and lattice polarized K3 surfaces.
The heterotic string compactified on $T^2$ has a large discrete
symmetry group SO(2, 18;Z), which acts on the scalars in the theory
in a natural way; there have been a number of attempts to construct
models in which these scalars are allowed to vary by using SO(2, 18;Z)invariant
functions. In our new work (which is joint work with David Morrison),
we give a more complete construction of these models in the special
cases in which either there are no Wilson lines–and SO(2, 2;Z)
symmetry– or there is a single Wilson line–and SO(2, 3;Z)
symmetry. In those cases, the modular forms can be analyzed in detail
and there turns out to be a precise theory of K3 surfaces with prescribed
singularities which corresponds to the structure of the modular forms.
This allows us to construct interesting examples of smooth Calabi–Yau
threefolds as elliptic fibrations over Hirzebruch surfaces from pencils
of irreducible genustwo curves.

Murthy, Sameer
NIKHEF, Amsterdam 
(I) Quantum black holes, wall crossing, and mock modular forms.
In the quantum theory of black holes in superstring theory,
the physical problem of counting the number of quarterBPS dyonic states
of a given charge has led to the study of Fourier coefficients of certain
meromorphic Siegel modular forms and to the question of the modular
nature of the corresponding generating functions. These Fourier coefficients
have a wallcrossing behavior which seems to destroy modularity. In
this talk I shall explain that these generating functions belong to
a class of functions called mock modular forms. I shall then discuss
some interesting examples that arise from this construction.
This is based on joint work with Atish Dabholkar and Don Zagier.
(II) Mathieu moonshine, mock modular forms and string theory.
I shall discuss a conjecture of Eguchi, Ooguri and Tachikawa
from 2010 that relates the elliptic genus of K3 surfaces and representations
of M24, the largest Mathieu group. The generating function of these
representations is a mock theta function of weight onehalf. After discussing
some properties of this function, I shall present a particular appearance
of this function in string theory that suggests a construction of a
nontrivial infinitedimensional M24module.
This is based on joint work with Jeff Harvey.

Pioline, Boris
University of Jussieu 
RankinSelberg methods for closed string amplitudes.
After integrating over location of vertex operators and supermoduli,
scattering amplitudes in closed string theories at genus $h \leq 3$
are expressed as an integral of a Siegel modular form on the fundamental
domain of Siegel’s upper half plane. I will describe techniques
to compute such modular integrals explicitly, by representing the integrand
as a Poincar´e series and applying the unfolding trick. The focus
will be mainly on genus one, but some results on higher genus will be
presented.
Based in part on work in collaboration with C. Angelantonj and I. Florakis.

Rose, Simon
Fields Institute 
Towards a reduced mirror symmetry for the quartic K3.
Mirror symmetry in terms of Yukawa couplings for a K3 is relatively
trivial, due to the triviality of its GromovWitten invariants. Using
reduced invariants, however, we can still tease out a lot of enumerative
details of these surfaces. As these reduced invariants satisfy the same
relations that ordinary invariants do, this raises the natural question:
Is there a reduced Bmodel theory?
In this talk we wil go over our current work on this project, which
is joint with Helge Ruddat.

Ruan, Yongbin
Michigan University 
(I) and (II): Mirror symmetry and modular forms.
Traditionally, we use mirror symmetry to map a difficult problem
(Amodel) to an easier problem (Bmodel). Recently, there is a great
deal of activities in mathematics to understand the modularity properties
of GromovWitten theory, a phenomenon suggested by BCOV almost twenty
years ago. Mirror symmetry is again used in a crucial way. However,
the new usage of mirror does not map a difficult problem to easy problem.
Instead, we make both side of mirror symmetry to work together in a
deep way. I will explain this interesting phenomenon in the talk. This
is a twoparts talk. In the first part, we will give an overview of
entire story. In the second part, we will focus on the appearance of
quasimodularity.

Wan, Daqing
UC Irvine 
(I) Rational points on a singular CY hypersurface.
The study of higher moments of Kloosterman sums naturally
leads to a singular CY hypersurface. In this talk, we explain how to
estimate the number of rational points on the singular CY hypersurface
via results on the Kloosterman sheaf.
(II) Mirror symmetry for the slope zeta function.
The slope zeta function is the slope part of the zeta functions
of a variety over a finite field.
It is an arithmetic object. We expect that the slope zeta function satisfies
the expected arithmetic mirror symmetry property for a mirror pair of
sufficiently large families of CY hypersurfaces. We shall explain some
evidence for this conjecture.

Whitcher, Ursula
WisconsinEau Claire 
Mirror quartics, discrete symmetries, and the congruent Zeta function.
We use GreenePlesserRoan and BerglundHuebschKrawitz mirror
symmetry to describe the structure of the congruent zeta function for
a set of pencils of quartic K3 surfaces which admit discrete group symmetries.

Yui, Noriko
Queen’s University 
Automorphy of CalabiYau threefolds of BorceaVoisin type.
CalabiYau threefold of BorceaVoisin type are constructed
as the quotients of products of ellptic curves and K3 surfaces by nonsymplectic
involutions. Resolving singularities, one obtains smooth CalabiYau
threefolds. We are interested in the modularity (automorphy) of the
Galois representations associated to these Calabi–Yau threefolds.
We establish the automorphy of some CalabiYau threefolds of BorceaVoisin
type.
This is a joint work with Y. Goto and R. Livn\'e.

Zagier, Don
MPIM Bonn and College de France 
(I) Quasimodular forms and holomorphic anomaly equation
Quasimodular forms are a special class of holomorphic functions
that are nearly modular and become modular after the addition of a
suitable nonholomorphic correction term. They are thus similar to,
but much simpler than, mock modular forms. They occur in mirror symmetry
in several ways, one of these being the socalled ”holomorphic
anomaly equation” which describes a sequence of quasimodular
forms with the nonmodularity of each form being defined inductively
in terms of its predecessors. We will describe how this works and
how one can understand the structure of the HAE in terms of deformations
of power series solutions to linear differential equations and ”bimodular
forms”, which are yet another type of nearly modular object.
This is joint work with Jan Stienstra.
(II) Some number theory coming from string amplitude calculations
Calculations of amplitudes in string theory lead in a natural way
to multiple zeta values at the ”tree” (genus 0) level and
to interesting modular functions at the ”1loop” (genus 1)
level. The talk will discuss various calculations related to this that
seem to have interesting arithmetic aspects, including certain very
specific rational linear combinations of multiple zeta values that rather
mysteriously occur in both the tree and 1loop level calculations, and
also some proven and conjectural identities for special values of KroneckerEisenstein
type lattice sums.

Zhou, Jie
Harvard University 
GromovWitten invariants and modular forms.
In this talk we shall solve the topological string amplitudes
in termsof quasi modular forms for some noncompact CY 3folds.
After a quick review of the polynomial recursion technique which is
used to solve the BCOV holomorphic anomaly equations, we will construct
the special polynomial ring which has a nice grading and show that topological
string amplitudes are polynomials of these generators. For the cases
in which the moduli space of complex structures could be identified
with a modular curve, this ring is exactly the differential ring of
quasi modular forms constructed out of periods. Moreover, the Fricke
involution serves as a duality relating the amplitudes at the large
complex structure limit and the conifold point. Combing the polynomial
recursion technique and the duality, we will then be able to express
the topological string amplitudes in terms of quasi modular forms. For
other cases, the special polynomial ring gives a generalization of the
ring of quasi modular forms without knowing much about the arithmetic
properties of the moduli space.

Participants as of September 5, 2013
* to be confirmed
Full Name 
University/Affiliation 
Arrival Date

Departure Date

Adebayo, Olasehinde 
Federal University of Technology Akure 
15Sep13

23Sep13

AmirKhosravi, Zavosh 
University of Toronto 
01Jul13

30Dec13

Candelas, Philip* 
University of Oxford 
16Sep13

21Sep13

Caviedes Castro, Alexander 
University of Toronto 


Ceballos, Cesar 
York University 
13Aug13

20Dec13

Clingher, Adrian 
University of Missouri  St. Louis 
12Sep13

19Sep13

Crooks, Peter 
University of Toronto 
10Sep13

01Dec13

de la Ossa, Xenia* 
University of Oxford 
16Sep13

21Sep13

Fei, Teng 
MIT 
08Sep13

21Sep13

Filippini, Sara Angela 
Fields Institute 
01Jul13

31Dec13

Fisher, Jonathan 
University of Toronto 
01Jul13

31Dec13

Gahramanov, Ilmar 
Humboldt University Berlin 


Gao, Peng 
John S. Toll Dr 
26Aug13

21Sep13

GarciaRaboso, Alberto 
University of Toronto 
01Aug13

31Dec13

Golyshev, Vasily 
Independent University of Moscow 
15Sep13

21Sep13

Goto, Yasuhiro 
Hokkaido University of Education 
15Sep13

21Sep13

Gualtieri, Marco 
University of Toronto 
05Sep13

05Dec13

Harder, Andrew 
University of Aberta 
26Aug13

20Sep13

Hosono, Shinobu 
University of Tokyo 
15Sep13

21Sep13

Kelly, Tyler 
University of Pennsylvania 
12Sep13

20Sep13

Koroteev, Peter 
Perimeter Institute for Theoretical Physics 
16Sep13

20Sep13

Kudla, Stephen* 
University of Toronto 
16Sep13

21Sep13

Li, Yingkun 
UCLA 
16Sep13

21Sep13

Luk, Kevin 
University of Toronto 


Malmendier, Andreas 
Colby College 
16Sep13

21Sep13

Molnar, Alexander 
Queen's University 
01Jul13

31Dec13

Murthy, Sameer 
National Institute for Nuclear Physics and High Energy Physics 
15Sep13

20Sep13

Overholser, Douglas 
University of California, San Diego 
01Jul13

31Dec13

Park, B. Doug 
University of Waterloo 
01Sep13

20Dec13

Perunicic, Andrija 
Brandeis University 
02Jul13

31Dec13

Pioline, Boris 
CERN 
16Sep13

20Sep13

Pym, Brent 
McGill University 


Rahmati, Mohammad Reza 
CIMAT 
16Sep13

22Nov13

Rayan, Steven 
University of Toronto 
15Jun13

30Jan14

Rose, Simon 
Fields Institute 
01Jul13

31Dec13

Ruan, Yongbin 
University of Michigan 
16Sep13

21Sep13

Ruddat, Helge 
Universität Mainz 
25Jun13

31Dec13

Schaug, Andrew 
University of Michigan 
15Sep13

20Oct13

Schütt, Matthias* 
Leibniz Universitaet Hannover 
16Sep13

21Sep13

Selmani, Sam 
McGill University 
15Sep13

20Sep13

Sere, Abdoulaye 
Polytechnic University of Bobo Dioulasso 
01Sep13


Silversmith, Robert 
University of Michigan 
15Sep13

21Sep13

Soloviev, Fedor 
University of Toronto 


Thompson, Alan 
Fields Institute 
01Jul13

30Dec13

van Garrel, Michel 
California Institute of Technology 
01Jul13

31Dec13

Wan, Daqing 
University of California 
15Sep13

20Sep13

Whitcher, Ursula 
University of WisconsinEau Claire 
13Sep13

17Sep13

Yui, Noriko 
Queen's University 
02Jul13

20Dec13

Zagier, Don 
MaxPlanckInstitut fur Mathematik 
16Sep13

21Sep13

Zhou, Jie 
Harvard University 
08Sep13

21Sep13

Zhu, Yuecheng 
University of Texas at Austin 
01Jul13

23Nov13

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