SCIENTIFIC PROGRAMS AND ACTIVITIES
|March 24, 2017|
Workshop on Arithmetic, Geometry and Physics around Calabi-Yau Varieties and Mirror Symmetry - July 23-29, 2001
Organizing and Scientific Committee:
Dr. Victor Batyrev (University of Tübingen)
A Calabi-Yau variety of dimension d is a complex manifold with trivial canonical bundle and vanishing Hodge numbers h i,0 for 0 < i < d. For instance, a dimension 1 Calabi-Yau variety is an elliptic curve, a dimension 2 Calabi-Yau variety is a K3 surface, and a dimension 3 is a Calabi-Yau threefold.
(A) One of the most significant developments in the last decade in Theoretical Physics (High Energy) is, arguably, string theory and mirror symmetry. String theory proposes a model for the physical world which purports its fundamental constituents as 1-dimensional mathematical objects "strings" rather than 0- dimensional objects "points". Mirror symmetry is a conjecture in string theory that certain "mirror pairs" of Calabi-Yau manifolds give rise to isomorphic physical theories. Calabi-Yau manifolds appear in the theory because in passing from the 10-dimensional space time to a physically realistic description in four dimension, string theory requires that the additional 6-dimensional space is to be a Calabi-Yau manifold.
Though the idea of mirror symmetry has originated in physics, in recent years, the field of mirror symmetry has exploded onto the mathematical scene. It has inspired many new developments in algebraic geometry, toric geometry, Riemann surfaces theory, infinite dimensional Lie algebras, among others. For instance, mirror symmetry has been used to tackle the problem of counting the number of rational curves on Calabi-Yau threefolds.
In the course of mirror symmetry, it has become more apparent that Calabi-Yau varieties enjoy tremendously rich arithmetic properties. For instance, arithmetic objects such as: modular forms, modular functions of one and more variables, algebraic cycles, L-functions, and p-adic L-functions, have popped up onto the scene. Also special classes of Calabi-Yau manifolds, e.g., of Fermat type hypersurfaces, or their deformations pertinent to mirror symmetry, offer promising testing grounds for physical predictions as well as rigorous mathematical analysis and computations.
(B) One of the most significant developments in the last decade in Arithmetic Geometry and Number Theory is the proof of the Taniyama-Shimura-Weil conjecture of the so-called modularity of elliptic curves defined over the field of rational numbers by A. Wiles and his disciples. Wiles' idea is to exploit 2-dimensional Galois representations arising from elliptic curves and modular forms of weight 2 on some congruence sup-groups of PSL(2,Z), and establish their equivalence. His method ought to be applied to explore arithmetic of Calabi-Yau threefolds. In particular, rigid Calabi-Yau threefolds defined over the field of rational numbers are equipped with 2-dimensional Galois representations, which are conjecturally equivalent to modular forms of one variable of weight 4 on some congruence subgroup of PSL(2, Z). For not necessarily rigid Calabi-Yau threefolds over the rationals, the Langlands Program predicts that there should be some automorphic forms attached to them. We plan to test the so-called modularity conjectures for Calabi-Yau varieties defined over the field of rational numbers, or more generally, over number fields, first trying to understand them for some special classes of Calabi-Yau threefolds, e.g., those mentioned in (A).
(C) There are a number of intriguing developments in the theory of algebraic cycles in the past 25 years, that not surprisingly, should open the door to an infusion of new techniques in the study of Calabi-Yau manifolds and mirror symmetry. The impact of classical Hodge theory as well as the p-adic Hodge cycles, is clearly evident. On the algebraic side, there is the relationship of algebraic K-theory and Chow groups of algebraic cycles, leading to the Bloch-Quillen-Gersten resolution description of Chow groups. There is also the more recent relationship of Bloch's higher Chow groups and higher K-theory (a higher Riemann-Roch theorem), and a conjectured "arithmetic index theorem". The influence of the work of Bloch and Beilinson on the subject of algebraic cycles is profound. For instance there are the fascinating Bloch-Beilinson conjectures on the existence of a natural filtration on the Chow groups, whose graded pieces can be described in terms of extension data, and their conjectures about injectivity of certain regulators of cycle groups of varieties over number fields. There is also the work of others on how conjecturally this filtration can be explained in terms of kernels of higher regulators and arithmetic Hodge structures. The Calabi-Yau manifolds present an ideal testing ground for some of these conjectures.
Geometry around mirror symmetry and string theory has been pursued by many mathematicians (complex geometers, toric geometers, and others), and great progress has been witnessed in understanding geometric aspects of the problem. In fact, recently a number of excellent books and survey articles have been published explaining complex geometric aspects of mirror symmetry on Calabi-Yau threefolds as well as on K3 surfaces.
Further, in the past two decades, a number of people who have studied that part of algebraic geometry dealing with Hodge theory and algebraic cycles, have found applications of their work in Quantum Cohomology, Mirror Symmetry and Calabi-Yau manifolds. One anticipates that these interactions between the various "schools" will blossom in the near future.
Arithmetic aspects on Calabi-Yau varieties and mirror symmetry, however,
are yet to be explored vigorously. For instance, Wiles' method should
be explored to establish the modularity for rigid Calabi-Yau threefolds
defined over the field of rational numbers a la Fontaine and Mazur.
Also, investigation on the intermediate Jacobians of Calabi-Yau threefolds
ought to be pursued using, for instance, p-adic Hodge theory. Recent
articles of P. Candelas et.al. on the computation of the zeta-functions
of Calabi-Yau manifolds over finite fields reveal a surprising connection
of mirror symmetry to p-adic L-functions (which are the essential ingredients
in Iwasawa theory).
The construction of algebraic cycles on Calabi-Yau threefolds (generalizing the method of Bloch), investigation of L-functions of Calabi-Yau threefolds a la the conjectures of Beilinson and Bloch, among others, ought to be pursued with more rigour and intensity.
Our goal is to bring together to the Fields Institute experts, recent Ph.D.'s and graduate students, working in physics, geometry and arithmetic around Calabi-Yau varieties and mirror symmetry, and to exchange ideas and learn the subjects first-hand mingling with researchers with different expertise. We expect these interactions to lead to progress in solving open problems in mathematics and physics as well as to pave way to new developments.
3. Expected participants
Expected recent Ph.D.'s at the workshop include:
We would like to have manuscripts by the end of December 2001.
Please send your manuscript in texformat and a hard copy to