
SCIENTIFIC PROGRAMS AND ACTIVITIES 

October 22, 2016  
Workshop on Arithmetic, Geometry and Physics around CalabiYau Varieties and Mirror Symmetry  July 2329, 2001
Organizing and Scientific Committee: Dr. Victor Batyrev (University of Tübingen) A CalabiYau 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 CalabiYau variety is an elliptic curve, a dimension 2 CalabiYau variety is a K3 surface, and a dimension 3 is a CalabiYau 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 1dimensional mathematical objects "strings" rather than 0 dimensional objects "points". Mirror symmetry is a conjecture in string theory that certain "mirror pairs" of CalabiYau manifolds give rise to isomorphic physical theories. CalabiYau manifolds appear in the theory because in passing from the 10dimensional space time to a physically realistic description in four dimension, string theory requires that the additional 6dimensional space is to be a CalabiYau 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 CalabiYau threefolds. In the course of mirror symmetry, it has become more apparent that CalabiYau varieties enjoy tremendously rich arithmetic properties. For instance, arithmetic objects such as: modular forms, modular functions of one and more variables, algebraic cycles, Lfunctions, and padic Lfunctions, have popped up onto the scene. Also special classes of CalabiYau 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 TaniyamaShimuraWeil conjecture of the socalled modularity of elliptic curves defined over the field of rational numbers by A. Wiles and his disciples. Wiles' idea is to exploit 2dimensional Galois representations arising from elliptic curves and modular forms of weight 2 on some congruence supgroups of PSL(2,Z), and establish their equivalence. His method ought to be applied to explore arithmetic of CalabiYau threefolds. In particular, rigid CalabiYau threefolds defined over the field of rational numbers are equipped with 2dimensional 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 CalabiYau threefolds over the rationals, the Langlands Program predicts that there should be some automorphic forms attached to them. We plan to test the socalled modularity conjectures for CalabiYau varieties defined over the field of rational numbers, or more generally, over number fields, first trying to understand them for some special classes of CalabiYau 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 CalabiYau manifolds and mirror symmetry. The impact of classical Hodge theory as well as the padic Hodge cycles, is clearly evident. On the algebraic side, there is the relationship of algebraic Ktheory and Chow groups of algebraic cycles, leading to the BlochQuillenGersten resolution description of Chow groups. There is also the more recent relationship of Bloch's higher Chow groups and higher Ktheory (a higher RiemannRoch 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 BlochBeilinson 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 CalabiYau manifolds present an ideal testing ground for some of these conjectures. 2. Objectives 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 CalabiYau 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 CalabiYau manifolds. One anticipates that these interactions between the various "schools" will blossom in the near future. Arithmetic aspects on CalabiYau varieties and mirror symmetry, however,
are yet to be explored vigorously. For instance, Wiles' method should
be explored to establish the modularity for rigid CalabiYau threefolds
defined over the field of rational numbers a la Fontaine and Mazur.
Also, investigation on the intermediate Jacobians of CalabiYau threefolds
ought to be pursued using, for instance, padic Hodge theory. Recent
articles of P. Candelas et.al. on the computation of the zetafunctions
of CalabiYau manifolds over finite fields reveal a surprising connection
of mirror symmetry to padic Lfunctions (which are the essential ingredients
in Iwasawa theory). The construction of algebraic cycles on CalabiYau threefolds (generalizing the method of Bloch), investigation of Lfunctions of CalabiYau 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 CalabiYau varieties and mirror symmetry, and to exchange ideas and learn the subjects firsthand 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: 4. Proceedings. We would like to have manuscripts by the end of December 2001. Please send your manuscript in texformat and a hard copy to

