Principal Investigator: Mark Gross
The principal investigator plans to study the geometry of mirror symmetry for Calabi-Yau manifolds. This will be done from the perspective of a program developed jointly with B. Siebert, which "tropicalizes" Calabi-Yau manifolds, using an algebro-geometric version of the Strominger-Yau-Zaslow conjecture. In particular, the principal investigator proposes to resolve a number of the remaining key steps of this program; these steps should be the component pieces of a complete conceptual proof of mirror symmetry at genus zero. These steps include a general gluing formula for the theory of logarithmic Gromov-Witten invariants and a generalization of the tropical vertex of the PI, Pandharipande and Siebert to all dimensions. In addition, the program with Siebert has led to the construction of canonical theta functions for lines bundles on (degenerating families) of Calabi-Yau manifolds. The existence of such theta functions is expected to have various applications, such as geometric compactifications of the moduli space of K3 surfaces (being pursued with P. Hacking and S. Keel) and a tropical approach to homological mirror symmetry (being pursued with Siebert and M. Abouzaid).
The work described in this proposal lies at the intersection of string theory and geometry. String theory replaces the traditional notion of the point particle with a small loop of string, moving through space-time. To make string theory compatible with quantum mechanics, space-time must be ten-dimensional. Since space-time appears four-dimensional, one expects six of these dimensions to be a very small "curled up" geometric object. These geometric objects are called Calabi-Yau manifolds. In the early 1990s, string theorists proposed a remarkable association between completely different Calabi-Yau manifolds: certain calculations extremely difficult to perform on one Calabi-Yau manifold could be completed by performing completely different, and much easier, calculations on a different Calabi-Yau manifold. This discovery was known as mirror symmetry. Since this time, many geometers have been trying to understand the mathematics behind this miraculous observation. New insights and explanations for this phenomenon are developing rapidly, and these insights are leading to new approaches to old problems in algebraic geometry. The proposed work aims to both further understanding of mirror symmetry and apply this understanding to produce new developments in algebraic geometry.
This award has funded research into a field of mathematics called "mirror symmetry". This is a subfield of geometry which traces its origin to string theory. String theory, whose development began in the 1970s, is an approach to reconciling quantum mechanics with Einstein's theory of gravity, the basic idea being that rather than envisioning a particle as a point, as is traditionally done, one should view a particle as a string, albeit a very short one. As it moves through space-time, the string vibrates, tracing out a surface. However, once one brings quantum mechanics into the picture, one discovers that one needs a ten-dimensional space-time to obtain a consistent theory. This is obviously at odds with the world we see around us, being visibly only four-dimensional. The solution proposed by string theorists to this apparent contradiction is that six of these dimensions are very small and cannot be directly observed. Furthermore, the most desirable kind of six-dimensional object from the point of view of string theory is something known as a Calabi-Yau manifold. Calabi-Yau manifolds can be extremely complicated objects, but they have a distinct advantage. Calabi-Yau manifolds are in fact objects known as algebraic varieties. This means they are defined by systems of polynomial equations. Such systems are precisely the objects studied by algebraic geometers. Around 1990, evidence from string theory emerged which suggested that Calabi-Yau manifolds should come in pairs, called mirror pairs, which yield the same four-dimensional physics. However, mirror pairs of Calabi-Yau manifolds are geometrically very different from each other, and the methods of extracting important physical quantities are quite different. The importance of this came to light in a 1990 paper of Candelas et al. They studied a mirror pair and showed that a certain expression known as the Yukawa coupling could be computed on the two members of the pair using completely different methods. The calculation on one member of the pair, while difficult, was doable. The calcuation on the other member, however, could in fact be translated into a question in algebraic geometry which had been considered extremely difficult, and involved the calculation of numbers of straight lines, conic sections, etc., contained in the Calabi-Yau manifold. These numbers were extremely difficult to calculate inside algebraic geometry, and the fact that physicists were able to calculate these numbers using the calculation for the mirror partner stunned mathematicians. Since then, there has been a great deal of work aimed at understanding the mysteries of mirror symmetry. In particular, much of my work since 1996 has been involved in one way or another with this goal. The research funded by this award follows the philosophy of using "tropical geometry" to understand mirror symmetry. The term "tropical" is in honor of a Brazilian mathematician who pioneered certain techniques which have evolved into a new kind of geometry. Tropical geometry studies geometric objects which are much easier to understand than the geometric objects defined using polynomial equations. Surprisingly, the very complicated geometry which occurs on both sides of the mirror symmetry picture cast a shadow in the tropical world. The ultimate explanation of mirror symmetry, my work suggests, is that both of the Yukawa couplings have a description in terms of tropical geometry, and these tropical descriptions coincide, hence giving a conceptual explanation of mirror symmetry. The general goal of the work during the award period was to develop further the correspondence between the tropical world and the more classical geometries described above. Indeed, I have made much progress in this direction over the last three years. As well as funding the research described above, the award also provided support for the doctoral training for four Ph.D. students, one of them female.