Calabi-Yau Varieties

University Of Pennsylvania, Philadelphia PA

Investigators

Abstract

Abstract 0074980--Grassi The research of the investigator is on the interface of algebraic geometry and the physics of string theory. Most of the work of the investigator involves the study of elliptically fibered Calabi-Yau varieties; the symmetry of the generic fiber is identified with a symmetry of a physical model. Since many Calabi-Yau varieties are in fact elliptically fibered, this extra structure gives way to new approaches to understand Calabi-Yau varieties in general. Certain relations in string theory, known as dualities, predict surprising connections among the mathematical properties of the Calabi-Yau varieties; vice-versa one can use mathematical comparisons between the Calabi-Yau varieties associated by the dualities to learn more about the dualities themselves. The investigator studies transformations of Calabi-Yau fourfolds and their "dual counterpart", Calabi-Yau threefolds with a gauge bundle, determined by the structure of the fourfold. In another project the P.I. investigates the singularities naturally associated to the gauge groups, and their relations to representation theory. This is also motivated by (and it has applications) to the physics of string theory. String theory is expected to explain the apparent contradictions that arise when one tries to combine relativity with quantum mechanics. In the model proposed by string theory, 6 (real) dimensions of our ten dimensional world have a very special structure, in particular they are complex algebraic varieties, called "Calabi-Yau". A general question is understanding properties of these manifolds, in order to get a better description of the physical theory. In practice, the interdisciplinary nature of this project has been mutually beneficial to physics and mathematics and has spurred much progress and activity in both fields. For example, it was shown that although there seems to be many different Calabi-Yau varieties which could describe the 6 extra dimensions, there are in fact surprising relations among them. This is also related to the presence of black holes and to special transformations in algebraic geometry. The research of the investigator involves studying these type of relations for ``elliptically fibered" Calabi-Yau varieties.