Symplectic Reflection Algebras and their Generalizations

University Of Chicago, Chicago IL

Investigators

Abstract

This research project is a step towards better mathematical understanding of some aspects of Mirror Symmetry. Mirror Symmetry predicts a natural correspondence between the moduli of deformations of a singular Calabi-Yau variety, on one hand, and the moduli of (Kahler structures on) resolutions of the mirror dual Calabi-Yau variety, on the other hand. However, there are plenty of examples of singular Calabi-Yau varieties which have either no deformations or no resolutions at all, or both. In such cases, Mirrror Symmetry predicts the existence of some sort of `noncommutative' substitutes for the nonexistent commutative deformations, resp. resolutions, that recovers the above mentioned duality between deformations and resolutions. The goals of the proposal could be briefly formulated as follows. 1. Generalize the techniques based on the notion of Symplectic reflection algebra introduced by P. Etingof and the PI during the last 5 years to other classes of singular symplectic varieties. 2. Develop general machinery of noncommutative deformations of symplectic resolutions analogous to the theory of Poisson deformations developed earlier by the PI and D. Kaledin. 3. Begin development of the theory of Calabi-Yau algebras. These algebras should replace symplectic reflection algebras in the case of not necessarily symplectic Calabi-Yau orbifolds. Noncommutative Geometry is a relatively new area of mathematics which arose in early 1980-s at the junction of classical geometry and quantum theory. Classical geometry which was created in its modern form in the 19-th century, is a mathematical theory which is designed to describe precisely various phenomena of Space and Time. It plays a fundamental role in the equations governing electric and magnetic forces, and is even more fundamental for Einstein's Relativity theory. The discovery of the famous `Uncertainty Principle' had clearly demonstrated that Classical geometry can not adequately describe all of the complexity of Quantum theory. The latter is one of the most important achievements of theoretical physics of the 20-th century, which lies at the foundation of this century's technological revolution, including laser technology and computers. It is a physical theory which allows to describe the behavior of very small objects, like atoms and electrons. Noncommutative Geometry is a new kind of geometry, that may also be called Quantum Geometry since it is designed to provide the adequate geometry at very small distances. Development of Noncommutative Geometry is thus absolutely fundamental for Quantum physics. The present proposal seeks to further develop this theory and to explore its interactions with other areas of mathematics and physics, including quite unexpected connections to Probability theory.