Beilinson-Drinfeld Grassmannians and chiral algebras in differential geometry

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This proposal is for obtaining support for my ongoing research in developing a theory and constructing interesting examples of chiral algebras (equivalently known as factorization algebras) in the setting of differential geometry. This research is in collaboration with professor Kobi Kremnizer (University of Oxford, UK). It is well known that in quantum Physics one cannot measure simultaneously the position and the momentum of a particle. This is the famous indeterminacy principle. However, this principle does not imply that there are no laws of Physics on the quantum level. One way to describe these laws is to use smearing, i.e. to view observables not as functions, but as functionals on test functions defined on the space-time. Such formulation leads to very rich algebraic structures, that encode independence of measurements that are performed far away from each other. In quantum physics such formulation is usually called algebraic quantum field theory. In mathematical setting these algebraic structures are called factorization algebras (there is an equivalent reformulation in term of chiral algebras). Under the name of vertex operator algebras they were known for more than 30 years, and in the 1990's a geometric formulation (chiral algebras) was introduced by A.Beilinson and V.Drinfeld. The work of Beilinson and Drinfeld is within algebraic geometry, and this limits in their method of constructing chiral algebras the possible dimension of the space-time to 2. In our research, we are constructing non-trivial examples of chiral algebras in differential geometry. Switching to differential geometry immediately removes the limit on dimension, but introduces many other problems. Some of them we have already solved, others are still a work in progress. The overall direction of this research is towards formulating an algebraic quantum field theory on a 4-dimensional space-time. This problem is open for at least two generations now, and we do not claim to be close to a solution. However, we like to have this challenge in mind to give a direction to our research. Our approach is through adapting the algebraic-geometric techniques of Beilinson and Drinfeld to differential geometry. Different from algebraic geometry objects in differential geometry are described not by polynomial rings but by rings of smooth functions. There are many differences between these two kinds of rings, for example infinitesimals are considerably more complicated in the case of rings of smooth functions. However, we were able to adapt enough of the algebraic-geometric techniques of Beilinson and Drinfeld to make it possible to construct a whole new class of examples of non-trivial chiral algebras.