Beilinson-Drinfeld Grassmannians and chiral algebras in differential geometry

这个提议是为了获得我正在进行的研究的支持，在微分几何的背景下，开发一个理论和构建手性代数（相当于被称为因子分解代数）的有趣的例子。这项研究是与Kobi Kremnizer教授（英国牛津大学）合作进行的。众所周知，在量子物理学中，不能同时测量粒子的位置和动量。这就是著名的测不准原理。然而，这一原理并不意味着在量子层面上没有物理定律。描述这些定律的一种方法是使用涂抹，即不将可观测物视为函数，而是将其视为在时空上定义的测试函数的函数。这样的公式导致了非常丰富的代数结构，编码了彼此相距很远的测量的独立性。在量子物理学中，这种表述通常称为代数量子场论。在数学上，这些代数结构被称为因式分解代数（在手性代数方面有一个等效的重新表述）。在顶点算子代数的名义下，它们被知道了30多年，在20世纪90年代，a . beilinson和V.Drinfeld引入了一个几何公式（手性代数）。Beilinson和Drinfeld的工作是在代数几何范围内，这限制了他们构造手性代数的方法中可能的时空维数为2。在我们的研究中，我们正在构造微分几何中手性代数的非平凡例子。切换到微分几何立即消除了尺寸上的限制，但引入了许多其他问题。有些问题我们已经解决了，有些问题还在进行中。本研究的总体方向是在四维时空上建立代数量子场论。这个问题至少已经持续了两代人，而我们并不声称已经接近解决这个问题。然而，我们希望将这一挑战牢记在心，为我们的研究指明方向。我们的方法是通过将Beilinson和Drinfeld的代数几何技术应用于微分几何。与代数几何不同，微分几何中的物体不是用多项式环而是用光滑函数环来描述的。这两种环有许多不同之处，例如，光滑函数环的无穷小要复杂得多。然而，我们能够适应足够的Beilinson和Drinfeld的代数-几何技术，使构建一个全新的非平凡手性代数的例子成为可能。

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.