Beilinson-Drinfeld Grassmannians and chiral algebras in differential geometry

微分几何中的 Beilinson-Drinfeld Grassmannians 和手性代数

• 批准号: RGPIN-2020-04845
• 负责人: Borisov, Dennis
• 金额: $ 1.31万
• 依托单位: University of Windsor
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=740253
• 关键词: Beilinson; Drinfeld; Grassmannians; chiral; algebras

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.

该建议是为了获得我正在进行的研究中的研究，并在差异几何形状的环境中构建了手性代数（等效地称为分解代数）的有趣示例。这项研究与Kobi Kremnizer教授（英国牛津大学）合作。众所周知，在量子物理学中，一个人不能同时测量粒子的位置和动量。这是著名的不确定性原则。但是，该原则并不意味着量子水平上没有物理定律。描述这些定律的一种方法是使用涂抹，即将可观察物视为函数，而是在时空上定义的测试功能上的功能。 这种公式导致非常丰富的代数结构，这些结构编码了彼此远距离执行的测量的独立性。在量子物理学中，这种制定通常称为代数量子场理论。在数学设置中，这些代数结构称为分解代数（在手性代数方面存在等效的重新制定）。以顶点操作员代数为名，他们闻名了30多年，在1990年代，A.Beilinson和V.Drinfeld引入了1990年的几何配方（手性代数）。 Beilinson和Drinfeld的工作在代数几何形状范围内，这限制了它们构建手性代数的方法，即时空可能的维度为2。在我们的研究中，我们正在构建不同几何学中手性代数的非平常实例。切换到差异几何形状立即消除了维度的限制，但引入了许多其他问题。其中一些我们已经解决了，另一些仍在进行中。这项研究的总体方向是在四维时空制定代数量子场理论。现在至少有两代人开放了这个问题，我们没有声称接近解决方案。但是，我们想牢记这一挑战，以指导我们的研究。我们的方法是通过调整贝林森和德林菲尔德的代数几何技术来适应差异几何形状。与差异几何形状中的代数几何对象不同，不是由多项式环描述，而是通过平滑函数的环描述。这两种环之间存在许多差异，例如，在平滑函数的环的情况下，无限量相比要复杂得多。但是，我们能够适应贝林森和德林菲尔德的代数几何技术，以使构建一个全新的非平凡性手性代数示例成为可能。