Geometric Aspects of Field Theories and Lattice Models

这些研究项目构成了几何学和理论物理学之间正在进行的有力互动的一部分。数学与其他科学的结合，以及科学与数学的结合，是富有成果的思想的持续源泉，对社会有长期的有益影响。我们可以通过回顾过去来衡量这些后果：我们目前的技术和经济依赖于过去几十年和几个世纪的基础研究。不同学科的融合是互利的。作为一名数学家，PI有一个特殊的目标，那就是将物理学的结构和直觉引入数学。这项研究有很多方面都是为了做到这一点。具体来说，我们将研究量子场论几何公式中的基础问题。这些项目是具体问题和一般结构调查的混合体。这些技术都是数学上的，常常受到物理学的启发。这项工作对纯几何以及物理问题的应用都有影响，例如物质的相分类。该奖项支持与PI一起工作的研究生参与其中的一些项目，他们也在这个广泛的领域内开展自己的单独项目。我们在由Segal和Atiyah在20世纪80年代发起的场论的公理系统中工作。从那以后，这些几何公理在许多方向上得到了改进和扩展，我们寻求继续这一过程。例如，通常的公理在单个流形或边界上评价理论，而许多计算涉及对流形族的评价，我们将进一步研究其公理化。在某些方面，场论类似于李群的表示。在李论中，酉结构起着重要的作用，在普通量子场论中，酉性也起着重要的作用。虽然几何公理，特别是拓扑场论的公理，以强形式包含局部性，但却没有相应的扩展的统一概念。这是我们将进一步调查的领域。一般理论的其他方面涉及到非拓扑可逆场论和拓扑模可逆理论。这个项目也有几条与具体理论相关的探究线。例如，我们计划将三维拓扑的chen - simons理论作为一个完全扩展的理论来构建。我们还旨在研究二维伊辛模型的几何化身中的动力学。最后，我们的目标是构建对应于可逆场理论的晶格模型，这是开发离散模型几何理论的更大努力的一部分。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

These research projects form part of an ongoing vigorous interaction between geometry and theoretical physics. The engagement of mathematics with other sciences, and of science with mathematics, is a continual source of fruitful ideas with long-term beneficial consequences for society. We can measure these consequences by looking backwards: our current technology and economy rely on the foundation of basic research from past decades and centuries. The mixing of different disciplines is mutually beneficial. As a mathematician the PI has a particular goal to bring structures and intuitions from physics into mathematics. The research has many facets designed to do just that. Specifically, we will work on foundational issues in geometric formulations of quantum field theory. The projects are a mix of specific problems and general structural investigations. The techniques are mathematical, often with inspiration from physics. This work has ramifications for pure geometry as well as applications to questions in physics, such as classification of phases of matter. This award supports graduate students working with the PI participate in some of these projects, and they also carry out their own separate projects within this broad field.We work within the Axiom System for field theories initiated by Segal and Atiyah in the 1980s. These geometric axioms have been refined and extended in many directions since, and we seek to continue this process. For example, the usual axioms evaluate a theory on a single manifold or bordism, whereas many computations involve evaluation on a family of manifolds, the axiomatics of which we will investigate further. In some ways a field theory is akin to a representation of a Lie group. In Lie theory unitary structures play a prominent role, and so too does unitarity in ordinary quantum field theory. While the geometric axioms, especially for topological field theories, include locality in a strong form, there is no corresponding extended notion of unitarity. This is an area we will investigate further. Other aspects of general theory to pursue relate to non-topological invertible field theories and theories which are topological modulo invertible theories. This project also has several lines of inquiry related to specific theories. For example, we have a plan to construct three-dimensional topological Chern-Simons theory as a fully extended theory. We also aim to investigate dynamics in a geometric incarnation of the two-dimensional Ising model. Finally, we aim to construct lattice models which correspond to invertible field theories, part of a larger effort to develop a geometric theory of discrete models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.