Geometric Aspects of Field Theories and Lattice Models

场论和晶格模型的几何方面

基本信息

批准号：
2005286
负责人：
David Ben-Zvi
金额：
$ 42.9万
依托单位：
University of Texas at Austin
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2005286&HistoricalAwards=false
关键词：
Geometric Aspects Field Theories Lattice

项目摘要

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.

这些研究项目构成了几何与理论物理学之间持续有力相互作用的一部分。数学与其他科学以及科学与数学的参与是不断富有成果的思想来源，对社会产生了长期的有益后果。我们可以通过向后看来衡量这些后果：我们当前的技术和经济依赖于过去几十年和几个世纪以来基础研究的基础。不同学科的混合是互惠互利的。作为数学家，PI具有将物理学的结构和直觉带入数学的特殊目标。这项研究有许多旨在做到这一点的方面。具体而言，我们将研究量子场理论几何公式中的基础问题。这些项目是特定问题和一般结构调查的组合。这些技术是数学上的，通常具有物理学的灵感。这项工作对纯几何形状以及对物理学的问题的应用（例如物质阶段的分类）的影响。该奖项支持与PI合作的研究生参与其中的一些项目，他们还在这个广阔的领域内进行了自己的独立项目。我们在Segal和Atiyah在1980年代发起的公理系统中工作。从那以后，这些几何公理已被许多方向进行了完善和扩展，我们试图继续这一过程。例如，通常的公理评估了单个歧管或边界的理论，而许多计算涉及对歧管家族的评估，我们将进一步研究其公理。在某些方面，田间理论类似于谎言群体的代表。谎言理论统一结构起着重要的作用，在普通量子场理论中，单位性也是如此。虽然几何公理，尤其是对于拓扑字段理论，包括以强形式的位置，但没有相应的扩展单位概念。这是我们将进一步研究的领域。追求的一般理论的其他方面与非易面的可逆领域理论和理论有关，这些理论和理论是拓扑模量可逆理论。该项目还具有与特定理论相关的几条查询。例如，我们有一个计划将三维拓扑结论Chern-Simons理论构建为完全扩展的理论。我们还旨在研究二维ISING模型的几何化身中的动力学。最后，我们旨在构建与可逆领域理论相对应的晶格模型，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NSF的法定任务，并被认为是通过基金会的智力优点和更广泛影响的审查标准通过评估来通过评估来支持的。