Collaborative Research: Factorization Homology, Deformation Theory, and Duality

合作研究：因式分解同调、变形理论和对偶性

• 批准号: 1812055
• 负责人: David Ayala
• 金额: $ 21.39万
• 依托单位: Montana State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2018
• 资助国家: 美国
• 起止时间: 2018-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812055&HistoricalAwards=false
• 关键词: Collaborative; Research; Factorization; Homology; Deformation

This project develops the geometric study of models of space-time, toward mathematical forms of fundamental concepts in particle physics. Two principal concepts from the theory of quantum fields are those of locality and duality. Locality generalizes the idea that there should not be so-called action at a distance in a complete quantum theory of nature. Duality occurs in two seemingly different quantum theories being fundamentally the same. The present project develops mathematical formulations of locality and duality in terms of a new theory of factorization homology. Factorization homology provides a mathematical means of articulating a physical theory on all of space-time in terms of physics in very small subregions, within which physics can be simplified in terms of combinatorics and higher-dimensional graph theory.This project develops the theory of factorization homology for variants of higher categories, and applications thereof to mathematical physics and differential topology. This theory can be thought of as the study of sheaves on moduli spaces of stratifications. One goal is to use factorization homology with adjoints to prove the cobordism hypothesis of Baez--Dolan and Lurie, which asserts that topological quantum field theories in the sense of Atiyah's axioms are uniquely determined by their value on a point. This would show that such a topological quantum field theory on space-time X comes from a sheaf on the moduli space of stratifications of X, and thus that the notion of locality can be understood in terms of the topology of such moduli spaces. A second goal is to show that this moduli space of stratifications of a manifold X satisfies an infinite-dimensional form of Poincar? duality. This would then give rise to a duality among topological physical theories on X, once expressed as sheaves on the aforementioned moduli space. This duality is a form of Koszul duality for topological physical theories.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.

这个项目发展了时空模型的几何研究，朝着粒子物理学基本概念的数学形式发展。量子场论的两个主要概念是定域性和对偶性。定域性概括了这样一种观点，即在完整的自然量子理论中不应该有所谓的超距作用。对偶性发生在两个看似不同但本质相同的量子理论中。本计画发展局部性与对偶性的数学公式，以一种新的因子分解同调理论为基础。因式分解同调提供了一种数学方法，可以在非常小的子区域内，根据物理学，阐明关于所有时空的物理理论，在这些子区域内，物理学可以根据组合学和高维图论来简化。本项目开发了更高类别变体的因式分解同调理论，并将其应用于数学物理学和微分拓扑学。这个理论可以被认为是研究层的模空间上的层。一个目标是使用因式分解同调与伴随证明的协边假说的贝兹-多兰和卢里，其中声称，拓扑量子场论的意义上的阿蒂亚的公理是唯一确定的价值点。这将表明，时空X上的这种拓扑量子场论来自于X的分层模空间上的层，因此定域性的概念可以用这种模空间的拓扑来理解。第二个目标是表明，这种模空间的分层流形X满足无限维形式的庞加莱？二元性 这将导致X上的拓扑物理理论之间的对偶性，一旦表示为上述模空间上的层。该奖项反映了NSF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。