Collaborative Research: Factorization Homology, Deformation Theory, and Duality

这个项目发展了时空模型的几何研究，朝着粒子物理基本概念的数学形式发展。量子场论中的两个主要概念是局域性和对偶性。局域性概括了在完整的自然量子理论中不应该存在所谓的远距离作用力的想法。对偶性发生在两个看似不同的量子理论中，它们本质上是相同的。本项目根据一种新的因式分解同调理论，发展了局部性和对偶性的数学公式。因式分解同调提供了一种在很小的子区域内以物理的形式表达关于所有时空的物理理论的数学方法，其中物理可以用组合学和高维图论来简化。这个项目发展了更高范畴的变体的因式分解同调理论，并将其应用于数学物理和微分拓扑学。这一理论可以看作是对分层模空间上的层的研究。一个目的是利用伴随的因式分解同调来证明Baez-Dolan和Lurie的协边假设，即Atiyah公理意义下的拓扑量子场论是由它们在一点上的值唯一确定的。这将表明，时空X上的这种拓扑量子场论来自于X的分层的模空间上的一捆，因此局域性的概念可以通过这种模空间的拓扑来理解。第二个目标是证明流形X的这个分层模空间满足Poincar？二元性。这将导致X上的拓扑物理理论之间的对偶性，一旦被表示为前述模空间上的层。这种二元性是拓扑物理理论的一种形式的Koszul二元性。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.