Mathematical Sciences: Topological Quantum Field Theories and Related Invariants in 3- and 4-Manifold Topology

9504423 Yetter本项目主要研究3维和4维的拓扑量子场论，主要从它们的构造和代数和组合方法的研究角度出发。特别是，研究人员致力于完成Crane和Frenkel的“龙卷风公式”的计算细节和示例构建，从“Hopf类别”给出的初始数据构建4D TQFT，并检查Crane/Yetter不变量的完全双分类概括。研究人员已经证明，任何3D打印（3D打印）。具有物理合理分解特性的TQFT（应由“完整的Donaldson/Floer和Seiberg/Witten理论”满足）具有相关的一元范畴和（内部）Hopf代数（见图4）。一元双范畴)和其中的形式Hopf范畴对象)，并正在研究这种现象的普遍性。研究人员希望在研究过程中能够找到4流形上对PL（等光滑）结构敏感的不变量的代数/组合结构。几个相关的研究方向也在进行中——构建2节和Seifert曲面的“量子”不变量，以及研究HOMFLY和Kauffman编织类别，以期为TQFT的构建找到新的初始数据。拓扑量子场论是研究3维和4维空间结构的一个活跃的新领域。他们的研究揭示了流形结构、经典代数结构、统计和量子物理学之间令人惊讶的联系，因为它们的拓扑性质和结构特性符合狄拉克的量子力学形式，所以被命名为流形。这个项目将有助于揭示隐含在我们生活的维度空间中的丰富代数结构。虽然研究人员的技术主要是代数的，对大多数物理学家来说是陌生的，但一些著名的相对论主义者表示相信，在这个项目中研究的这种结构是发现和理解备受寻求的量子引力理论的关键。* * *

9504423 Yetter This project deals with topological quantum field theories in dimensions 3 and 4, primarily from the point of view of their construction and study by algebraic and combinatorial means. In particular, the investigators are engaged in the completion of computational details and the construction of examples of the "Tornado Formula" of Crane and Frenkel, a construction of 4D TQFT's from initial data given by a "Hopf category," and the examination of a fully bicategorical generalization of the Crane/Yetter invariant. The investigators have already shown that any 3D (resp. 4D) TQFT with physically reasonable factorization properties (which should be satisfied by the "full Donaldson/Floer and Seiberg/Witten theories) has an associated monoidal category and (internal) Hopf algebra (resp. monoidal bicategory) and formal Hopf category object therein) and are studying how general this phenomenon is. It is the investigators' hope that algebraic/combinatorial constructions for invariants sensitive to PL (equiv. smooth) structure on 4-manifolds may be found in the course of the project. Several related lines of research-- constructions of "quantum" invariants of 2-knots and Seifert surfaces, and the study of the HOMFLY and Kauffman skein categories with a view to finding new initial data for the construction of TQFT's--are also being pursued. Topological quantum field theories are a lively new area of research into the structure of spaces of dimensions 3 and 4. So named because of their topological nature and their structural properties, which fit into Dirac's formalism for quantum mechanics, their study has revealed surprising connections between the structure of manifolds, classical algebraic structures, and statistical and quantum physics. This project will contribute to the unfolding of the rich algebraic structures implicit in spaces of the dimension(s) in which we live. Although the investigators' techniques are primarily algebraic in fl avor and foreign to most physicists, some noted relativists have expressed the belief that constructions of the sort studied in this project hold the key to finding and understanding the much sought quantum theory of gravity. ***