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

9504423然而，这个项目主要从构造的角度讨论3维和4维的拓扑量子场论，并用代数和组合的方法进行研究。特别是，研究人员致力于完成计算细节和构建Crane和Frenkel的“龙卷风公式”的例子，从“Hopf范畴”给出的初始数据构建4D TQFT，以及检查Crane/Yetter不变量的完全双范畴推广。调查人员已经证明，任何3D(分别为。4D)具有物理上合理的因式分解性质的TQFT(应由“完全Donaldson/Floer和Seiberg/Witten理论”满足)有相伴随的么半群范畴和(内部)Hopf代数(分别.其中的形式Hopf范畴对象)和形式Hopf范畴对象)，并且正在研究这种现象有多普遍。研究人员希望不变量的代数/组合结构对PL(Equiv.在项目过程中，可能会发现4-流形上的光滑)结构。几个相关的研究方向--2-纽结和Seifert曲面的“量子”不变量的构造，以及HOMFLY和Kauffman Skein范畴的研究，以期为TQFT的构造找到新的初始数据--也在进行中。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。由于它们的拓扑性质和结构性质符合狄拉克的量子力学形式，它们的研究揭示了流形结构、经典代数结构以及统计和量子物理之间令人惊讶的联系。这个项目将有助于揭示蕴含在我们生活的维度(S)空间中的丰富的代数结构。尽管研究人员的技术基本上是代数的，对大多数物理学家来说是陌生的，但一些著名的相对主义者已经表示，这个项目中研究的那种结构是发现和理解备受关注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. ***