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

小行星9504423 这个项目涉及三维和四维的拓扑量子场论，主要是从它们的构造和研究的角度，通过代数和组合的手段。 特别是，调查人员从事完成的计算细节和建设的例子“龙卷风公式”的起重机和Frenkel，一个建设的四维TQFT的初始数据给出的“霍普夫类别”，并检查一个完全bicategorical推广的起重机/Yetter不变。 研究人员已经表明，任何3D（或 4D）TQFT具有物理上合理的因子分解性质（这应该由“完全唐纳森/Floer和Seiberg/维滕理论”满足），有一个相关的monoidal范畴和（内部）Hopf代数（分别为. monoidal bicategory）和形式的霍普夫范畴对象），并正在研究这种现象有多普遍。 研究人员希望，对PL（equiv.）敏感的不变量的代数/组合构造。光滑）结构上的4流形可能会发现在该项目的过程中。 几个相关的研究-建设的“量子”不变量的2节和塞弗特表面，并研究的HOMFLY和考夫曼绞类别，以期找到新的初始数据建设的TQFT的-也正在进行中。 拓扑量子场论是研究三维和四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、统计和量子物理之间令人惊讶的联系。 这个项目将有助于丰富的代数结构隐含在我们生活的维度空间的展开。 虽然研究者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者表示相信，在这个项目中研究的那种结构是发现和理解备受追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. ***