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. ***
