A rudimentary theory of topological 4D gravity
In the Segal-Atiyah approach to abstract topological quantum field theory, J. Milnor's classical concept of a cobordism category provides an essential ingredient. Recently, \textit{U. Tillmann} [Invent. Math. 130, No. 2, 257-275 (1997; Zbl 0891.55019)] has shown, in this context, the utility of certain closely related 2-categories. Based upon Tillmann's ideas, the paper under review presents an outline of another theory of topological gravity, together with some instructive examples of such a theory. More precisely, the author first defines the notion of a gravity 2-category as a generalizaton of a cobordism category, introduces then theories of topological gravity as monoidal functors from a topological gravity category to the category of dualizable objects over a ring-spectrum and exhibits, subsequently, various concrete topological examples of them. Also, the relations of these possible theories of topological gravity to other, already existing topological quantum field theories is briefly sketched. Alltogether, this paper must be seen as an outline of an interesting and seemingly promising new approach to topological quantum field theory. It is merely strategic, speculative and fragmentary, on the one hand, but nevertheless highly enlightening, inspiring and guiding, on the other hand.
scientific article; zbMATH DE number 1818580
scientific article