Mathematical Sciences: Construction of a Geometric Category Representing H4(M;Z), and Its Implications

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Over the next three years, McLaughlin plans to find a category representing H4(M;Z), in a manner similar to the interpretation of H2(M;Z) as isomorphism classes of line bundles, and H3(M;Z) as equivalence classes of gerbs. He will investigate the consequences for string theory, elliptic cohomology, algebraic K-theory, knot theory, and the problem of defining degree three, non-abelian, sheaf cohomology. The basic idea is to use a smooth version of Deligne cohomology, and the transgression to the free loop space. This will be interpreted as a generalized holonomy map. The category will be constructed by deducing which geometric structure has a given gerb as its holonomy. McLaughlin discovered the object representing the first Pontrjagin class in this way. This work will give a new geometric insight into Deligne cohomology, as the objects constructed actually lie in this more refined theory. There will in turn, be consequences for algebraic K-theory, using Beilinson's regulator. The very existence of the transgression raises the possibility of a deep relationship between Chen's iterated integrals and Deligne cohomology, refining the work of Getzler, Jones and Petrack. McLaughlin has obtained preliminary formulae in this direction, which emphasize the extraordinary role played by the cup product. This program offers a candidate for the geometric cocycle in elliptic cohomology. The object in question is a "fake" infinite-dimensional vector bundle associated to a gerb. It looks locally like a vector bundle but glues together in a different way. This approach has the advantage of reconciling all the standard conjectures about elliptic cohomology and will be a priority in the proposed research. This "fake" bundle can also be studied as a substitute for the string bundle on loop space, opening the way for a rigorous index theory of strings. Finally, in the case of a three- manifold M, this object actually transgresses to give a new structure on the space of knots in M. The implications for knot theory will be fully investigated. The overall objective of this project is to unify several deep and currently disparate areas of topology. This will be done by showing how analytic techniques employed in their study are different aspects of a common ancestor. Applications are envisioned to the theory of knots and to other active areas of geometry and topology.