Principal Investigator: Charles D. Frohman
The principal investigator will extend and interpret quantum invariants of three-manifolds utilizing the geometry of character varieties. The ideas he is exploring will combine tools from gauge theory, representation theory, homological algebra, and three-manifold topology. This entails work on several problems. With Kania-Bartoszynksa he will define a quantum invariant of three-manifolds which will be a real analytic function on the open interval (-1,1). The invariant will be obtained by heat kernel regularization of the divergent formula for the Turaev-Viro invariant. The power series expansion at 0, will be in terms of weighted signed counts of surfaces carried by a spine of the manifold. The normalized limit as you approach 1, will yield the total Reidemeister torsion of the SU(2)-character variety of the fundamental group of the manifold. Using ideas from matrix models he will develop an analogous invariant based on the SL(2,C)-character variety of the three-manifold. With his students he will continue to study the connection between the A-polynomial and quantum invariants, and explore the knot and link homology theories of Khovanov and Khovanov-Rozansky. With Oliver Dasbach and Marta Asaeda he is looking at homology theories underlying the Alexander polynomial. Finally, given a three-manifold and a Heegaard splitting there is an algebra which is the Kauffman bracket skein module of the Heegaard surface and a bimodule over that algebra built from the skein modules of the two handle-bodies. With Mike McLendon, he will study whether this homology is a three-manifold invariant, and if it is, what its relation to Khovanov homology is.
The rational understanding of the path integrals of Richard Feynman stand as one of the major unresolved problems of mathematics. Using his integrals Feynman was able to make computations in quantum electrodynamics that far exceeded previous work. The tools he developed allowed the construction of modern integrated circuits. The rules that physicists use for computing path integrals have never been made completely rigorous. The major thrust of Frohman's work in recent years has been about these integrals in a simplified setting where things can actually be computed. Specifically, the Yang-Mills measure in the Kauffman bracket skein module assigns to a gauge field on the space of flat connections on a surface a number. The formula for the measure coincides with an asymptotic expansion that appears throughout the physical literature. However, in this situation it is actually a convergent series. Frohman is using this formula, and the estimates he used to prove it converged, to pursue the analytic study of three-manifold invariants that were before only computable using algebraic and combinatorial methods. The goal of the project is to reveal the geometric and toplogical nature of quantum invariants of three-manifolds to the end of increasing the understanding of three-manifolds, representation theory and quantum gravity.
