Quantum Topology in Dimension Three

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AbstractAward: DMS-0508635Principal Investigator: Charles D. FrohmanThe principal investigator will extend and interpret quantuminvariants of three-manifolds utilizing the geometry of charactervarieties. The ideas he is exploring will combine tools fromgauge theory, representation theory, homological algebra, andthree-manifold topology. This entails work on severalproblems. With Kania-Bartoszynksa he will define a quantuminvariant of three-manifolds which will be a real analyticfunction on the open interval (-1,1). The invariant will beobtained by heat kernel regularization of the divergent formulafor the Turaev-Viro invariant. The power series expansion at 0,will be in terms of weighted signed counts of surfaces carried bya spine of the manifold. The normalized limit as you approach 1,will yield the total Reidemeister torsion of the SU(2)-charactervariety of the fundamental group of the manifold. Using ideasfrom matrix models he will develop an analogous invariant basedon the SL(2,C)-character variety of the three-manifold. With hisstudents he will continue to study the connection between theA-polynomial and quantum invariants, and explore the knot andlink homology theories of Khovanov and Khovanov-Rozansky. WithOliver Dasbach and Marta Asaeda he is looking at homologytheories underlying the Alexander polynomial. Finally, given athree-manifold and a Heegaard splitting there is an algebra whichis the Kauffman bracket skein module of the Heegaard surface anda bimodule over that algebra built from the skein modules of thetwo handle-bodies. With Mike McLendon, he will study whether thishomology is a three-manifold invariant, and if it is, what itsrelation to Khovanov homology is.The rational understanding of the path integrals of RichardFeynman stand as one of the major unresolved problems ofmathematics. Using his integrals Feynman was able to makecomputations in quantum electrodynamics that far exceededprevious work. The tools he developed allowed the construction ofmodern integrated circuits. The rules that physicists use forcomputing path integrals have never been made completelyrigorous. The major thrust of Frohman's work in recent years hasbeen about these integrals in a simplified setting where thingscan actually be computed. Specifically, the Yang-Mills measure inthe Kauffman bracket skein module assigns to a gauge field on thespace of flat connections on a surface a number. The formula forthe measure coincides with an asymptotic expansion that appearsthroughout the physical literature. However, in this situation itis actually a convergent series. Frohman is using this formula,and the estimates he used to prove it converged, to pursue theanalytic study of three-manifold invariants that were before onlycomputable using algebraic and combinatorial methods. The goal ofthe project is to reveal the geometric and toplogical nature ofquantum invariants of three-manifolds to the end of increasingthe understanding of three-manifolds, representation theory andquantum gravity.