Quantum Topology in Dimension Three

项目编号：dms -0508635项目负责人：Charles D. frohman项目负责人将利用特征变量的几何形式对三流形的量子不变量进行扩展和解释。他正在探索的思想将结合规范理论、表示理论、同调代数和三流形拓扑等工具。这需要解决几个问题。利用Kania-Bartoszynksa，他将定义一个三流形的量子不变量，它将是开区间（-1,1）上的实解析函数。该不变量将通过对Turaev-Viro不变量的发散式进行热核正则化得到。幂级数在0处的展开式，将是由流形脊带的曲面的加权带符号计数表示的。当你接近1时的归一化极限，将产生流形基本群的SU(2)-特征变体的总Reidemeister扭转。利用矩阵模型的思想，他将基于三流形的SL（2，C）字符变体开发一个类似的不变量。他将与他的学生继续研究a多项式与量子不变量之间的联系，并探索Khovanov和Khovanov- rozansky的结和链接同调理论。他与oliver Dasbach和Marta Asaeda一起研究Alexander多项式的同调理论。最后，给定一个三流形和Heegaard分裂，有一个代数，它是Heegaard曲面的Kauffman支架串模和由两个柄体串模构建的代数上的双模。与迈克·麦克伦登一起，他将研究这种同调是否是三流形不变量，如果是，它与霍万诺夫同调的关系是什么。对理查德·费曼的路径积分的合理理解是数学中尚未解决的主要问题之一。利用他的积分，费曼能够在量子电动力学中进行计算，远远超过了以前的工作。他开发的工具使现代集成电路的构建成为可能。物理学家用来计算路径积分的规则从来都不是完全严格的。弗洛曼近年来的主要工作是在简化的情况下研究这些积分，在这种情况下，事情实际上是可以计算的。具体来说，在考夫曼支架绞丝模块中，Yang-Mills测量给一个平面连接空间上的规范场分配了一个数字。该测量公式与整个物理文献中出现的渐近展开一致。然而，在这种情况下，它实际上是一个收敛级数。Frohman正在使用这个公式，以及他用来证明它收敛的估计，来进行三流形不变量的分析研究，这些不变量以前只能用代数和组合方法计算。该项目的目标是揭示三流形量子不变量的几何和拓扑性质，以增加对三流形、表示理论和量子引力的理解。

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.