On Noncommutative Residue Gravitational Action Conformal Invariant And Geometric Quantization Formula

The present thesis is devoted to studying the noncommutative residue, gravitationalaction, conformal invariant and geometric quantization formula. Especially in recentyears, noncommutative geometry has very great development and plays a prominent rolein geometry, topology, number theory, as well as physics. Noncommutative residue comesfrom the important research results of Adler and Wodzicki, Wodzicki residue (noncom-mutative residue) appears in the computing Chern-Connes character formula of spectraltriple, and plays a very important role in noncommutative geometry. For an even dimen-sional compact oriented conformal real manifold without boundary, a canonical Fredholmmodule was constructed and a conformal invariant was defned by Connes. Connes alsogave an explanation of the Polyakov action and its4-dimensional analogy by using hisconformal invariant. In addition, Connes’ result was generalized to the higher dimen-sional case and an explicit expression of the Connes’ invariant in the fat case was givenby Ugalde. Chapter2generalizes the result of Connes to manifolds with boundary, doubleconformal invariants are constructed for real manifolds and complex manifolds.More importantly, Connes made a challenging observation that the noncommutativeresidue of the square of the inverse of the Dirac operator was proportional to the Einstein-Hilbert action, Kastler, Kalau and Walze gave some proof of this theorem respectively,which we call the Kastler-Kalau-Walze theorem. Ponge explained how to defne “lowerdimensional” volumes of any compact Riemannian manifold as the integrals of local Rie-mannian invariants. Fedosov etc. defned a noncommutative residue for manifolds withboundary and proved that it was a unique continuous trace. Gilkey, Branson and Fullingobtained a formula about heat kernel expansion coefcients of nonminimal Operator. Forspin manifold with boundary and associated with Dirac operator, Wang gave an opera-tor theoretic explanation of the gravitational action and proved a Kastler-Kalau-Walzetype theorem. For an even dimensional compact oriented real manifold, Ackermann and Tolksdorf proved a generalized version of the well-known Lichnerowicz formula for thesquare of the most general Dirac operator with torsion. Chapter3,4and5deal with thenoncommutative residue for manifolds with boundary associated to some operators andgive some operator theoretic explanation of the gravitational action.In addition, the Atiyah-Segal-Singer equivariant index theorems play an importantrole in diferential geometry. In1982, a fascinating conjecture appeared about group ac-tions. Guillemin and Sternberg gave a precise mathematical formulation of Dirac’s ideathat “quantization commutes with reduction”, in which they defned the former as geo-metric quantization. For even dimensional spincmanifolds, Fuchs proved a Kostant typeformula and got a cutting formula by the Kostant formula. Liu and Wang extended theFreed odd index theorem to the equivariant case and proved the Atiyah-Hirzebruch typetheorems for odd spin manifolds. Chapter6focus on several odd geometric quantizationformulas related to equivariant odd index theorems.The thesis is comprised of six chapters. Chapter1starts with a short introduction toBoutet de Monvel’s calculus and the noncommutative residue on manifolds with boundary.Chapter2generalizes the result of Connes. Fredholm module associated to the d(ˉ)operator are constructed, and double conformal invariants for real manifolds and complexmanifolds are obtained.Chapter3introduces the Kastler-Kalau-Walze type theorem associated to nonmin-imal operators by heat equation asymptotics on compact manifolds with boundary, andan operator theoretic explanation of the gravitational action is given.Chapter4analyses the lower dimensional volumes associated to sub-Dirac operatorsfor foliation with spin leave.Chapter5, Kastler-Kalau-Walze type theorems associated to Dirac operators withtorsion on compact manifolds with boundary are obtained.In Chapter6, we mop up several odd geometric quantization formulas by equivariantodd index theorems.