Augmented Spinor Space And Riemann-Roch Operator

Pauli martices in Physics equal to spinor space in Mathematics, In Physics,Pauli matrices are{P_i|P_iP_j + P_jP_i = -2δ_ij, P_i^-1=(P|-)_i^t}For any two Pauli matrics systems {P_i}, {P|^<sub>i}, by Schur lemma,there exists an unitary matrix T satisfyingTP_iT^-1 = (P|^<sub>)iwe say these two Pauli matrices systems are equal. T satisfying this equality is unique up to a multiple e^iθ, so there not only exist a transformation between two Pauli matrices systems ,but also an uncertain factor e^iθ, so corresponding to spinor space ,iso-morphism between any two spinor spaces is unique up to a multiple e^iθ.For studying spinor space,we define an augmented spinor structure on spinor space,thus we can get the uniqueness of Schur isomorphism, this new concept hope to be helpful to Physics,in the last part,we give the relation between Riemann-Roch operator and Dirac operator.Firstly,we introduce Clifford algebra in detail and it's some subgroup:Spin, Spin^C. At the same time, we give the action of these group and it's lie group on space. Further-more,we can get the relation between Spin, Spin^C and SO(2n).Secondly,we translate Pauli matrices into spinor space in geometry language.By Schur lemma,we can get the isomorphism between two spinor space is unique to e^iθ multiple.Through adding some structure :Jack-map and Jack-orientation, we can get the uniqueness of isomorphism.Consequently,we give a naive model: Grassmann algebra. Tensor product of spinor space can extend low dimension to higher dimentinal space.Lastly,we study the relation between Riemann-Roch operator and Dirac operator.If there is a U(n) structure on M,by imbedding i : U(n) → Spin^C(2n), we can get it's Spin^C Structure .At the same time,we discuss the relation among Levi-civita connec-...