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Spinc Structure
In spin geometry, a spinᶜ structure (or complex spin structure) is a special classifying map that can exist for orientable manifolds. Such manifolds are called spinᶜ manifolds. C stands for the Complex number, complex numbers, which are denoted \mathbb and appear in the definition of the underlying Spinc group, spinᶜ group. In four dimensions, a spinᶜ structure defines two complex plane bundles, which can be used to describe negative and positive Chirality (physics), chirality of Spinor, spinors, for example in the Dirac equation of relativistic quantum field theory. Another central application is Seiberg–Witten theory, which uses them to study 4-manifold, 4-manifolds. Definition Let M be a n-dimensional orientable manifold. Its tangent bundle TM is described by a classifying map M\rightarrow\operatorname(n) into the Classifying space for SO(n), classifying space \operatorname(n) of the special orthogonal group \operatorname(n). It can factor over the map \operatornam ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin Geometry
In mathematics, spin geometry is the area of differential geometry and topology where objects like spin manifolds and Dirac operators, and the various associated index theorems have come to play a fundamental role both in mathematics and in mathematical physics. An important generalisation is the theory of symplectic Dirac operators in symplectic spin geometry and symplectic topology, which have become important fields of mathematical research. See also * Contact geometry * Symplectic topology * Spinor * Spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\co ... * Spin manifold Books * * Differential topology Differential geometry {{physics-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |