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Teorema do índice para operadores de Dirac
- Autor/a
- Franco Sanmartín, Carlos
This work describes a proof for the Atiyah-Singer index theorem related to Dirac operators, obtained using physical ideas, like the Schwartz kernel’s asymptotic expansion of the heat equation and the Getzler symbolic calculus. This theorem concerns compact, evendimensional, oriented Riemannian manifolds without boundary, and equipped with certain Clifford bundles. The corresponding Dirac operator acts over its smooth sections; more precisely, it applies the space of even sections to the space of odd sections, with respect to a grading of the Clifford bundle. It is an elliptic operator, and thus it is Fredholm; what means that the kernel and cokernel are finite dimensional. So the analytical index can be defined as the difference between these two dimensions. The index theorem gives an equality between this analytical index and the evaluation of a certain characteristic class, the Â-genus, in the fundamental homology class of the manifold (integral of the Â-genus over the manifold). Considering arbitrary coefficients, the index theorem is a very general result which includes other important theorems like Gauss-Bonnet, signature and Riemann-Roch ones, and it is useful in Geometry, Topology and Theoretical Physics.The proof presented here involves a wide variety of different concepts, like curvature, characteristic classes, Clifford algebras, Dirac operators, spin structures, Sobolev spaces, spectral decomposition, Hodge decomposition, trace-class operators, Schwartz kernels, asymptotic expansions and symbolic calculus. Consequently, a lot of geometric, topological, analytical and algebraic tools are required.
Data sheet
- Edition
- 1
- Publication place
- Santiago de Compostela
- Publication Year
- 2018
- Serie
- 135a Publicaciones del Departamento de Geometría y Topología
- Availability
- Si
This work describes a proof for the Atiyah-Singer index theorem related to Dirac operators, obtained using physical ideas, like the Schwartz kernel’s asymptotic expansion of the heat equation and the Getzler symbolic calculus. This theorem concerns compact, evendimensional, oriented Riemannian manifolds without boundary, and equipped with certain Clifford bundles. The corresponding Dirac operator acts over its smooth sections; more precisely, it applies the space of even sections to the space of odd sections, with respect to a grading of the Clifford bundle. It is an elliptic operator, and thus it is Fredholm; what means that the kernel and cokernel are finite dimensional. So the analytical index can be&