Explicit Computations of Examples in Spin Geometry - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T00:34:50Z http://mathoverflow.net/feeds/question/116133 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/116133/explicit-computations-of-examples-in-spin-geometry Explicit Computations of Examples in Spin Geometry mkreisel 2012-12-12T01:26:48Z 2012-12-12T05:36:14Z <p>I have been trying to learn about spin geometry, Dirac operators, and index theory by reading Lawson/Michelson's "Spin Geometry" and Friedrich's "Dirac Operators in Riemannian Geometry." Both are abstract, and basically no explicit examples are worked all the way through.</p> <p>For example, I have been trying to find the spinor bundles, Dirac operators, and various indices for relatively simple manifolds: spheres and tori. However often these computations are detailed and even when I get to the end, it's not clear that I've done it correctly. </p> <p>Is there another book, or perhaps online notes, which have a bunch of examples worked through in detail so that I can make sure what I'm doing is correct and also have a bank of examples to look at as I progress?</p> http://mathoverflow.net/questions/116133/explicit-computations-of-examples-in-spin-geometry/116144#116144 Answer by MTS for Explicit Computations of Examples in Spin Geometry MTS 2012-12-12T05:36:14Z 2012-12-12T05:36:14Z <p>Appendix A to Chapter 9 of the book <em>Elements of Noncommutative Geometry</em> by Gracia-Bondia, Varilly, and Figueroa is titled "Spin geometry of the Riemann sphere". It is 15 pages long and goes into quite some detail. (Some might call that level of detail excruciating, but YMMV.)</p> <p>As Paul Siegel notes, computations on homogeneous spaces can be done quite effectively using representation theory. Some years ago, in the course of learning about that approach, I wrote up an account of the construction of the spinor bundle, Dirac operator, etc on \$S^2\$, viewed as the homogeneous space \$SU(2)/U(1)\$. If you're interested, email me (you can find my email address at my website, linked in my profile) and I can send it to you.</p>