Tuesday, January 06, 2009

Geometry and Topology for Theoretical Physicists

Happy 2009 to all U-duality readers! Today was the first day of class for the Winter term of Caltech's math sequence for theoretical physicists. This math sequence consists of three classes covering the modern mathematics a young string theorist should know. For the Fall term the main textbook was Nakahara's Geometry Topology and Physics, 2nd Edition.

This term, the main text is Morita's Geometry of Differential forms.

Today's lecture jumped right in to Čech cohomology, which can be thought of as a type of 'unifying' cohomology in the sense that it is equivalent to de Rham, simplicial and singular cohomology for well-behaved choices of topological space X. de Rham cohomology arises most frequently for physicists, but since it involves solving differential equations, it is not always the most pleasant for computations. Simplicial cohomology is probably the most computation-friendly cohomology, so Čech cohomology can be thought of as a kind of 'bridge' for translating physics problems from de Rham to a simpler (pun intended) simplicial cohomology setting. The proof of the equivalence of de Rham and Čech cohomology is most enlightening in this respect and involves forming a square of commutative (or anticommutative if you wish) mappings between vector spaces of bi-degree forms e.g. (p,q where total degree is p+q), with the vertical direction increasing via the de Rham coboundary operator, d, and horizontal direction via the Čech coboundary operator, delta. One can then define a new coboundary operator D=d+delta and show it satisfies D^2=0. I'll sketch the proof in more depth in another post, for those interested. I'll also refer the interested reader to another classic text on cohomology, namely Bott and Tu's, Differential Forms in Algebraic Topology.