Brodzki et al on KK theory, and was struck by the increasingly categorical nature of the C* approach. ]]>

I’d expect that errors would be made, as the ekpyrotic scenario requires new math and physics. In short, the full M-theory is needed. Colliding 3-branes seems to involve some hardcore math. I’m not even sure people agree as to how to properly describe D-brane charge. Does one use K-theory or K-homology? I personally prefer the K-homology route, as it employs Fredholm modules, which are related to spectral triples in noncommutative geometry.

Urs talked about this on 07-26-2006, in the topic K-Theory for Dummies, II. The K-Matrix theory paper he cites by Asakawa is one I read back in 2004. They use Fredholm modules to describe D-brane configurations in 10D spacetime. Explicitly, the eigenvalues of the scalar fields Phi^{mu} (mu=0,1,…,9), are supposed to describe the positions of non-BPS instantons in R^10. So their spectral triples are of the form (A,H,T), where A=C(M) is a C*-algebra (algebra of scalar fields), and spec(A)=M describes the embedding of a higher dimensional D-brane worldvolume in R^10.

As the Phi^{mu} are Hermitian matrices belonging to the adjoint rep of U(N), I’m sure you can guess how primitive idempotents and ribbon graphs arise in K-Matrix theory. Even more, I’m sure you know what spec(A)=M looks like. ðŸ˜‰

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