From: baez@galaxy.ucr.edu (John Baez) Subject: Re: Elementary QFT queries Date: 27 Oct 2000 10:10:45 GMT Newsgroups: sci.physics.research Summary: [missing] In article , A.J. Tolland wrote: >Topological K-Theory is a generalized cohomology theory, a way of >computing topological invariants of smooth manifolds. (IIRC, there is >also something called algebraic K-theory. Any relation?) Sure, these theories are close cousins. Remember the relation between topological spaces and algebras: starting from a space we can form the algebra of continuous complex-valued functions on it, and by restricting our class of spaces and algebras sufficiently, we can obtain a perfect correspondence between (sufficiently nice) spaces and (sufficiently nice) algebras. The Gelfand-Naimark theorem is one way of making this precise. But most importantly, this relation suggests that anything we can do for topological spaces, we can also do - with enough cleverness - for algebras. In particular, if we're smart, we can take the definition of topological K-theory, run it through the space/algebra correspondence, and pull out a definition of K-theory for algebras. K-theory is all about vector bundles on topological spaces: to define K(X) for a space X, you just take the category of vector bundles over X, decategorify it, and thrown in formal additive inverses. How can we copy this for an algebra? Well, we need to figure out how to describe a vector bundle over X in terms of the algebra A of continuous functions on X. Here's the trick: the set M of all continuous sections of a vector bundle over X is a *module* over the algebra A. I.e., you can add elements of M, and multiply them by elements of A, and all the obvious rules hold. And if X is reasonably nice (say a compact Hausdorff space), M is actually a "finitely generated projective module" over A - a particular nice sort of module that algebraists swoon over. Even better, for such spaces X, every finitely generated projective module over A comes from a vector bundle over X in this way! This is called Swan's theorem. So, this tells us how to define K(A) when A is an algebra. Just take the category of finitely generated projective A-modules, decategorify it, and throw in formal additive inverses! In other words, algebraic K-theory is defined exactly like topological K-theory, with finitely generated projective modules serving as the fill-in for vector bundles. Well, at least that's the beginning of the story. As you may know, for spaces we have not only K(X) = K^0(X), but a bunch of other K-groups K^i(X), which all fit together in a nice big package. Defining these "higher K-groups" in the algebraic context is a bit of work. Quillen did it in the late 1960s. By the way, the space/algebra correspondence works most easily when the algebras happen to be *commutative*, but Connes and others have figured out how to do lots of it for *noncommutative* algebras too. In particular, Connes' "noncommutative geometry" ideas - which are becoming rather popular in particle physics and string theory - are nicely tied to earlier work on K-theory for noncommutative algebra. All the stuff you know and love about topological K-theory, like the Chern character, has analogs in this noncommutative setting! I bet string theorists are already starting to ponder this stuff.... Hmm. This thread is titled "elementary QFT queries", and here we are, sailing off into the scientific ionosphere! I'd better quit.