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One of the many wonderful things about K-theory, relative to other generalized cohomology theories, is that it can be defined for not-necessarily-commutative C*-algebras. The resulting construction, in addition to reproducing topological K-theory on compact Hausdorff spaces $X$ via the commutative C*-algebras $C(X)$, can also reproduce equivariant K-theory via crossed products $C(X) \rtimes G$ and twisted K-theory via C*-algebras constructed from the global sections of certain bundles of C*-algebras over $X$, both of which will be noncommutative in general.

It is tempting to wonder if an analogous theory can be developed for tmf. This theory should have (at least) two parts:

  • A contravariant functor associating to any compact Hausdorff space $X$ a "higher C*-algebra."
  • A covariant functor associating to any "higher C*-algebra" its "operator tmf."

At the very least, the composition of these functors should produce $\text{tmf}(X)$. There are various other things it's natural to ask for. To get equivariant tmf, higher C*-algebras should admit crossed products coming from actions of 2-groups. To get twisted tmf, bundles of certain higher C*-algebras on $X$ should be classified by $H^4(X, \mathbb{Z})$.

Has a proposal for such a theory been given in the literature?

One candidate here is that "higher C*-algebra" should mean tensor C*-category, and that we should associate to a compact Hausdorff space $X$ the tensor C*-category of Hilbert C*-modules over $C(X)$, or equivalently of Riemannian vector bundles over $X$. Then we should take "higher modules" over this object in some refined sense.

Another candidate is that "higher C*-algebra" should mean some variant of conformal nets in the sense of Bartels-Douglas-Henriques, which I guess are more like "higher von Neumann algebras." One reason to believe this is that (at least if I understand what I have been told) there is a conformal net whose automorphism group has the homotopy type $B^3 \mathbb{Z}$, which is the right homotopy type for bundles of that conformal net to be classified by $H^4(X, \mathbb{Z})$. But I don't know what to associate to a compact Hausdorff space.

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    $\begingroup$ What's a tmf? [That was my question, but for some reason comments require at least fifteen characters.] $\endgroup$
    – David Handelman
    Commented Dec 12, 2014 at 0:14
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    $\begingroup$ @David: tmf stands for "topological modular forms." It is a "higher" version of K-theory. For a pointer to the literature see, for example, ms.unimelb.edu.au/~nganter/talbot. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 12, 2014 at 0:30
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    $\begingroup$ Some people seem to think that vertex algebras should play the role of C*-algebras for TMF. Vertex algebras are known to be related to lots of things that are known to be related to TMF - the Witten genus, conformal field theory, string structures, and so on - but I'm not sure there's any more evidence than that. $\endgroup$
    – Paul Siegel
    Commented Dec 12, 2014 at 1:59
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    $\begingroup$ I have shown in my dissertation that any cohomology theory for $C^{\ast}$-algebras that has the property that the inclusion $A \subset M_2(A)$ induces an isomorphism must be a $bu$-module, i.e., it admits some self-map of degree two. So any generalization of $tmf$ cannot be nicely compatible with notions like Morita invariance etc. However, there is unpublished work using unitary vertex superalgebras as some sort of categorification of complex Clifford algebras, this goes in a new and promising direction, see maths.gla.ac.uk/~cvoigt/research.xhtml $\endgroup$
    – Andreas Thom
    Commented Dec 12, 2014 at 9:25
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    $\begingroup$ @Andreas: thanks for the suggestions! In fact this theory should not assign anything to an arbitrary C*-algebra, the problem being that the category of modules (whatever one means by this) over an arbitrary C*-algebra lacks a natural tensor structure. This theory should at best assign things to commutative C*-algebras and maybe Hopf C*-algebras. $\endgroup$
    – Qiaochu Yuan
    Commented Dec 12, 2014 at 18:10

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