It was established by Atiyah, Bott and Shapiro that one can recover the periodicity of K-theory from the Morita periodicity of Clifford algebras. This has since been translated to a correspondence between KO spectrum and 1d supersymmetric quantum field theories. Motivated by the work of Witten, Segal anticipated a 2-dimensional analogue for elliptic cohomology, which he called an elliptic object. This idea was later refined by Stolz and Teichner, who conjectured that there should be a similar correspondence between 2d (0,1) supersymmetric QFTs and topological modular forms (TMF). For this reason, one really would like to think of TMF as a sort of "categorified K-theory". This has turned out to be an incredibly rich and difficult problem.

To upgrade the 1-dimensional K-theory picture to the two dimensional one, maybe one should first ask what the categorified analogue of a Clifford algebras is. From the physical perspective, two reasonable proposals are the free fermion conformal nets and free fermion vertex operator algebras (VOAs). Douglas and Henriques show that you can recover a geometric model for the string group from free fermion conformal nets, and with a little bit of care working with vertex algebra extensions, one should be able to show that the (crossed-braided super modular) category of fermion VOA representations also has the correct homotopy type to give a model for the string group.

Now from the pure-math side, one might try to replace bundles of Clifford modules with a more direct categorification. To do this, one should note that one can see KO-theory as classifying graded Clifford module bundles up to some parity reversal action. Really, this is some reprhasing of Karoubi K-theory to match up with physical supersymmetry intuition. Since an invertible algebra is necessarily central simple, and all of the central simple super algebras in super vector spaces are Clifford algebras, one can instead see K-theory as classifying bundles of modules over invertible super algebras (again, up to parity reversal). We call these module bundles invertibly twisted vector bundles. To categorify this, we replace super vector bundles by 2-vector bundles: bundles of superlinear categories.

The question then becomes "What is an invertible superlinear monoidal category?" or maybe "What is an invertible superlinear braided category?". These are certainly super fusion categories, since any invertible algebra is also dualizable. In the non-super case, invertible fusion categories are already classified. Just as in the 1-dimensional case, these necessarily have trivial (Drinfild) centre. More interestingly, in the braided case, if one also finds that the (Muger) centre must be trivial, then they exactly get invertible supermodular categories, and the categories of free fermion VOA representations are certainly examples. Then in trying to "categorify" K-theory in two different ways, we have arrived back at the same thing!

It's maybe naive to hope that one really finds that TMF classifies module bundles over super modular categories, but it would be interesting to begin by classifying these crossed braided super modular categories up to Morita equivalence (which is probably interesting in its own right!), and seeing if the magic number 576 (the periodicity of TMF) or any other hints of the homotopy type of TMF show up.

Citations

• Clifford modules- Atiyah, Bott, and Shapiro
• Elliptic genera and quantum field theory- Witten
• Elliptic cohomology- Segal
• What is an elliptic object?- Stolz and Teichner
• Geometric string structures- Douglas and Henriques
• Lectures on twisted K-theory and orientifolds- Freed
• Invertible fusion categories- Sanford and Snyder