The n-Category Café

Here are a few talks that look fun to me:

• Ed Witten will give us a “new look” at the Jones polynomial, an invariant of knots related to quantum field theory.
• Steve Awodey and Vladimir Voevodsky will talk about homotopy type theory.
• Martin Hyland will talk about functional interpretations of type theory.
• Anton Zeilinger will talk about experiments with entangled photons.
• Eric Allender will talk about relationships between computational complexity and algorithmic information theory.
• Maxim Kontsevich, Ed Witten, Sergei Gukov and Dirk Kreimer will talk about connections between number theory and physics — I don’t see titles for these talks yet!

Among $n$-Café regulars, David Corfield, Minhyong Kim, Jamie Vicary and Jeffrey Morton will be there. Who else?

I’ll be there, and I’ll talk about categorified quantum mechanics:

• Spans and the categorified Heisenberg algebra.

You may remember an old puzzle here on the $n$-Café, about two different ways to ‘categorify the Heisenberg algebra’. The first showed up in the work of Domenico Fiorenza, Chris Rogers and Urs Schreiber on higher symplectic geometry: just as the Heisenberg Lie algebra arises naturally from a symplectic vector space, we get a Lie 2-algebra from a ‘2-plectic vector space’, and so on. The second, a version of the associative Heisenberg algebra where the canonical commutation relations hold only up to isomorphism, showed up in Khovanov’s work on representation theory, and again—in a somewhat different guise—in Jeffrey Morton and Jamie Vicary’s work on spans of groupoids as a categorified version of operators between Hilbert spaces.

I have nothing to say about the relation between these two gadgets; indeed they seem quite different to me. The first is not really ‘categorifying’ the Heisenberg Lie algebra; it’s using higher categorical ideas to generalize the Heisenberg Lie algebra from the symplectic case to the $n$-plectic case. The second is really categorifying the associative Heisenberg algebra, in that we can apply a standard decategorification process and get back that algebra.

But I’ll say a bit about how the second one shows up in physics, since this had been rather unclear up to now. Briefly, whenever we have a 4-dimensional (or higher-dimensional) TQFT, we get ‘2-Fock space’: a 2-Hilbert space that describes collections of particle-like topological defects. And then, we get an action of Morton and Vicary’s categorified Heisenberg algebra on this 2-Fock space.

One goal of my talk is to discuss some work by my former students, since it fits together in nice patterns that I haven’t seen people talking about. I especially mean these papers:

• Jeffrey Morton and Jamie Vicary, The categorified Heisenberg algebra I: a combinatorial representation.
• Jeffrey Morton, 2-Vector spaces and groupoids.
• Jeffrey Morton, Categorified algebra and quantum mechanics.
• Alex Hoffnung, Spans in 2-categories: a monoidal tricategory.
• Mike Stay, Compact closed bicategories.