## Spans and the Categorified Heisenberg Algebra

Heisenberg reinvented matrices while discovering quantum mechanics, and the algebra generated by annihilation and creation operators obeying the canonical commutation relations was named after him. It turns out that matrices arise naturally from 'spans', where a span between two objects is just a third object with maps to both those two. In terms of spans, the canonical commutation relations have a simple combinatorial interpretation.

More recently, Khovanov introduced a 'categorified' Heisenberg algebra, where the canonical commutation relations hold only up to isomorphism, and these isomorphisms obey new relations of their own. The categorified Heisenberg algebra naturally acts on the '2-Fock space' describing collections of particles in a 4-dimensional topological quantum field theory.

The meaning of the new relations in the categorified Heisenberg algebra was initially rather mysterious. However, Jeffrey Morton and Jamie Vicary have shown that they again have a nice interpretation in terms of spans. We can begin to formalize this using the work of Alex Hoffnung and Mike Stay, who have shown that spans of groupoids are morphisms in a symmetric monoidal bicategory.

I gave a talk on this at Oxford and Université Paris 7, and a 3-hour minicourse on it in Lanzhou. Here's the talk:

• Spans and the Categorified Heisenberg Algebra: slides and video.
The minicourse goes into more detail; you can see the slides here:
• Part 1 – How Jeffrey Morton and Jamie Vicary categorified the Heisenberg algebra using spans of groupoids.

• Part 2 – The definition of symmetric monoidal bicategory, following Mike Stay.

• Part 3 – How the categorified Heisenberg algebra manifests in linear algebra and the theory of Young diagrams.

For more on this subject try these papers:

For more on the idea of using spans in quantum mechanics, try this talk of mine:

Creation and annihilation spans by Jeffrey Morton, truncated shuffle polytope by Mike Stay, Young diagrams by Jeffrey Morton and Jamie Vicary
baez@math.removethis.ucr.andthis.edu