First off, a nice recent XKCD comic about height.
I’ve been busy of late starting up classes, working on a paper which should appear on the archive in a week or so on the groupoid/2-vector space stuff I wrote about last year. I resolved the issue I mentioned in a previous post on the subject, which isn’t fundamentally that complicated, but I had to disentangle some notation and learn some representation theory to get it figured out. I’ll maybe say something about that later, but right now I felt like making a little update. In the last few days I’ve also put together a little talk to give at Octoberfest in Montreal, where I’ll be this weekend. Montreal is a lovely city to visit, so that should be enjoyable.
A little while ago I had a talk with Dan’s new grad student – something for a class, I think – about classical and modern differential geometry, and the different ideas of curvature in the two settings. So the Gaussian curvature of a surface embedded in has a very multivariable-calculus feel to it: you think of curves passing through a point, parametrized by arclength. The have a moving orthogonal frame attached: unit tangent vector, its derivative, and their cross-product. The derivative of the unit tangent is always orthogonal (it’s not changing length), so you can imagine it to be the radius of a circle, with length , the radius of curvature. Then you have curvature along that path. At any given point on a surface, you get two degrees of freedom – locally, the curve looks like a hyperboloid or an ellipse, or whatever, so there’s actually a curvature form. The determinant gives the Gaussian curvature . So it’s a “second derivative” of the surface itself (if you think of it as ). The Gaussian curvature, unlike the curvature in particular directions, is intrinsic – preserved by isometry of the surface, so it’s not really dependent on the embedding. But this fact takes a little thinking to get to. Then there’s the trace – the scalar curvature.
In a Riemannian manifold, you need to have a connection to see what the curvature is about. Given a metric, there’s the associated Levi-Civita connection, and of course you’d get a metric on a surface embedded in , inherited from the ambient space. But the modern point of view is that the connection is the important object: the ambient space goes away entirely. Then you have to think of what the curvature represents differenly, since there’s no normal vector to the surface any more. So now we’re assuming we want an intrinsic version of the “second derivative of the surface” (or n-manifold) from the get-go. Here you look at the second derivative of the connection in any given coordinate system. You’re finding the infinitesimal noncommutativity of parallel transport w.r.t two coordinate directions: take a given vector, and transport it two ways around an infinitesimal square, and take the difference, get a new vector. This all is written as a (3,1)-form, the Riemann tensor. Then you can contract it down and get a matrix again, and then contract on the last two indices (a trace!) and you get back the scalar curvature again – but this is all in terms of the connection (the coordinate dependence all disappears once you take the trace).
I hadn’t thought about this stuff in coordinates for a while, so it was interesting to go back and work through it again.
In the noncommutative geometry seminar, we’ve been talking about classical mechanics – the Lagrangian and Hamiltonian formulation. So it reminded me of the intuition that curvature – a kind of second derivative – often shows up in Lagrangians for field theories using connections because it’s analogous to kinetic energy. A typical mechanics Lagrangian is something like (kinetic energy) – (potential energy), but this doesn’t appear much in the topological field theories I’ve been thinking about because their curvature is, by definition, zero. Topological field theory is kind of like statics, as opposed to mechanics, that way. But that’s a handy simplification for the program of trying to categorify everything. Since the whole space of connections is infinite dimensional, worrying about categorified action principles opens up a can of worms anyway.
So it’s also been interesting to remember some of that stuff and discuss it in the seminar – and it was inially suprising that it’s the introduction to “noncommutative geometry”. It does make sense, though, since that’s related to the formalism of quantum mechanics: operator algebras on Hilbert spaces.
Finally, I was looking for something on 2-monads for various reasons, and found a paper by Steve Lack which I wanted to link to here so I don’t forget it.
The reason I was looking was that (a) Enxin Wu, after talking about deformation theory of algebras, was asking after monads and the bar construction, which we talked about at the UCR “quantum gravity” seminar, so at some point we’ll take a look at that stuff. But it reminded me that I was interested in the higher-categorical version of monads for a different reason. Namely, I’d been talking to Jamie Vicary about his categorical description of the harmonic oscillator, which is based on having a monad in a nice kind of monoidal category. Since my own category-theoretic look at the harmonic oscillator fits better with this groupoid/2-vector space program I’ll be talking about at Octoberfest (and posting about a little later), it seemed reasonable to look at a categorified version of the same picture.
But first things first: figuring out what the heck a 2-monad is supposed to be. So I’ll eventually read up on that, and maybe post a little blurb here, at some point.
Anyway, that update turned out to be longer than I thought it would be.