As promised in the previous post, here is a little writeup of the second conference I was at recently…

Connections in Geometry and Physics

The conference at PI was an interestingly varied cross-section of talks, with a good many of them about geometry which, to be honest, is a little over my head.  Ostensibly about “connections”, the talks actually ranged quite widely, which was interesting, and reminded me I have a lot af geometry to catch up on.  A lot of talks had to do with structures at various places along the heirarchy: (1) symplectic manifolds, (2) Kähler manifolds, and (3) Calabi-Yau manifolds.  These last are interesting to string theorists and others, in part because they satisfy a form of Einstein’s equations, while also carrying a bunch of extra structure.

Now, at least I know what all the above things are: Symplectic manifolds $(M,\omega)$ have the “symplectic form” $\omega$, a non-degenerate exact 2-form (a canonical example being $\sum dp^i \wedge dq^i$ in the cotangent space to $\mathbb{R}^n$, which happens to be the configuration space for a particle moving in $\mathbb{R}^n$ – symplectic forms often show up on configuration spaces).  A Kähler manifold is symplectic, but also has a complex structure (i.e. a way to multiply tangent vectors by $i$), which preserves the symplectic form, and a metric, which gets along with both of the above.  If the metric satisfies Einstein’s equations and is flat (this really amounts to the connection to “connections”, since this is the same as there being some flat connections, namely the Levi-Civita connection), then $M$ is a Calabi-Yau manifold.

Anyway, this sets up the kind of geometry a lot of people were talking about, and while I didn’t exactly have the background to follow everything, I got a sense of what kinds of questions people are interested in, which was good.  A lot of questions have to do with Lagrangian submanifolds of any of the above (from symplectic through Calabi-Yau).  These are submanifolds where the symplectic form gives zero when applied to any tangent, and which have the highest possible dimension consistent with this property (namely $n$, if the original thing is $2n$-dimensional).  Another theme which came up several times – for example, in the talk by Denis Auroux – has to do with “mirror symmetry” for Kähler manifolds (and Calabi-Yaus), which has to do with finding a “mirror” for the manifold $M$, called $\check{M}$ where the complex geometry on the mirror corresponds to the symplectic geometry on $M$, and vice versa.

There were some talks in the direction of physics.  One of the most obviously physical was Niky Kamran’s, talking about a project he’s worked on with F. Finster, J. Smoller, and S-T. Yau, about long-time dynamics of particles satisfying the Dirac equation, living on a background geometry described by the Kerr metric – which describes a rotating black hole.  Since I worked with Niky on a related project for my M.Sc (my thesis was basically a summary putting together a bunch of results by these same four people), I followed this talk better than many of the others.

Working on this project, I got a strong sense of how important symmetry is in studying a lot of real-world problems.  One of the essential facts about the Kerr metric is that it’s very symmetric: it’s stable in time, and rotationally symmetric.  Actually, all the black-hole solutions to Einstein’s equations are quite symmetric – there is only a small family of solutions, parametrized by mass and angular momentum (and electrical charge).  The symmetry makes differential equations written in terms of this metric much nicer – you can split things into the radial and angular parts, for example – and in particular, the wave equations Niky was talking about are integrable just because of this symmetry, so it’s possible to get exact analytic results.  (Other approaches to this kind of problem get results only numerically and approximately, but can deal with much more general backgrounds.)  The starting point (which basically is what my thesis summarizes) is to show that there are no “bound states” for the Dirac equation.  Fermions (which is what it describes) are most familiar to us in bound states: in shells orbiting the nucleus of an atom.  But if the attractive force pulling on them is gravity, rather than electical charge, this situation isn’t stable.  The work Niky was talking about deals with what happens instead: what are the long-term dynamics of a fermion near a rotating black hole?

They use spectral methods – basically, Fourier analysis – to find out.  The Dirac equation is a wave equation (for a spinor field), and you can look at the different frequencies, and get an estimate of how fast they decay.  (Since there aren’t stable orbits, the strength of the spinor field has to decay over time.)  In fact, they get a sharp estimate of the order (namely $t^{-5/6}$).  Basically, one should imagine that the wave is a superposition of “ripples” – some radiating outward from the event horizon, and some converging toward it.  Put in terms of a particle – an electron, say, or a neutrino – this says it will either fall into the black hole, or (if it has enough energy) escape off to infinity.

There were some other physics-ish talks, such as that by James Sparks, on the geometry of the “AdS/CFT” correspondence.  This correspondence has to do with two kinds of quantum field theory.  The “AdS” stands for “Anti de Sitter”, which is a sort of geometric structure for a manifold which resembles a hyperboloid – actually, all the unit vectors in $\mathbb{R}^6$ where the metric has signature (4,2): that is, the metric is something like $\Delta(1,1,1,1,-1,-1)$.  This hyperboloid is 5-dimensional, and has a metric with one timelike dimension.  Plain old “de Sitter” space is a similar thing, but using a metric with signature (5,1).  It’s possible to define some field theory on AdS space, called supersymmetric supergravity.  This theory turns out to have exactly the same algebra of observables as a different theory, “CFT” or conformal field theory, on the (conformal) boundary of Anti de Sitter space.  Sparks told us about a geometric interpretation of this.

Then there was Sergei Gukov, with a talk called “Brane Quantization”, based on this work with Ed Witten.  He was a little reticent to actually describe how this “brane quantization” actually works, preferring to refer us to that paper, but gave us a very nice, and relatively comprehensible overview of different approaches to quantizing a symplectic manifold.  (As I said, they tend to show up as configuration spaces in classical physics. A basic problem of quantization is how to turn the algebra of functions on a symplectic manifold $(M,\omega)$ into an algebra of operators on a Hilbert space $\mathcal{H}$.)  In particular, he contrasted their method with geometric quantization (which needs to make some arbitrary choices, then takes $\mathcal{H}$ to be a space of sections of some line bundle on $M$ with a connection whose curvature is $\omega$), and with deformation quantization (which needs no special choices, but only constructs an algebra of operators by algebraic deformation, and not actually $\mathcal{H}$ itself, which some people, but not Sergei Gukov, find satisfactory).  The basic idea of Brane quantization seems to be that $M$ gets complexified (somehow – it might be either impossible, or non-unique), and then studying something called an A-model of the result.  This is apparently related to, for example, Gromov-Witten theory, which I’ve written about here recently.

Finally, I’ll mention a few other talks which stood out as rather different from the rest.  Veronique Godin talked about “Relative String Topology” – string topology being a way of studying space by looking at embeddings of the circle (or of paths) into it – that is, its loop space (or path space).  Usually, invariants that come from path spaces only detect the homotopy type of the original spaces – in particular, they’re not helpful as knot invariants.  Godin talked about a clever way to detect more structure by means of an $A_{\infty}$-coalgebra structure on the cohomology groups of the path space.  The “relative” part means one’s looking at a manifold $M$ with embedded submanifold $N$ (for example, $N$ is a knot in $M=\mathbb{R}^3$), and considering only paths starting and ending on $N$.  (This is how one can get a coalgebra structure – turning one path into two paths if it crosses through $N$ again is a comultiplication – this extends to chains in the cohomology).

Chris Brav gave a talk about how braid groups act on derived categories, which I didn’t entirely follow, but subsequently he did explain to me in a pretty comprehensible way what people are trying to accomplish when they look at derived categories.  At some point I’ll have to think about this more carefully and maybe post about it.  But roughly, it’s the same sort of “nice categorical properties” I mentioned in the previous post, about smooth spaces.  Looking at derived categories of sheaves on a space, makes the objects seem more complicated, but it also makes them behave better with respect to taking things like limits and colimits.

Benjamin Young prefaced his talk, “Combinatorics Inspired by Donaldson-Thomas Theory” by pointing out that he’s a combinatorialist, not a geometer.  But Donaldson-Thomas invariants are apparently a kind of “signed count” of some geometric structures (as are a lot of invariants – the same kind of “weighted count” invariants appear in Gromov-Witten and Dijkgraaf-Witten theory, just for instance).  So he described some geometry relating to “brane tilings” – basically, embedding certain kinds of graphis in a torus – and how they give rise to structures that correspond to certain kinds of Young diagrams (“not the same Young”, he added, perhaps unnecessarily, but it got a chuckle anyway).  So the counts can be turned into a combinatorial problem of counting those Young diagrams with the appropriate sign, which can be done using a generating function.

So in any case, this conference had a whole range of talks, from several different fields.  While I found myself lost in a number of talks, I was also quite fascinating with how wide a range of topics were embraced under its umbrella – “connections” indeed!  So in the end this was one of those conferences which opened my eyes to a wider view of the field, which was certainly a good reason to go!