Musings

The starting point is the observation that, on a Kähler manifold, the nonzero components of the Ricci tensor are $$R_{i\overline{\jmath}} = - \partial_i \partial_{\overline{\jmath}} \log det g_{k\overline{l}}$$ On a general Kähler manifold, the Ricci form, $R = i R_{i\overline{\jmath}} dz^i\wedge d\overline{z}^j$, is closed, but not necessarily exact (since $\log det g_{k\overline{l}}$ is only locally defined). On one with trivial canonical class, however, the Ricci form is exact. Choose some fiducial Kähler metric, $\tilde{g}_{i\overline{\jmath}}$, in the desired Kähler class. It won’t be Ricci-flat, but its Ricci form will satisfy $$\tilde{R}_{i\overline{\jmath}} = - \partial_i \partial_{\overline{\jmath}} \tilde{F}$$ for some globally defined function, $\tilde{F}$. Moreover, it differs from the desired metric by $$g_{i\overline{\jmath}} = \tilde{g}_{i\overline{\jmath}} + \partial_i \partial_{\overline{\jmath}} \phi$$ for some globally-defined function $\phi$.

The Ricci flatness equation then reads

where $\lambda$ is a constant which relates the coordinate volume to the metric volume of $g$. More invariantly, $$J^n = (-1)^n n! \lambda \Omega\wedge \overline{\Omega}$$ where $J= i g_{i\overline{\jmath}} dz^i\wedge d\overline{z}^j$ is the Kähler form and $\Omega$ is the holomorphic $n$-form. On the manifolds Toby and Matt work with, the Jacobian of the holomorphic change of coordinates between patches is 1, and they can take $\Omega=dz^1\wedge\dots dz^n$ on each patch.

Anyway, (1) has the form of a Monge-Ampère equation for the unknowns $\phi,\lambda$. This was the equation for which Yau proved existence. It’s pretty nonlinear-looking, but heck, it’s just a PDE in just one variable, which is a lot better than the Einstein equation we started with.

Moreover, if $\tilde{g}_{i\overline{\jmath}}$ is close to being Ricci-flat, then we can expand the determinant and the leading term is just a Poisson equation. So similar methods can be used to solve it. They use Gauss-Seidel iteration, and find solutions for the metric of a family of Kummer surfaces in a couple of days running.

The Kummer surface is a particular class of K3 surfaces, realized as $T^4/\mathbb{Z}_2$, with the 16 fixed-point of the $\mathbb{Z}_2$ action blown up. The neighbourhood of each erstwhile fixed point looks like the total space of $\mathcal{O}(-2)\to \mathbb{P}^1$, and can be covered with two patches, whose coordinates are related by $$(y',w')= (1/y, y^2 w)$$ The “torus patch” has coordinates $(z_1, z_2)$, with the identification $(z_1,z_2)\sim (-z_1,-z_2)$. On the overlap with the patch replacing the fixed point at the origin, $$(z_1,z_2) = \pm \sqrt{2w} (y,1) = \pm \sqrt{2 w'} (1,y')$$ (and similarly for the other 15 fixed points).

So, you see, the setup is pretty concrete, and while they made some simplifications (a cubical torus, equal blowup parameters for the 16 fixed points), grinding out a solution was well within their grasp.

They’ve made available some pretty movies which allow the suitably imaginative to “visualize” the shape of a Ricci-flat Kummer surface.

So, what can we do with their results (aside from admiring the movies)?

Well, for one thing, they can calculate more. They can compute the spectrum of various Laplacians on K3. Thus, we can read off the spectrum of conformal weights of primary fields in the K3 conformal field theory.

Time will tell, but I’m gonna wager that interesting physics of these conformal field theories can be extracted by combining their numerical results with exact results from special points (like the orbifold limit, $T^4/\mathbb{Z}_2$) along a family of K3s.

Moving up to complex dimension 3, I’m a little more pessimistic. The Ricci-flat metric isn’t the exact metric of the $\sigma$-model, because of the 4-loop (and beyond) $\beta$-function. Previously, I’ve always waved this off; since there was no prayer of determining the Ricci-flat metric, who cared if the exact metric differed slightly from it? The best we were ever going to do was extract RG-invariant aspects of the physics, those that were insensitive to the precise form of the metric.

Now, perhaps, we need to get a bit more serious.