Musings

Numerical Calabi-Yau Metrics

A while back, I wrote about the work of Headrick and Wiseman on numerical Ricci-flat metrics on K3. The big limitation in extending their work to complex dimension-3 (or to less-symmetrical K3s) was simply a matter of storage.They needed to store the values of the Kähler potential for each point on the grid which, while doable in real dimension-4, was prohibitive in real dimension-6. Though their numerics were very efficient, their calculations were highly storage-limited.

I bumped into Matt today, and he told me about a recent paper by Simon Donaldson which seems to alleviate the storage problem. The trick is that, if you expect the Kähler potential to be fairly smooth, you are storing a lot of redundant information by recording its value at each point of the grid. Instead, you can get a good approximation with much lower storage requirements by storing its coefficients in a basis of “harmonic functions.”

Donaldson works with a projective embedding of the manifold, and expands in a basis of harmonic functions on the ambient projective space. $$e^K = \sum_{n=1}^{\infty}\frac{h_{A\overline{B}} p^A(z)p^{\overline{B}}(\overline{z})}{\Vert z\Vert^{2n}}e^{K_0}$$ Here $z_a$ are homogeneous coordinates of the ambient projective space (sections of the hyperplane bundle), $p^A$ are a basis of homogeneous polynomials of degree $n$ in the $z_a$, $K_0$ is the fiducial Kähler potential (perhaps the one induced from the Fubini-Study metric of the ambient projective space) and $\Vert z\Vert^2= g^{a\overline{b}}z_a\overline{z}_b$ for some suitable positive-definite matrix, $g^{a\overline{b}}$. Instead of storing the values of $e^K$ at each grid point, we store the values of the constants, $h_{A\overline{B}}$. For reasonably slowly-varying functions, this is vastly more efficient.

Toby and Matt are working on combining Donaldson’s proposal for efficiently storing the Kähler potential, with their own algorithm for solving the Monge-Ampère equation. Provided that they don’t sacrifice too much speed in translating back and forth from Donaldson’s nonlocal variables, $h_{A\overline{B}}$, this should be a big step forward.