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March 21, 2006

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= n=1 h AB¯p A(z)p B¯(z¯)z 2ne K 0 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 az_a are homogeneous coordinates of the ambient projective space (sections of the hyperplane bundle), p Ap^A are a basis of homogeneous polynomials of degree nn in the z az_a, K 0K_0 is the fiducial Kähler potential (perhaps the one induced from the Fubini-Study metric of the ambient projective space) and z 2=g ab¯z az¯ b\Vert z\Vert^2= g^{a\overline{b}}z_a\overline{z}_b for some suitable positive-definite matrix, g ab¯g^{a\overline{b}}. Instead of storing the values of e Ke^K at each grid point, we store the values of the constants, h AB¯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 AB¯h_{A\overline{B}}, this should be a big step forward.

Posted by distler at March 21, 2006 6:49 PM

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Re: Numerical Calabi-Yau Metrics

The problem of building Calabi-Yau metrics on threefolds (6 real dimensions), has been recently solved in the work by Douglas, Karp, Lukic and Reinbacher, building on the Donaldson’s ideas. As an example, explicit Ricci-Flat metrics on Quintic threefolds where built.

The introduction of an original Monte-Carlo method to perform integrals on algebraic varieties combined with the Donaldson’s algorithm, makes this method the easiest to implement, fastest to run and most economical (since the point of view of data storage).

Furthermore, a natural extension of these beautiful geometrical algorithms allows one to solve the hermitian Yang-Mills equations. Thus, using this technology one can write down the explicit 4 dimensional effective action coming from Calabi-Yau compactifications of string theory.

Posted by: Sergio Lukic on March 26, 2007 10:33 PM | Permalink | Reply to this

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