The 24-Cell and Calabi-Yau Threefolds with Hodge Numbers (1,1)He constructs the "simplest" six-dimensional Calabi-Yau manifold we know.
Click to zoom in.
This diagram shows all the known Calabi-Yau spaces with small Hodge numbers, h^{1,1}+h^{1,2} < 25. The colors indicate how those spaces were constructed. Note that both of these numbers have to be positive. The world's catalog of known Calabi-Yau spaces indicates that h^{1,1}+h^{1,2} < 503 for all Calabi-Yau shapes.
The left-right symmetry of the graph above is nothing else than the mirror symmetry: it exchanges h^{1,1} and h^{1,2} which means that it preserves their sum (y coordinate on the graph) but changes the sign of their difference (x coordinate).
The new entries, showed as purple disks or pieces of disks on my version of the graph above, were constructed by Braun as free quotients of the 24-cell hypersurface; this hypersurface is a "Platonic 24-hedron" in 4 spatial dimensions, analogous to the regular Platonic polyhedra in 3 dimensions. (There are six such polytopes in 4 dimensions.) Its boundary is composed out of 24 mundane octahedra.
The 24-cell polytope. Click to see Wikipedia.
If you look, the minimum entry with the values h^{1,1} = h^{1,2} = 1 was missing so far. Now the hole is filled. Note that the anthropic people wouldn't be interested in such manifolds because they're too constrained; there are not too many cycles that may carry a large number of fluxes. The misanthropic people such as myself think that these "minimal" surfaces are the most important ones - and arguably the most relevant ones physically and cosmologically - because they are very constrained. ;-)
The singer, Jonathan Mann, wants to understand string theory and maths. He wants to hang up with Edward Witten, too. :-) Thanks to Sarah Kavassalis.
