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Mapping Calabi-Yau threefolds by the degenerated representatives

The chemistry Nobel prize goes to the de facto invention of lithium-ion batteries. It's not a terribly huge advance in pure science but as applied science, it's been literally game-changing, of course. Although Czechia has a top European reservoir of lithium, I do hope that we will switch to lithium-free batteries at some point.



The first hep-th paper today is
Classifying Calabi-Yau threefolds using infinite distance limits
by Grimm and two co-authors (NL/CH/UK) that elaborates on a nice and clever way to map the landscape of the Calabi-Yau threefolds. Their excitement is seen on the fact that they worked hard enough to post the preprint at the top – it was posted 3 seconds after the new arXiv.org day started.



They look at special points in the landscape of these 6-real-dimensional manifolds that may be very useful for "navigation" in that landscape – at the degeneration limits.



The paper has 41 pages and contains lots of technicalities that at most "dozens of people in the world" will understand at the research level so that they may write followup papers. But let me say some general comments.

First, I find it extremely natural that they use the "infinite volume limits". We did want to map the extreme directions of "tori" in M-theory or type II string theory – click at the picture at the top – and Grimm et al. also look at the extreme points in the Kähler moduli space that are infinitely far from the "totally smooth and generic" manifolds.



Just to be clear, they do much more than just "scaling the Calabi-Yau threefolds to a large volume". They scale different Kähler moduli differently, like we did – there are many directions in the moduli space in which you may go, and their classification is really the point of the paper.

They find out that when you go to some extreme boundaries of the Kähler and complex structure moduli spaces, the third cohomology gets split into the Hodge parts (3-0 and 2-1) in different ways that depend on the direction. This allows them to determine the 3-vs-3 intersection numbers from some new starting point, from certain diagrams. And lots of figures connect the different topologies – through conifold transitions and del-Pezzo-six transitions, among many other things.

From some viewpoints, you could say that they solve just some non-fundamental mathematical problem – the space of Calabi-Yau threefolds is a complicated solution to some conditions and the result is whatever it is. However, it is also very possible that a proper understanding of the solution to this problem – a good way to map the shapes of the compact Ricci-flat manifolds – is equivalent to a fundamental law of physics.

Why?

Because we know from dualities that these spaces have many interpretations. For example, the moduli space of M-theory or type II string theory on K3 surfaces – basically the space of shapes of these curved K3 surfaces, with some fluxes and decorations or without them – is equivalent to the moduli space of heterotic strings on tori (the relationship is known as the string-string duality). And there are other dualities of this form.

So the shapes of K3 – and similarly the Calabi-Yau threefolds – may look like "composite objects created by gluing pieces of space" that are not fundamental (and that's surely how naive physicists such as those in loop quantum gravity want to look at the space; the most naive ones want to construct the space out of a discrete LEGO) – but string theory indicates that the whole shapes are also some manifestations of some underlying non-local, non-geometric rules and diagrams.

In other words, the normal picture is that the construction of a Calabi-Yau threefold is a complicated task involving the gluing of regions of space so that the whole manifold is Ricci-flat etc. It's complicated and that's why the landscape of the allowed shape is rather complicated, too. But the point is that there could be very different ways to construct this whole space. You start with some "beacons" which symbolize the degenerated Calabi-Yaus or something like that, some special locations or valleys in the landscape, and you find some graph-theory or similar rules how these beacons may be connected. And the actual geometries are just some "details that live on top of the combinatorial problem" and that are guaranteed to exist – perhaps some solutions to a minimization problem.

It's rather clear that at least partially, it must be possible to describe the space of Calabi-Yau threefolds in this way. The manifolds are the extrema of the Einstein-Hilbert action and the action depends on the configuration space of all manifolds in a way that may be simplified a little bit. Its topology may be described by some diagrams.

All these clever methods to think about these matters make it sort of ambiguous whether the spacetime, the world sheet, the AdS boundary, the configuration space of fields, or some other space is more fundamental. Physicists are increasingly dealing with all of them "on equal footing". And the general rules of quantum gravity may be rules that apply to all these spaces simultaneously. This feeling of mine is still largely just an emotion, perhaps a wishful thinking, but one that is supported by some partial evidence.

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