Thursday, December 25, 2008

Emergent spacetime from modular motives

Once again, I believe that the best hep-th article today is the first one. Rolf Schimmrigk wrote about
Emergent spacetime from modular motives (PDF)
Schimmrigk's work may sound as hardcore mathematics - and it arguably is - but don't forget that there are millions of people who are familiar with his name: the readers of "The Elegant Universe" by Brian Greene. ;-)

The term "motive" sounds very mysterious, mathematically obscure, and these papers contain a lot of other jargon of category theory (also known as "general abstract nonsense"), too. However, the goal of all of these fancy things is to actually make things more physical, not less physical. For example, there are all kinds of homologies that are morally equivalent - including de Rham cohomology, Betti cohomology, and Čech cohomology (yes, the last name of Eduard Čech is a Czech name and it means Czech).

While a physicist would simply adopt these notions as being equivalent, mathematicians can't do so. Instead, Alexander Grothendieck had to visualize an enigmatic object, the motive, that unifies all the moral and morally equivalent ideas underlying various kinds of homology. The motive is supposed to unify all essential features of a manifold. You can see the "very big picture", quasi-religious sentiments penetrating all these words that make the situation analogous to M-theory before it was demystified.

After all, "motives" begin with an "M", too, so M-theory could be a M(otive) theory, too. :-)

However, many of these notions in the theory of motives have been given rigorous definitions and exact results have been proven. Schimmrigk is now using the spiritual flavor of motives ;-) to reconstruct spacetime geometry from a conformal field theory. So you should immediately notice that his emergent geometry is linked to perturbative string theory, not necessarily to "all of string theory", and as we know, the geometry that emerges from a particular CFT is not unique (due to mirror symmetry and related dualities).

It turns out that the world sheet modular invariance, despite its being superficially unrelated to any spacetime geometry because of its "internal world sheet character", is actually the key tool to reconstruct a spacetime geometry from an abstract CFT. So far this stuff looks like babbling so get ready for a cultural shock, an example. 

Take a degree-twelve hypersurface in the weighted projective space CP_(2,2,2,3,3). Note that this projective space is complex four-dimensional, so the hypersurface is a complex three-dimensional manifold. Schimmrigk's story associates an Omega-motivic L-function described by the series
LOmega(X312,s) = 1 + 6/13- 150/25+ 94/37- 497/49- 1210/61s + 582/73s + ...
as seen in equation 87. That's a pretty shocking generalization of the Riemann zeta function with seemingly random mutations of all the numerical coefficients. (The denominators jump by 12 so they're not quite random.) ;-) Nevertheless, this function seems to know about the generic K3 fibers of this Calabi-Yau three-fold (the hypersurface) and many other things, too. Besides integers, you may also find "i" and "sqrt(3)" in many formulae.

This branch of mathematics connects number theory and geometry and you can see that it is full of seemingly numerological links between random numbers and homological features of a manifold. There's a lot of theorems in this line of reasoning whose purpose may look unclear at the beginning, but if you master all of it, you will learn the properties of specific integers so intimately that you will become able to reconstruct any kind of higher-dimensional geometry allowed by string theory from these properties. ;-)

In other words, our ideas about the continuous geometries are flooded by a lot of intuitive stuff that obscures the true essence of geometry. If these big ideas are correct and if we look carefully enough, everything boils down to number theory and algebra. While these constructions are perturbative in nature (in terms of the string coupling), it is plausible that they already capture most of the essential processes by which the Calabi-Yau geometry emerges non-perturbatively, too.

Note that monstrous moonshine and its generalizations link number theory and group theory on one side and complex analysis on the other side. All these a priori isolated branches of mathematics are going to unify more tightly than ever before. These links belong among the deepest insights of mathematics and it is very conceivable that a proper understanding of their origin as well as implications will be needed for a truly complete demystification of string theory as a theory of everything - all of natural sciences as well as all of mathematics.

And that's the memo.

Bonus: incompatible arrows of time

As a Christmas gift, Sean Carroll is finally beginning to understand why everything he has written about the arrow of time before December 2008 was complete rubbish: he just understood that one can't have incompatible arrows of time coexisting in the same Universe (and carried by subsystems that interact with each other). See e.g. this article in which your humble correspondent was trying to explain the very same elementary point to Sean Carroll and others exactly a year ago.

Every Universe that at least qualitatively resembles ours inevitably has a universal arrow of time (the thermodynamical arrow of time also inevitably coincides with the decoherence arrow and other logical arrows of time, by similar arguments), and by a conventional definition of the past and the future, we may always guarantee that the entropy is low in the past and high in the future. I am not sure whether Sean Carroll has already gotten *that* far ;-) but he is already reaching the previous paragraph which is encouraging.

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