Thursday, September 12, 2019

Moore & ladies: high-Hodge vacua are less numerically dominant than thought

There are many interesting new hep-th papers today. The first author of the first paper is Heisenberg, Kallosh and Linde have a post-Planck update on the CMB – a pattern claimed to be compatible with the KKLT, there are 17 new papers omitting cross-listings, but I choose the second paper:
Flux vacua: A voluminous recount
If we overlook the title that tries to please Al Gore if not Hillary Clinton (too late), Miranda Cheng, Greg Moore, and Natalie Paquette (Amsterdam-Rutgers-Caltech) work to avoid an approximation that is often involved while counting the flux vacua – you know, the computations that yield the numbers such as the insanely overcited number of 10500.

In particular, the previous neglected "geometric factor" is \[

\frac{1}{\pi^{m/2}}\int \det ({\mathcal R} + \omega\cdot 1)

\] OK, something like a one-loop measure factor in path integrals. This factor influences some density of the vacua. Does the factor matter?

They decide that sometimes it does, sometimes it does not. More curiously, they find out that this factor tends to be an interestingly behaved function of the topological invariants. It's intensely decreasing towards a minimum, as you increase some topological numbers, and then it starts to increase again.

Curiously enough, the minimum is pretty much reached for the values of the topological numbers that are exactly expected to be dominant in the string compactifications. In this sense, the critical dimensions and typical invariants in string theory conspire to produce the lowest number of vacua that is mathematically possible, at least when this geometric factor is what we look at.

This is a "hope" I have been explicitly articulating many times – that if you actually count the vacua really properly, or perhaps with some probability weighting that has to be there to calculate which of them could have arisen at the beginning, the "most special or simplest" vacua could end up dominating.

They're not quite there but they have some substantial factor that reduces – but not sufficiently reduces – the number of vacua for very large Hodge numbers i.e. in this sense "complicated topologies of the compactification manifolds". I mean large Hodge numbers. Note that large Hodge numbers (which may become comparable to 1,000 for Calabi-Yau threefolds) are really needed to get high estimates of the number of vacua such as that 10500. You need many cycles and many types of fluxes to obtain the high degeneracies.

Wouldn't it be prettier if the Occam-style vacua with the lowest Hodge numbers were the contenders to become the string vacuum describing the world around us? There could still be a single viable candidate. I have believed that the counting itself is insufficient and the Hartle-Hawking-like wave function gives a probabilistic weighting that could pick the simplest one. They have some evidence that previously neglected effects could actually suppress the very number of the vacua with the large Hodge numbers or other signs of "contrivedness".

Clearly, everyone whose world view finely depends on claims about "the large number of string vacua" has the moral duty to study the paper and perhaps to try to go beyond it.

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