Musings

Let $X$ be a simply-connected Calabi-Yau manifold. We’re going to consider compactifications of string theory on $X$ down to 4 dimensions. Let $\mathcal{M}$ be the space or Riemannian metrics on $X$ (modulo diffeomorphisms). Let $\mathcal{C}$ be the group of Weyl rescalings, $g\to e^{\psi} g$, of the metric on $X$. Let $\mathcal{C}_1(g)$ be the group of Weyl rescalings which preserve the total volume of $X$. We always have an isomorphism $\mathcal{C}_1\sim \mathcal{C}/\mathbb{R}$, but — of course — the precise realization of $\mathcal{C}_1$ depends on the metric, $g$. Anyway, HHM seem to want to claim a sort of uniformization theorem

the space of constant scalar curvature metrics on $X$. In two dimensions, this space is finite-dimensional; here, of course, it is still infinite dimensional. This space contains (and this is HHM’s big point) metrics of both positive an negative scalar curvature, separated by a codimension-1 hypersuface of metrics of vanishing scalar curvature. But, say HHM, these latter metrics all have non-vanishing Ricci curvature.

The Ricci-flat metrics are found on some finite-dimensional hypersurface in the interior of the region of negative scalar curvature.

The first question one might ask at this point is: Why the quotient by $\mathcal{C}_1$? (Or, equivalently, why the restriction to $\mathcal{M}_c$?) In string theory, Weyl-deformations of the metric on $X$ are physical. But several of their arguments rely heavily on the restriction to constant scalar curvature metrics. Maybe this is just a technicality.

Now, say the volume of $X$ is large compared to the string scale. Kaluza-Klein excitations of the 10D supergravity fields on $X$ are much lighter than stringy excitations. So the 4D effective description (if we ignore, for the moment, the fields other than the metric) involves an infinite-component $\sigma$-model whose target space is $\mathcal{M}$. This $\sigma$-model comes equipped with a potential which (again, ignoring $\alpha '$ corrections) which is just

This thing has a local minimum at the Ricci-flat metrics, but can be arbitrarily negative for large, positive scalar curvature.

Now, infinite-component field theories are rather nasty, and do all kinds of things which violate our usual intuitions. But HHM say “don’t worry, be happy!” and focus on just one “mode” which parametrizes some path from the Ricci-flat metric to the region of positive scalar curvature.

We can then build configurations where the scalar curvature of $X$ is positive over some large region in 4D space, and then returns to the Ricci-flat metric at infinity. These configurations (which they claim can be stable) can have some ominous implications, as HHM discuss.

But

1. There’s really no approximation in which one can truncate to a finite number of 4D scalar fields. Once you turn on this “mode” by an appreciable amount, they all couple in.
2. Once you get any distance away from the Ricci-flat locus, even though the scalar curvature may remain small, the other components of the Riemann tensor may become arbitrarily large, vitiating the approximation that we are going to ignore $\alpha '$ corrections.
3. Indeed, once you go away from the Ricci-flat locus, the dilaton develops a tadpole, making (if I got the sign right) quantum corrections important.

I’d be quite interested to see if these configurations can be constructed as solutions to the 10D supergravity equations. (One could the check whether the supergravity approximation was, indeed, valid.) But the “4D effective theory” arguments seem pretty dubious to me.