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Type IIB string theory with the cosmological constant \(10^{-144}\)

A Cornell-Northeastern-MIT quintuplet (Demirtas et al.) released two new hep-th papers (cross-listed in hep-ph)

A Cosmological Constant That is Too Small (short)
Small Cosmological Constants in String Theory (long)
Demirtas, Kim, McAllister, Moritz, and Rios-Tascon claim to solve "one of the layers of the cosmological constant problem" as we have carefully distinguished it for over two decades.

There are still some doubts whether the basic modern cosmological model is right – one in which the acceleration of the expansion is due to dark energy (70% of energy density in the Universe) which behaves like the cosmological constant or strictly is the cosmological constant. We will assume that the Universe has a positive cosmological constant, which has been the professionals' (I mean competent top-down theorists') preferred belief since the telescope observations of the late 1990s.

In the 4D Planck units, the apprent cosmological constant is tiny, like \[ \Lambda \sim + 10^{-123} \] The inverse square root of this number approximately gives the linear size of the Universe which is \(10^{60}\) in the Planck units (that is a radius). The value of \(\Lambda\) above has some awkward, not so hidden, features. First, it is positive. Second, it is really tiny. Its being positive means (as we know from other sources, meaning particle physics) that our Universe cannot be unbroken supersymmetric. The tiny value indicates a naturalness-like problem: it seems extremely unlikely, like \(Prob\sim 10^{-123}\), that a positive dimensionless number ends up being as small as \(10^{-123}\).

In reality, the absence of very light superpartners means that supersymmetry is broken much more than needed to generate this tiny cosmological constant. The most straightforward order-of-magnitude estimates of the cosmological constant as a function of the observed SUSY breaking (meaning the observed superpartner masses) would produce \(\Lambda\) that is much closer to one than \(10^{-123}\) is, and consequently a much smaller Universe than \(10^{60}\) in the Planck units.

Various approximations (like non-gravitational or effective quantum field theory) and various "strategies to think and estimate" within them, like the naturalness argumentation, imply (to one level of belief or another) that it should be impossible to derive \(10^{-123}\) as the "only right" value of the cosmological constant. Many people have adopted the multiverse/landscape view that there are googols of vacua in string theory (the number is at least roughly the inverse cosmological constant), some of them end up having a tiny positive constant like ours by a coincidence, and those are overrepresented in the "research derived from observations" because life and observers are much more likely to arise in such universes (and they may be totally banned elsewhere). The more you emphasize that something like humans may be born in a world like ours and it should determine which of the vacua are likely as an arena for cosmology, the more "anthropic" you are. Clearly, many of the "hardcore anthropic" believers end up saying very wrong and stupid things, even basic things about the probability calculus.

But this whole scheme may be completely incorrect and the new papers we discuss returned to a previous stage of the research, namely to an intermediate question
Can string theory refuse to respect the simplified probability estimates (mostly based on quantum field theory and naturalness) and rather naturally produce a tiny numerical value of the cosmological constant that looked unlikely?
And their answer is Yes, it can. It is far from even a semi-realistic vacuum because the spectrum is unrealistic, some supersymmetry is unbroken, and correspondingly, the sign of the cosmological constant is negative, not positive. Many readers (and us) understand numbers so let me show you the value of the cosmological constant that they have calculated for a supersymmetric AdS background in type IIB string theory:\[ V_0 = -3M_{pl}^2 e^{\mathcal K} |W|^2 \approx -1.68\times 10^{-144} M_{pl}^4 \] The first equality is a standard supergravity formula for the vacuum energy. To get a small number, you might be tempted to use the exponential but you wouldn't get very far. Instead, they succumb to the fact that the exponential of the Kähler potential is "pretty much" comparable to one and they end up finding the configurations with a tiny superpotential \(W\) instead.
The trick behind the smallness is that the perturbative part of \(W\) cancels completely – which is much more possible than in field theory because the terms are determined by integers – and the small part comes from non-perturbative terms, here the world sheet instantons.
To get the cosmological constant around \(10^{-144}\), they need world sheet instantons coming from a rather large (but not implausibly large) compactification manifold. Its volume in the string units turns out to be around 945. You are surely interested in the precise geometry. It's a threefold hypersurface \(X\) in a toric variety \(V\) constructed out of a reflexive polytope \(\Delta\) and you need a \(8\times 4\) matrix of vertices, you can determine the topological invariants, calculate \(g_s\sim 0.01,\) and you need to add fluxes and orientifold planes, to make it yummy. The work and expertise needed to settle this higher-dimensional geometry correctly clearly look much higher than the "creative" usage of this geometry for their clever phenomenological scenario – but the latter activity, while less intellectually demanding, may be more revolutionary and groundbreaking.

But this seems to be a controllable, supersymmetric, compactification with a cosmological constant whose absolute value is as "shockingly" tiny as the observed one, or a bit smaller. Clearly, they may find a whole class of such tiny-cosmological-constant vacua (but the number of elements in the class is much lower than the googols you need in the statistical, anthropic strategy to establish their existence).

The supersymmetric and \(AdS_4\) character of their vacuum may be considered a disadvantage because it's clearly not ready to be used directly for our Universe. From a conceptual perspective, when the goal is to make a solid step in the understanding of the cosmological constant problem, you may find the unbroken supersymmetry a great advantage because it makes the whole vacuum controllable and calculable. The results seem much more reliable than any results involving supersymmetry-breaking vacua.

So again, they negate or circumvent the "probabilistic estimates" of the typical magnitude of the cosmological constant.

They obtain a tiny (negative) vacuum energy because the superpotential is tiny. And the superpotential is tiny because the perturbative terms exactly cancel – which is possible because many quantities are discrete, determined by integers such as the topological invariants of the compactification manifold; and because the volume of the manifold is naturally "rather large" (of order one thousand), much like the largest Hodge numbers (which are several hundreds), and this "relatively large size" of the manifold, correlated with the complex topology ("several hundreds" is naturally the highest possible Hodge number \(h^{1,1}\) of Calabi-Yau threefolds that may mathematiucally exist) is enough to make the world sheet instantons "exponentially small" and that is what ultimately produces the tiny superpotential and therefore a tiny cosmological constant.

They cannot make a "directly analogous" construction of a SUSY-breaking de Sitter vacuum right now. No one can really reliably calculate de Sitter (or positive cosmological-constant) vacua in string theory at this moment, with no further refinements needed in this sentence, and claims about dS vacua almost always encode the authors' prejudices (some of which seem stupid). But these authors believe that a similar general Ansatz or strategy might lead to tiny positive cosmological constants and perhaps a realistic dS vacuum, too: perturbative terms exactly cancel which is not "unnatural in string theory" because it only amounts to some cancellation of (a priori) integers; and the observed tiny value therefore comes from non-perturbative (and therefore exponentially small) corrections.

Because SUSY seems broken intensely in our Universe, I think that their general strategy includes the implicit statement that string theory produces some discreteness-based cancellation of the vacuum energy whose strength is much stronger than the cancellations derived purely from supersymmetry. This ability would amount to another "string miracle" but one may argue that qualitatively similar string miracles are already known. The real punch line is a more general one: Top-down theorists as a community should avoid premature conclusions such as "the anthropic scenario is made unavoidable by the tiny value of the vacuum energy" because qualitatively different scenarios seem possible, at least if we look at slightly different classes of vacua than those that we need for the world around us.

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