Moduli stabilisation and the statistics of axion physics in the landscape

String theory realisations of the QCD axion are often said to belong to the anthropic window where the decay constant is around the GUT scale and the initial misalignment angle has to be tuned close to zero. In this paper we revisit this statement by studying the statistics of axion physics in the string landscape. We take moduli stabilisation properly into account since the stabilisation of the saxions is crucial to determine the physical properties of the corresponding axionic partners. We focus on the model-independent case of closed string axions in type IIB flux compactifications and find that their decay constants and mass spectrum feature a logarithmic, instead of a power-law, distribution. In the regime where the effective field theory is under control, most of these closed string axions are ultra-light axion-like particles, while axions associated to blow-up modes can naturally play the role of the QCD axion. Hence, the number of type IIB flux vacua with a closed string QCD axion with an intermediate scale decay constant and a natural value of the misalignment angle is only logarithmically suppressed. In a recent paper we found that this correlates also with a logarithmic distribution of the supersymmetry breaking scale, providing the intriguing indication that most, if not all, of the phenomenologically interesting quantities in the string landscape might feature a logarithmic distribution.

Misaligned supersymmetry and open strings

The study of non-supersymmetric string theories is shedding light on an important corner of the string landscape and might ultimately explain why, so far, we did not observe supersymmetry in our universe. We review how misaligned supersymmetry in closed-string theories leads to a cancellation between bosons and fermions even in non-supersymmetric string theories. We then show that the same cancellation takes place for open strings by studying an anti-Dp-brane placed on top of an Op-plane in type II string theory. Misaligned supersymmetry consists in cancellations between bosons and fermions at different energy levels, in such a way that the averaged number of states grows at a rate dominated by a factor $$ {\mathrm{e}}^{C_{\mathrm{e}\mathrm{ff}}\sqrt{n}} $$ e C eff n , with Ceff< Ctot, where Ctot is the inverse Hagedorn temperature. We prove the previously conjectured complete cancellation, i.e. we prove that Ceff = 0, for a vast class of models.

Landscape of modular symmetric flavor models

We study the moduli stabilization from the viewpoint of modular flavor symmetries. We systematically analyze stabilized moduli values in possible configurations of flux compactifications, investigating probabilities of moduli values and showing which moduli values are favorable from our moduli stabilization. Then, we examine their implications on modular symmetric flavor models. It is found that distributions of complex structure modulus τ determining the flavor structure are clustered at a fixed point with the residual ℤ3 symmetry in the SL(2, ℤ) fundamental region. Also, they are clustered at other specific points such as intersecting points between |τ|2 = k/2 and Re τ = 0,±1/4,±1/2, although their probabilities are less than the ℤ3 fixed point. In general, CP-breaking vacua in the complex structure modulus are statistically disfavored in the string landscape. Among CP-breaking vacua, the values Re τ = ±1/4 are most favorable in particular when the axio-dilaton S is stabilized at Re S = ±1/4. That shows a strong correlation between CP phases originated from string moduli.

Calabi-Yau Spaces in the String Landscape

Calabi-Yau spaces, or Kähler spaces admitting zero Ricci curvature, have played a pivotal role in theoretical physics and pure mathematics for the last half century. In physics, they constituted the first and natural solution to compactification of superstring theory to our 4-dimensional universe, primarily due to one of their equivalent definitions being the admittance of covariantly constant spinors. Since the mid-1980s, physicists and mathematicians have joined forces in creating explicit examples of Calabi-Yau spaces, compiling databases of formidable size, including the complete intersecion (CICY) data set, the weighted hypersurfaces data set, the elliptic-fibration data set, the Kreuzer-Skarke toric hypersurface data set, generalized CICYs, etc., totaling at least on the order of 1010 manifolds. These all contribute to the vast string landscape, the multitude of possible vacuum solutions to string compactification. More recently, this collaboration has been enriched by computer science and data science, the former in bench-marking the complexity of the algorithms in computing geometric quantities, and the latter in applying techniques such as machine learning in extracting unexpected information. These endeavours, inspired by the physics of the string landscape, have rendered the investigation of Calabi-Yau spaces one of the most exciting and interdisciplinary fields.

A counterexample to the Nelson-Seiberg theorem

We present a counterexample to the Nelson-Seiberg theorem and its extensions. The model has 4 chiral fields, including one R-charge 2 field and no R-charge 0 filed. Giving generic values of coefficients in the renormalizable superpotential, there is a supersymmetric vacuum with one complex dimensional degeneracy. The superpotential equals zero and the R-symmetry is broken everywhere on the degenerated vacuum. The existence of such a vacuum disagrees with both the original Nelson-Seiberg theorem and its extensions, and can be viewed as the consequence of a non-generic R-charge assignment. Such counterexamples may introduce error to the field counting method for surveying the string landscape, and are worth further investigations.