On the Explicit Construction and Statistics of Calabi-Yau Flux Vacua

TL;DR

This work constructs explicit flux vacua for type IIB string theory on a two-moduli Calabi–Yau orientifold and tests the Ashok–Douglas density against Monte Carlo vacuum statistics. By solving the F-term conditions for the dilaton and complex-structure moduli in terms of flux quanta, and using the periods of the Calabi–Yau, the authors compare the vacuum distribution to the density $\det(-\mathcal{R}-\omega)$, under a tadpole bound $N_{\rm flux}\le L$. They examine regions near the Landau–Ginzburg point and near the conifold locus, finding good agreement with the density, a clustering of vacua near the conifold, and scaling of the vacuum count with flux distance $L$ that improves with larger flux ranges. The results imply a statistical preference for small D3-brane gauge groups and indicate that while large SUSY breaking scales are common, small $|W|$ vacua exist but are not generic, highlighting both the power and limitations of current statistical approaches to the string landscape.

Abstract

We explicitly construct and study the statistics of flux vacua for type IIB string theory on an orientifold of the Calabi-Yau hypersurface $P^4_{[1,1,2,2,6]}$, parametrised by two relevant complex structure moduli. We solve for these moduli and the dilaton field in terms of the set of integers defining the 3-form fluxes and examine the distribution of vacua. We compare our numerical results with the predictions of the Ashok-Douglas density $\det (-R - ω)$, finding good overall agreement in different regions of moduli space. The number of vacua are found to scale with the distance in flux space. Vacua cluster in the region close to the conifold singularity. Large supersymmetry breaking is more generic but supersymmetric and hierarchical supersymmetry breaking vacua can also be obtained. In particular, the small superpotentials and large dilaton VEVs needed to obtain de Sitter space in a controllable approximation are possible but not generic. We argue that in a general flux compactification, the rank of the gauge group coming from D3 branes could be statistically preferred to be very small.

Figures (9)

• Figure 1: Number of vacua with $\vert \psi \vert < r$. The value of $N$ plotted includes a weighting due to the $SL(2,\mathbb{Z})$ copies of each vacuum within the range of fluxes sampled. The dots represent the numerical results, and the continuous line the (rescaled) numerical integration of $\int_{\vert \psi \vert < r} d \mu$, where $d \mu$ is the index density. The flux range used was (-20,20).
• Figure 2: The weighted number of vacua, $N$, with $\vert \phi \vert < r$. The dots represent the numerical results and the continuous line the numerical integration of $\int_{\vert \phi \vert < r} d \mu$, rescaled by the same factor as in diagram \ref{['psismallresults']}.
• Figure 3: The weighted number of vacua $N$ with $N_{flux} < L$. $4.5 \times 10^6$ sets of flxues were generated, with values of $L$ equally distributed between $0$ and $800$ and the range of fluxes being $(-20,20)$. The results are fit by $N \sim L^{4.3}$.
• Figure 4: The value of $(2 \pi )^4 e^K |W|^2$ in units of $(\alpha')^2$ for vacua in the vicinity of $\psi = \phi = 0$, for $(2 \pi)^4 e^K |W|^2 < 100000$. The flux range was (-20, 20), and the vacua satisfied the conditions (\ref{['solnconds']}).
• Figure 5: The value of $Z$ for vacua near the conifold. We have restricted to $|Z| < 0.0001$ and have rescaled the above plot by $10^4$. The flux range used was (-40,40).