A Calculable Toy Model of the Landscape

TL;DR

The paper constructs a concrete, calculable field-theoretic analogue of the string landscape using multiple U(1) gauge factors, FI terms, and a Wilson-line superpotential to generate vast numbers of vacua with varied SUSY and symmetry properties. It analyzes a two-U(1) toy model in depth, showing an energy-dependent vacuum structure split into non-overlapping pie-slice regions, and demonstrates how RG flow can rotate boundaries and cause boundary crossings, potentially triggering phase transitions in the early universe. It then generalizes to U(1)^n with deconstructed higher-dimensional flux interpretations, addresses mixed anomalies via axionic Green-Schwarz mechanisms, and explores the role of soft masses and supergravity in stabilizing vacua and enabling cosmological-constant cancellations. A universal IR fixed point, $\overline{Y}=\sqrt{10/3}$, indicates low-energy physics can become insensitive to UV landscape details, while the framework provides a tractable, scalable platform for studying vacuum statistics, phase structure, and cosmological implications in landscape scenarios.

Abstract

Motivated by recent discussions of the string-theory landscape, we propose field-theoretic realizations of models with large numbers of vacua. These models contain multiple U(1) gauge groups, and can be interpreted as deconstructed versions of higher-dimensional gauge theory models with fluxes in the compact space. We find that the vacuum structure of these models is very rich, defined by parameter-space regions with different classes of stable vacua separated by boundaries. This allows us to explicitly calculate physical quantities such as the supersymmetry-breaking scale, the presence or absence of R-symmetries, and probabilities of stable versus unstable vacua. Furthermore, we find that this landscape picture evolves with energy, allowing vacua to undergo phase transitions as they cross the boundaries between different regions in the landscape. We also demonstrate that supergravity effects are crucial in order to stabilize most of these vacua, and in order to allow the possibility of cancelling the cosmological constant.

Figures (9)

• Figure 1: The vacuum structure of the Fayet-Iliopoulos (FI) model as a function of the FI coefficient $\xi$. The different classes of vacua outlined in the text (and here denoted {$\emptyset$}, {$+$}, and {$-$} respectively) occupy non-overlapping regions along the $\xi$-axis. In the regions corresponding to {$+$} and {$-$} vacua, we have indicated the existence of an unstable {$\emptyset$} extremum in parenthesis. We have also shaded the regions corresponding to vacua with at least one non-zero vacuum expectation value, corresponding to vacua exhibiting $R$-symmetry breaking.
• Figure 3: The "landscape" of our toy model with two $U(1)$ gauge fields and three chiral superfields, sketched for $\lambda=1$. The different classes of vacua (labelled according to which chiral fields receive non-zero vacuum expectation values) occupy the non-overlapping regions indicated above in bold type, while unstable extrema in each region are indicated in smaller type. We have shaded the regions corresponding to vacua with two non-zero vacuum expectation values (Classes {12} and {23}); these are regions in which $R$-symmetry is potentially broken.
• Figure 4: (a) Left figure: Same landscape as in Fig. \ref{['fig1']}, now sketched for $\lambda=1/2$. The {1}, {2}, and {3} regions have each become smaller, remaining equal in size, while the {12} and {23} regions (as well as the new {13} region) have expanded to fill in the gaps. Just as for $\lambda=1$, the different classes of vacua occupy non-overlapping regions in this landscape, and we have indicated the unstable extrema in parentheses. (b) Right figure: As $\lambda\to 0$, the {1}, {2}, and {3} regions disappear entirely, leaving a landscape entirely dominated by non-overlapping ${\bf \{{12}\}}$, {23}, and {13} regions. These regions ultimately represent flat directions in the $\lambda=0$ limit.
• Figure 5: Same landscape as in previous figures, now sketched for $\lambda=4/3$. Regions {1}, {2}, and {3} have each become larger, with a new overlap region (indicated with double cross-hatching) between the {1} and {3} regions. Class {13} extrema exist but are now unstable in this overlap region, while stable Class {12} and {23} vacua continue to populate the gaps between Regions {1} and {2}, and Regions {2} and {3}, respectively. There continue to be no overlaps amongst Regions {1}/ {12}/ {2} or Regions {2}/ {23}/ {3}.
• Figure 7: The probability that a given extremum in our toy model is stable, plotted as a function of the Yukawa coupling $\lambda$ and integrated over all $(\xi_1,\xi_2)$. This function increases monotonically from $7/24$ when $\lambda=0$, and asymptotes to $1/2$ as $\lambda\to\infty$. Note that in all cases, this probability is significantly greater than we would have naı vely expected based on a random assignment of signs for the eigenvalues of the six-dimensional mass matrix.