Flux compactifications grow lumps

The simplest flux compactifications are highly symmetric—a $q$-form flux is wrapped uniformly around an extra-dimensional $q$-sphere. In this paper, we investigate solutions that break the internal $\mathrm{SO}(q+1)$ symmetry down to $\mathrm{SO}(q)\times {\mathbb{Z}}_2$; we find a large number of such lumpy solutions, and show that often at least one of them has lower vacuum energy, larger entropy, and is more stable than the symmetric solution. We construct the phase diagram of lumpy solutions, and provide an interpretation in terms of an effective potential. Finally, we provide evidence that the perturbatively stable vacua have a nonperturbative instability to spontaneously sprout lumps. We give an estimate of the decay rate and argue that generically it is exponentially faster than all other known decays.

Figure 1

A cartoon of the phase digram of ellipsoidal solutions for the case ${\mathrm{\Lambda }}_D>0$ and $q\ge 4$. For each value of the conserved flux number $n$, there are two solutions: a symmetric solution, where the internal manifold is a perfect sphere, and a lumpy one, where the internal manifold is either a prolate or an oblate ellipsoid. There are two critical values of $n$: first, $n=n_c$, at which the warped solution crosses through the symmetric one; second, $n=n_{\max }$, at which both solutions disappear spontaneously, indicated by the star. Left: A plot of ellipticity against $n$. Arrows indicate directions of decreasing free energy; they point towards the solution with smaller ${\mathrm{\Lambda }}_{\mathrm{eff}}$ and away from the solution with larger ${\mathrm{\Lambda }}_{\mathrm{eff}}$. Right: A plot of the effective cosmological constant ${\mathrm{\Lambda }}_{\mathrm{eff}}$ against $n$. The dominant solution, the one with smaller ${\mathrm{\Lambda }}_{\mathrm{eff}}$, is shown with a solid line and the subdominant solution is shown with a dashed line; the two solutions switch at $n=n_c$.

Figure 2

A cartoon of the effective potential in the ellipticity direction. We will argue that the effective potential tends to $+\infty$ as the internal manifold becomes increasingly M&M shaped, and to $-\infty$ as the internal manifold becomes increasingly football shaped. The symmetric solution is always an extremum of this effective potential. Whether the warped solution is football or M&M shaped, and whether it is dominant or subdominant is determined solely by the classical stability of the symmetric solution. When the symmetric solution has negative mass squared (left panel), the warped solution has lower free energy and is M&M shaped; when the symmetric solution has positive mass squared (right panel), the warped solution has higher free energy and is football shaped. All these solutions have an instability (either perturbative or nonperturbative) to becoming increasingly football shaped.

Figure 3

The Freund-Rubin solutions. Left: When ${\mathrm{\Lambda }}_D<0$, there is a single solution for each value of $n$, and it is always ${\mathrm{AdS}}_p$ (${\mathrm{\Lambda }}_{\mathrm{eff}}<0$). Increasing $n$ increases ${\mathrm{\Lambda }}_{\mathrm{eff}}$ towards zero from below and causes the flux density $\rho$ to fall towards an asymptote. Right: When ${\mathrm{\Lambda }}_D>0$, there are either two solutions or no solutions, depending on $n$. The small-volume branch is drawn in green; it always has smaller ${\mathrm{\Lambda }}_{\mathrm{eff}}$, and is stable against total-volume perturbations. It is ${\mathrm{AdS}}_p$ for $n<n_{\text{Mink}}$ and ${\mathrm{dS}}_p$ for $n_{\text{Mink}}<n<n_{\max }$. The large-volume branch is drawn in orange; it is unstable to total-volume perturbations and is always ${\mathrm{dS}}_p$. As $n$ is increased past $n_{\max }$, the two branches merge, annihilate, and disappear. As $n\to 0$, the green lines tend to ${\mathrm{\Lambda }}_{\mathrm{eff}}\to -\infty$. The behavior of the solution in this limit is independent of ${\mathrm{\Lambda }}_D$; we refer to this limit as the “nothing state.”

Figure 4

The effective potential for the radion field $R(x)$ using an ansatz that assumes spherical symmetry of the internal dimensions. The extrema of this potential correspond to the Freund-Rubin solutions discussed in the previous section. Left: When ${\mathrm{\Lambda }}_D\le 0$, there is a single minimum for each value of $n$; increasing $n$ causes the solution to move out to larger $R$ and less negative ${\mathrm{\Lambda }}_{\mathrm{eff}}/M_p^p$. Right: When ${\mathrm{\Lambda }}_D>0$, the number of extrema depends on $n$. For small $n$, the effective potential has a minimum and a maximum. The maximum is always de Sitter, and the minimum can be either de Sitter or AdS. Increasing $n$ causes the minimum and maximum to merge and annihilate.

Figure 5

Equation (19) gives the critical value of ${\rho }^2{R}^2$ at which the $\ell$th spherical harmonic develops a negative mass squared. The larger $\ell$ is, the larger the value of ${\rho }^2{R}^2$ at which the mode becomes negative. When ${\mathrm{\Lambda }}_D=0$, there is a fluctuation with $\ell =q-1$ that is exactly massless, and there are fluctuations with $\ell =2$ through $\ell =q-2$ that have negative mass squareds. When ${\mathrm{\Lambda }}_D>0$, increasing $R$ tends to push more modes to positive mass squareds. When ${\mathrm{\Lambda }}_D<0$, increasing $R$ tends to push more modes to negative mass squareds; in fact, as $R\to \infty$, there are negative mass squared modes for all values of $\ell$.

Figure 6

Four sample solutions to Eqs. (22)–(24), all with $p=q=4$. The top solution, labeled “symmetric” [${\mathrm{\Lambda }}_D/M_D^D=1$, $Q=446$ and $M_D\mathrm{\Phi }(\pi /2)=5.10$] is of the Freund-Rubin type, where $R$ and $\mathrm{\Phi }$ are constant. The next two solutions, labeled “football” [${\mathrm{\Lambda }}_D/M_D^D=1$, $Q=34.4$ and $M_D\mathrm{\Phi }(\pi /2)=3.36$] and “M&M” [${\mathrm{\Lambda }}_D/M_D^D=1$, $Q=58.6$ and $M_D\mathrm{\Phi }(\pi /2)=2.55$], both have a single extremum between $\theta =0$ and $\theta =\pi$. These solutions are ellipsoidal in shape and related to the $\ell =2$ instability. The last solution, labeled “Higher $\ell$” [${\mathrm{\Lambda }}_D/M_D^D=-1$, $Q=5.83$ and $M_D\mathrm{\Phi }(\pi /2)=1.37$] has three extrema between $\theta =0$ and $\theta =\pi$. There can be many solutions with the same value of $n$; the top three solutions all come from the same theory, and have $n=.91n_{\max }$. The symmetric solution lies on the small-volume Freund-Rubin branch of Fig. 8, the M&M lies on the small-volume lumpy branch, and the football lies on the large-volume lumpy branch. The third column gives the internal geometry as an embedding space: when the plotted curve is rotated around the $X_1$ axis, the induced metric on the resulting surface is equal to the metric of the lumpy internal manifold. Football solutions are prolate and $R$ reaches a minimum at the equator ($\theta =\pi /2$); M&M solutions are oblate and $R$ reaches a maximum at the equator. The field $\mathrm{\Phi }$ always has the opposite behavior, so that the flux density always gets larger where $R$ gets smaller, and vice versa.

Figure 7

The order parameter $ϵ$, which measures the lumpiness of the solution, is defined by the length of the blue curve that runs along a line of longitude divided by the length of the red curve that runs around the equator. M&M-shaped solutions have $ϵ<1$; football-shaped solutions have $ϵ>1$; Freund-Rubin solutions have $ϵ=1$ by definition.

Figure 8

A phase diagram of the ellipsoidal solutions for the case $p=q=4$ and ${\mathrm{\Lambda }}_D>0$. Symmetric solutions have $ϵ=1$; for $n<n_{\max }$ there are two symmetric solutions—a small-volume solution and a large-volume solution—and for $n>n_{\max }$ there are no symmetric solutions. We find that each symmetric solution is accompanied by an ellipsoidal solution with $ϵ\ne 1$, which has roughly the same internal volume. Whenever the ellipsoidal solution has $ϵ>1$, it also has $\mathrm{\Delta }({\mathrm{\Lambda }}_{\mathrm{eff}}/M_p^p)\equiv ({\mathrm{\Lambda }}_{\mathrm{eff}}/M_p^p{)}_{\text{lumpy}}-({\mathrm{\Lambda }}_{\mathrm{eff}}/M_p^p{)}_{\text{symmetric}}>0$ and vice versa. To produce this phase diagram, we numerically found lumpy solutions at a large number of values of $Q$, and then numerically integrated each solution to find $n$ and ${\mathrm{\Lambda }}_{\mathrm{eff}}$. (The region with very small $Q$ is hard to access numerically; this region corresponds to the large-volume ellipsoidal solutions with small $n$. We were not able to find solutions in this region, and so we have drawn a straight dashed line that continues the trend.)

Figure 9

The phase diagram of ellipsoidal solutions. In all cases, solutions have $ϵ<1$ if and only if $\mathrm{\Delta }({\mathrm{\Lambda }}_{\mathrm{eff}}/M_p^p)<0$ and they have $ϵ>1$ if and only if $\mathrm{\Delta }({\mathrm{\Lambda }}_{\mathrm{eff}}/M_p^p)>0$. When ${\mathrm{\Lambda }}_D<0$, increasing $n$ makes the solution more and more M&M shaped; when ${\mathrm{\Lambda }}_D=0$ the additional scaling symmetry means that $ϵ$ stays constant with $n$ and only the total internal volume changes; and when ${\mathrm{\Lambda }}_D>0$, there are two ellipsoidal solutions that come together, merge, and annihilate as $n$ is increased. The case $q=3$ is special because there is no ellipsoidal solution when ${\mathrm{\Lambda }}_D=0$; this is related to the fact that the $\ell =2$ “danger mode” is precisely massless. For all higher $q$, the ${\mathrm{\Lambda }}_D=0$ ellipsoidal solution is M&M shaped. As $n\to 0$, all three cases merge to a single solution whose volume and shape is independent of ${\mathrm{\Lambda }}_D$. To make this figure, we numerically found solutions for a large number of values of $Q$ with $q=3$, 4 and 5, and with ${\mathrm{\Lambda }}_D<0$, ${\mathrm{\Lambda }}_D=0$ and ${\mathrm{\Lambda }}_D>0$. The specific plots shown here are for $p=4$ and $q=3$ on the left and $p=4$ and $q=4$ on the right. We conjecture that this qualitative behavior continues for $q\ge 4$. (As with Fig. 8, numerics prevented us from finding solutions with very small $Q$ and ${\mathrm{\Lambda }}_D>0$, and so as before we have drawn a straight dashed line that continues the trend. Also, in the ${\mathrm{\Lambda }}_D>0$ case, the symmetric branch ends spontaneously at the same $n=n_{\max }$ as the lumpy branch.)

Figure 10

A branch of lumpy solutions associated with $\ell =4$ deformations of the Freund-Rubin solutions; the plots of $R(\theta )$ and $\mathrm{\Phi }(\theta )$ for these solutions have three extrema. The geometry of the internal manifold is shown in embedding coordinates: rotating the surface around the $X_1$ axis gives a surface with an induced metric that matches the lumpy internal metric. These solutions are for the case $p=q=4$ and ${\mathrm{\Lambda }}_D<0$. This case has a critical value of $n=n_{c,\ell =4}$ at which the $\ell =4$ danger mode develops a negative mass squared. When $n<n_{c,\ell =4}$, the solutions we find always are energetically subdominant and diamond shaped; when $n>n_{c,\ell =4}$, the solutions are energetically favored and box shaped. This type of transition is qualitatively similar to that observed for the ellipsoids of the previous section.