Field dynamics and tunneling in a flux landscape

We investigate field dynamics and tunneling between metastable minima in a landscape of type IIB flux compactifications, utilizing monodromies of the complex structure moduli space to continuously connect flux vacua. After describing the generic features of a flux-induced potential for the complex structure and type IIB axiodilaton, we specialize to the mirror quintic Calabi-Yau to obtain an example landscape. Studying the cosmological dynamics of the complex structure moduli, we find that the potential generically does not support slow-roll inflation and that in general the landscape separates neatly into basins of attraction of the various minima. We then discuss tunneling, with the inclusion of gravitational effects, in many-dimensional field spaces. A set of constraints on the form of the Euclidean paths through field space are presented, and then applied to construct approximate instantons mediating the transition between de Sitter vacua in the flux landscape. We find that these instantons are generically thick wall and that the tunneling rate is suppressed in the large-volume limit. We also consider examples where supersymmetry is not broken by fluxes, in which case near–Bogomolnyi-Prasad-Sommerfeld thin-wall bubbles can be constructed. We calculate the bubble-wall tension, finding that it scales like a D- or NS-brane bubble, and comment on the implications of this correspondence. Finally, we present a brief discussion of eternal inflation in the flux landscape.

Figure 1

Here, we depict one of the monodromies of a two-dimensional torus. On the top left, the torus can be represented by a region of the complex plane with opposite edges of the rectangle identified. Twisting the torus by an angle of $2\pi$ (bottom) is equivalent to taking $U\to U+1$ (top). The result is a torus that is identical in shape, but with transformed cycles (bottom right).

Figure 2

The complex structure moduli space of the mirror quintic. The plot shows two of the three singular points: the large complex structure point $z=0$ and the conifold point $z=1$. The two branch cuts correspond to the large complex structure and the conifold monodromies.

Figure 3

The potential for the mirror quintic complex structure modulus is multivalued due to the three-cycle monodromies. The panels show two minima connected by a conifold monodromy. The upper minimum $V_F$ has flux configuration $F=[-2,-6,-9,-1]$; $H=[-1,0,-7,0]$ and the lower minimum $V_T$ has the flux configuration $F=[-3,-6,-9,-1]$; $H=[-1,0,-7,0]$. The polar coordinates used in the right panel are $R^2=z_I^2+(z_R-1{)}^2$ and $\tan (\Phi )=z_I/(z_R-1)$.

Figure 4

The metric $K_{z\overline{z}}$ on the complex $z$-plane is almost rotationally symmetric around the LCS point. The metric diverges near the LCS point $z=0$ and near $z=1$ where $K_{z\overline{z}}\sim \ln |z-1|$.

Figure 5

Here, we show the potential corresponding to a flux configuration in the upper minimum (min 1, indicated by the filled circle to the right) given by $F=[3,-4,-1,-2]$; $H=[1,0,5,0]$ (example 1 in Table ). The lower minimum, min 2, is denoted by the filled circle on the left. Some sample trajectories are shown as solid lines, with initial positions denoted by small filled circles (the initial velocity is zero in these examples). The numerical evolution cannot be tracked into the near-conifold region, where the trajectory on the bottom right disappears. The potential splits into three basins of attraction, whose boundaries are denoted by the dashed lines. Zero-velocity initial conditions in basin 1 will yield trajectories that end in min 1; trajectories that begin in basin 2 will end in min 2; and trajectories that begin in basin 3 will fall to the next sheet down, possibly reaching a minimum should one exist.

Figure 6

A potential for a field $\varphi$ with two minima and one maximum.

Figure 7

Two minima with flux configurations $F=[0,-3,5,6]$; $H=[1,1,1,0]$ and $F=[-6,-3,5,6]$; $H=[1,1,1,0]$ (top) and $F=[3,-4,-1,-2]$; $H=[1,0,5,0]$ and $F=[1,-4,-1,-2]$; $H=[1,0,5,0]$ (bottom) connected by an instanton. On the far left, a contour plot of the potential in the polar representation is shown. The next cell depicts the potential obtained by sampling along the dashed line in the contour plot as a function of the canonical field $x$. The endpoints of the instanton connecting the two minima are denoted by the dots. On the far right, the instanton solution $x(\chi )$, $r(\chi )$ is displayed.

Figure 8

Two zero-energy minima of the flux-induced potential for the mirror quintic. Although the minima on the two sheets are connected by a smooth potential there is no tunneling between them.