Cost of seven-brane gauge symmetry in a quadrillion F-theory compactifications

We study seven-branes in $O(10^{15})$ four-dimensional F-theory compactifications where seven-brane moduli must be tuned in order to achieve non-Abelian gauge symmetry. The associated compact spaces $B$ are the set of all smooth weak Fano toric threefolds. By a study of fine-star-regular triangulations of three-dimensional reflexive polytopes, the number of such spaces is estimated to be $5.8\times 10^{14}\lesssim N_{\text{bases}}\lesssim 1.8\times 10^{17}$. Typically hundreds or thousands of moduli must be tuned to achieve symmetry for $h^{11}(B)<10$, but the average number drops sharply into the range $O(25)–O(200)$ as $h^{11}(B)$ increases. For some low-rank groups, such as $SU(2)$ and $SU(3)$, there exist examples where only a few moduli must be tuned in order to achieve seven-brane gauge symmetry.

Figure 1

The logarithm of the average numbers of triangulations $N_T$ per polytope when using different methods for $h_B^{11}\le 10$. The red curve denotes the exact result, i.e. method $A$, whereas the blue curve estimates the number of FRSTs using method $B$.

Figure 2

The logarithm of the average number of triangulations $N_T$ using method $B$. A line fits the data to high accuracy for $7\le h^{11}(B)\le 22$. The gray stars shown on the plot are explained in the description of method $C$.

Figure 3

Shapes of facets.

Figure 4

Costs of unweighted tunings. The shaded area indicates there are no varieties with $h^{11}(B)=32,33$.

Figure 5

Average costs of tunings after weighting by the numbers of FRSTs of the polytopes. The plot is divided into four regions. The labels $A$, $B$, $C$ and $D$ denote the different methods we apply in the corresponding regions to get the numbers of FRSTs of the polytopes in the weighting process. Top: Weighted average costs for all $h^{11}(B)$. Bottom: Focusing on the region $h^{11}(B)\ge 20$.

Figure 6

Average difference between costs of tuning on $D_{\mathrm{vert}}$ and $D_{\mathrm{int}}$.

Figure 7

Ratio between costs of tuning on $D_{\mathrm{vert}}$ and $D_{\mathrm{int}}$. The dashed line in the plot denotes $y=1$ which shows that the costs of tuning on $D_{\mathrm{vert}}$ are always larger than those on $D_{\mathrm{int}}$.