Abstract
In this paper, we explicitly construct the smooth compact base threefold for the elliptic Calabi-Yau fourfold with the largest known h1,1 = 303 148. It is generated by blowing up a smooth toric “seed” base threefold with (E8, E8, E8) collisions. The 4d F-theory compactification model over it has the largest geometric gauge group, {E}_8^{2561}times {F}_4^{7576}times {G}_2^{20168}times mathrm{SU}{(2)}^{30200}, and the largest number of axions, 181 820, in the known 4d mathcal{N} = 1 supergravity landscape. We also prove that there are at least 110015 048≈ 7.5 × 1045 766 different flip and flop phases of this base threefold. Moreover, we find that many other base threefolds with large h1,1 in the 4d F-theory landscape can be constructed in a similar way as well.
Highlights
In this paper, we explicitly construct the smooth compact base threefold for the elliptic Calabi-Yau fourfold with the largest known h1,1 = 303 148. It is generated by blowing up a smooth toric “seed” base threefold with (E8, E8, E8) collisions
We find that many other base threefolds with large h1,1 in the 4d F-theory landscape can be constructed in a similar way as well
If we allow bases that only support non-flat fibration over complex curves, the lower bound of such toric threefolds was proved to be 43 × 2.96 × 10755 in [21], and further estimated to be ∼ 103,000 in [22]
Summary
On Btoric, there are still a number of non-toric (4, 6)-curves located on E8 divisors, which needs to be blown up They are analogous to the (4, 6)-points on the (−9)/(−10)/(−11) curves, in the elliptic Calabi-Yau threefold cases [7, 8]. Note that the locations of these non-toric curves depend on the coefficients of f and g, which correspond to the complex structure moduli of the elliptic Calabi-Yau fourfold. After these non-toric blown ups, we get the non-toric base threefold B3 with h1,1(B3) = h1,1(Btoric) + 619 = 181 819. It is possible to construct a smooth threefold base that supports a flat fibration X4
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