Cost of seven-brane gauge symmetry in a quadrillion F-theory compactifications
Cost of seven-brane gauge symmetry in a quadrillion F-theory compactifications
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万亿次 F 理论紧致化中七膜规范对称性的成本
DOI：
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发表时间：
2016
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通讯作者：
Jiahua Tian
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作者：
James Halverson;Jiahua Tian
我们研究了$O(10^{15})$四维f理论紧化中的七膜，其中为了实现非阿贝尔规范对称，必须对七膜模进行调谐。所关联的紧空间$B$是所有光滑弱范诺环三倍的集合。通过对三维自反多面体的精细星形正则三角剖分的研究，估计这类空间的数量为$5.8\乘以10^{14}\lesssim N_\text{bases}\lesssim 1.8\乘以10^{17}$。通常，为了实现$h^{11}(B)<10$的对称性，必须调整数百或数千个模，但随着$h^{11}(B)$的增加，平均值急剧下降到$O(25)$-$O(200)$的范围。对于一些低秩群，例如$SU(2)$和$SU(3)$，存在只需调整几个模就能达到七膜规范对称的例子。
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.