标题： 具有增强对称性的通量景观不在$SL(2, \mathbb{Z})$椭圆点上 显示英文标题

标题： Flux Landscape with Enhanced Symmetry Not on $SL(2, \mathbb{Z})$ Elliptic Points

摘要： 我们研究了在两个环面orientifold上SUSY闵可夫斯基F项方程的解结构，其中包含$h^{2, 1} = 1$。 在我们之前的研究\cite{Ishiguro:2020tmo}基础上，固定通量D3-膜电荷的上限$N_{\rm flux}$，我们得到了每种情况下的整个景观以及物理上不同的解的退化分布。 与我们之前的研究不同，我们考虑了一个非可分解的环面orientifold及其景观，在该景观中$SL(2, \mathbb{Z})$被破坏为某个同余子群，这在以往的研究中已经有所了解。 我们发现，复结构模量$U$并不是整个对偶群所享有，而是其与一个“缩放”外自同构群的外半直积。 基本区域被扩大以包括$|U| < 1$区域。 此外，我们发现高退化性出现在椭圆点上，不是$SL(2, Z)$的椭圆点，而是外自同构群的椭圆点。 此外，$\mathbb{Z}_2$-增强的对称性在椭圆点上实现。 外自同构群是例外的，因为它与背景三循环的辛基变换一致，而不是$SL(2, \mathbb{Z})$的外自同构群。 我们还将这一结果与其他可分解环形定向物的景观进行了比较。 显示英文摘要

摘要： We study structures of solutions for SUSY Minkowski F-term equations on two toroidal orientifolds with $h^{2, 1} = 1$. Following our previous study \cite{Ishiguro:2020tmo}, with fixed upper bounds of a flux D3-brane charge $N_{\rm flux}$, we obtain a whole Landscape and a distribution of degeneracies of physically-distinct solutions for each case. In contrast to our previous study, we consider a non-factorizable toroidal orientifold and its Landscape on which $SL(2, \mathbb{Z})$ is violated into a certain congruence subgroup, as it had been known in past studies. We find that it is not the entire duality group that a complex-structure modulus $U$ enjoys but its outer semi-direct product with a "scaling" outer automorphism group. The fundamental region is enlarged to include the $|U| < 1$ region. In addition, we find that high degeneracy is observed at an elliptic point, not of $SL(2, Z)$ but of the outer automorphism group. Furthermore, $\mathbb{Z}_2$-enhanced symmetry is realized on the elliptic point. The outer automorphism group is exceptional in the sense that it is consistent with a symplectic basis transformation of background three-cycles, as opposed to the outer automorphism group of $SL(2, \mathbb{Z})$. We also compare this result with Landscape of another factorizable toroidal orientifold.