Flux Landscape with enhanced symmetry not on SL(2, ℤ) elliptic points

A bstract We study structures of solutions for SUSY Minkowski F-term equations on two toroidal orientifolds with h 2 , 1 = 1. Following our previous study 1, with fixed upper bounds of a flux D3-brane charge N 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 , 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, 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 , ℤ). We also compare this result with Landscape of another factorizable toroidal orientifold.