We study conformal field theories for strings propagating on compact, seven-dimensional manifolds with G 2 holonomy. In particular, we describe the construction of rational examples of such models. We argue that analogues of Gepner models are to be constructed based not on N = 1 minimal models, but on Z 2 orbifolds of N = 2 models. In Z 2 orbifolds of Gepner models times a circle, it turns out ...

Let f(x) = x+ax+bx+cx ∈ Z[x] and consider the hypersurface of degree five given by the equation Vf : f(p) + f(q) = f(r) + f(s). Under the assumption b 6= 0 we show that there exists Q-unirational elliptic surface contained in Vf . If b = 0, a < 0 and −a 6≡ 2, 18, 34 (mod 48) then there exists Q-rational surface contained in Vf . Moreover, we prove that for each f of degree five there exists Q(i...

In this paper, we compute the minimum discriminants of imprimitive degree 10 fields for different combinations of Galois group and signature. We use class field theory when there is a quintic subfield, and a Martinet search in the more difficult case where there is only a quadratic subfield. If L is a degree n number field, let DL be its discriminant, and dL := |DL| its absolute discriminant. L...