Flux Vacua, Asymptotic Periods, and Relatively Anosov Representations

Source abstract

We develop a topological method to detect classical flux vacua of M-theory on Calabi-Yau fourfolds with a single complex structure modulus using only asymptotic period data and monodromy information. To this end, we express the index of the F-term of the flux superpotential as a winding number along the boundary of the universal cover of the complex structure moduli space. The index evaluates to a sum over zeros and poles of a suitably normalized boundary F-term, where local contributions can be computed algebraically on a dense set of boundary points. Assuming a conjectured bound on the remaining contributions, sufficiently many sign changes of the boundary F-term guarantee a flux vacuum in the interior. We illustrate the method on the mirror Fermat sextic with explicit examples and a systematic flux scan. A main technical ingredient in our construction is a continuity result for projective period maps: we prove that the non-holomorphic (3,1)-period map extends continuously to the boundary of the covering space if the monodromy representation is relatively Anosov. In general, this property does not hold for holomorphic (4,0)-periods.