Revisiting arithmetic solutions to the $W=0$ condition

Abstract

The gravitino mass is expected not to be much smaller than the Planck scale for a large fraction of vacua in flux compactifications. There is no continuous parameter to tune even by hand, and it seems that the gravitino mass can be small only as a result of accidental cancellation among period integrals weighted by integer-valued flux quanta. DeWolfe et.al. (2005) proposed to pay close attention to vacua where the Hodge decomposition is possible within a number field, so that the precise cancellation takes place as a result of algebra. We focus on a subclass of those vacua---those with complex multiplications---and explore more on the idea in this article. It turns out, in Type IIB compactifications, that those vacua admit non-trivial supersymmetric flux configurations if and only if the reflex field of the Weil intermediate Jacobian is isomorphic to the quadratic imaginary field generated by the axidilaton vacuum expectation value. We also found that flux statistics is highly enriched on such vacua, as F-term conditions become linearly dependent.

28+25 pages, a few references added and a few typos corrected

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Cited by in corpus (7)

• String-theory Realization of Modular Forms for Elliptic Curves with Complex Multiplication
• A counterexample to the Nelson-Seiberg theorem
• Purely non-perturbative AdS vacua and the swampland
• W=0 Complex Structure Moduli Stabilization on CM-type K3 x K3 Orbifolds:---Arithmetic, Geometry and Particle Physics---
• A formal notion of genericity and term-by-term vanishing superpotentials at supersymmetric vacua from R-symmetric Wess-Zumino models
• Novel counterexample to the Nelson-Seiberg theorem
• A note on varieties of weak CM-type