# Revisiting arithmetic solutions to the $W=0$ condition

## TL;DR

This paper investigates special flux vacua in string theory where algebraic structures enable precise cancellations, revealing conditions for supersymmetric flux configurations and enriched flux statistics in Type IIB compactifications.

## Contribution

It identifies conditions under which complex multiplication vacua admit supersymmetric fluxes and explores the algebraic nature of these solutions in flux compactifications.

## Key findings

- Supersymmetric flux configurations occur iff the reflex field matches the axidilaton field.
- Flux statistics are significantly enriched in these special vacua.
- Linearly dependent F-term conditions characterize these vacua.

## 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.