Freed-Witten anomaly in general flux compactification

Abstract

Turning on a Neveu-Schwarz\char21{}Neveu-Schwarz (NS-NS) three-form flux in a compact space drives some D-branes to be either Freed-Witten anomalous or unstable to decay into fluxes by the appearance of instantonic branes. By applying $T$-duality on a toroidal compactification, the NS flux is transformed into metric fluxes. We propose a $T$-dual version of the Atiyah-Hirzebruch Spectral Sequence upon which we describe the Freed-Witten anomaly and the brane-flux transition driven by NS and metric fluxes in a twisted torus. The required conditions to cancel the anomaly and the appearance of new instantonic branes are also described. In addition, we give an example in which all D6-branes wrapping Freed-Witten anomaly-free three-cycles in ${\stackrel{\texttildelow{}}{T}}^{6}/{\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ are nevertheless unstable to decay into fluxes. Even more we find a topological transformation between Ramond-Ramond, NS-NS, and metric fluxes driven by a chain of instantonic branes.