D6-branes and torsion

TL;DR

The paper analyzes D6-branes in four-dimensional type IIA flux vacua on twisted six-tori, showing that brane charges, moduli, and the discrete D-brane spectrum are governed by torsion in cohomology, ${\rm Tor}\, H^n({\cal M}, \mathbb{Z})$ and ${\rm Tor}\, H_1(\Pi_3, \mathbb{Z})$. By employing a Maurer–Cartan framework with integer twistings ${\omega}^k_{ij}$, it demonstrates that torsion factors such as ${\rm Tor}\, H^n({\tilde{T}}^6, \mathbb{Z})$ arise and modify RR tadpole conditions and D6-brane charges, including the Freed-Witten anomaly when D6-branes wrap non-closed chains. Open-string moduli can be lifted by torsion, giving a D6-brane discretum with discrete Wilson lines valued in ${\mathbb{Z}}_{M_i}$ and potentially light lifted modes; the masses scale with the geometric flux quanta and the large complex structure limit can render these modes lighter than KK scales. The analysis also clarifies how torsion constrains chirality in flux vacua: torsional charges tend to yield non-chiral sectors, while realistic chiral models require branes carrying non-torsion charges, a conclusion reinforced by a type IIB–type IIA dictionary via twisted-torus duals.

Abstract

The D6-brane spectrum of type IIA vacua based on twisted tori and RR background fluxes is analyzed. In particular, we compute the torsion factors of the (co)homology groups H_n and describe the effect that they have on D6-brane physics. For instance, the fact that H_3 contains Z_N subgroups explains why RR tadpole conditions are affected by geometric fluxes. In addition, the presence of torsional (co)homology shows why some D6-brane moduli are lifted, and it suggests how the D-brane discretum appears in type IIA flux compactifications. Finally, we give a clear, geometrical understanding of the Freed-Witten anomaly in the present type IIA setup, and discuss its consequences for the construction of semi-realistic flux vacua.