Chiral fermions and anomaly cancellation on orbifolds with Wilson lines and flux

We consider six-dimensional supergravity compactified on orbifolds with Wilson lines and bulk flux. Torus Wilson lines are decomposed into Wilson lines around the orbifold fixed points, and twisted boundary conditions of matter fields are related to fractional localized flux. Both, orbifold singularities and flux lead to chiral fermions in four dimensions. We show that in addition to the standard bulk and fixed point anomalies the Green-Schwarz term also cancels the four-dimensional anomaly induced by the flux background. The two axions contained in the antisymmetric tensor field both contribute to the cancellation of the four-dimensional anomaly and the generation of a vector boson mass via the Stueckelberg mechanism. An orthogonal linear combination of the axions remains massless and couples to the gauge field in the standard way. Furthermore, we construct convenient expressions for the wave functions of the zero modes and relate their multiplicity and behavior at the fixed points to the bulk flux quanta and the Wilson lines.

Figure 1

A torus lattice ${\mathbb{Z}}^2$ and its fundamental domain (gray); the canonical basis of 1-cycles ${\mathcal{T}}_{1,2}$ is depicted by dashed arrows.

Figure 2

The fundamental domain of $T^2/{\mathbb{Z}}_2$; the black dots denote the orbifold fixed points. The canonical basis of the orbifold 1-cycles is presented in (a). (b) shows the projection of the torus 1-cycles ${\mathcal{T}}_{1,2}$, and (c) illustrates the decompositions ${\mathcal{T}}_1={\mathcal{C}}_3+{\mathcal{C}}_4$ and ${\mathcal{T}}_2={\mathcal{C}}_1+{\mathcal{C}}_3$.

Figure 3

The (absolute square of the) $P$-even, constant wave function for $N=0$ (a), and the linear independent wave functions for $N=2$ and $k_m=0$, (b) to (d). The scale is chosen such that the wave functions integrate to $1/2$ on the orbifold.

Figure 4

The (absolute square of the) two possible wave functions for $N=2$ and nontrivial Wilson line configurations, classified by $k_m$ are depicted. Empty circles denote fixed points at which the wave function vanishes. $j=0(1)$ is shown on the left(right).