Loop Fayet-Iliopoulos terms in $T^2/Z_2$ models: Instability and moduli stabilization

We study Fayet-Iliopoulos (FI) terms of six-dimensional supersymmetric Abelian gauge theory compactified on a $T^2/Z_2$ orbifold. Such orbifold compactifications can lead to localized FI-terms and instability of bulk zero modes. We study 1-loop correction to FI-terms in more general geometry than the previous works. We find induced FI-terms depend on the complex structure of the compact space. We also find the complex structure of the torus can be stabilized at a specific value corresponding to a self-consistent supersymmetric minimum of the potential by such 1-loop corrections, which is applicable to the modulus stabilization.

Figure 1

The loop diagram that generates the FI-term.

Figure 2

The configuration of brane modes.

Figure 3

Torus of $\tau =e^{i\pi /3}$.

Figure 4

A regularization of ${\delta }^{(2)}(x,y)$.