One-Loop Kahler Metric of D-Branes at Angles

TL;DR

This work computes one-loop corrections to the Kahler metric of D-brane moduli in toroidal orientifolds with branes at angles, separating the ${\mathcal{N}}=1$ (intersection) and ${\mathcal{N}}=2$ (parallel) sectors. Using carefully constructed complex vertex operators adapted to brane angles and a detailed analysis of annulus and Möbius-strip amplitudes, the authors show that all ${\mathcal{N}}=1$ contributions cancel, a result not anticipated from field-theory nonrenormalization theorems. By contrast, ${\mathcal{N}}=2$ corrections persist in the parallel sector and are captured by explicit moduli-dependent logarithms and lattice sums. The findings indicate a string-level nonrenormalization in minimally supersymmetric settings and motivate further study of higher-genus effects, magnetized branes, and potential phenomenological implications for D-brane inflation and flavor physics, while ensuring tadpole cancellation remains a guiding consistency condition.

Abstract

We evaluate string one-loop contributions to the Kahler metric of D-brane moduli (positions and Wilson lines), in toroidal orientifolds with branes at angles. Contributions due to bulk states in the loop are known, so we focus on the contributions due to states localized at intersections of orientifold images. We show that these quantum corrections vanish. This does not follow from the usual nonrenormalization theorems of supersymmetric field theory.

Figures (11)

• Figure 1: Tilted coordinates for a brane with wrapping number $(n,m)=(2,1)$. The complex coordinates $Z_{\theta}^i$ and $\Psi_{\theta}^i$ also include the length $L_i$ and perpendicular distance $D_i$.
• Figure 2: Covering space. Since $|n+mU|=|n+m\bar{U}|$, dividing $n+mU$ by $n+m\bar{U}$ gives just the angle $e^{2i\theta}$. In this example $(n,m)=(2,1)$.
• Figure 3: As the physical cylinder worldsheet only extends between $0$ and $\pi$, the quasiperiodicity of our extended fields on the covering torus lies in the "unphysical" region (in the sense of the method of images). Notice that the branes at $\sigma=0$ and $\sigma=\pi$ may be at angles, (i.e. $\theta_0\neq\theta_{\pi}$) but this is not drawn in the figure.
• Figure 4: Integration region for $\nu_{\mathcal{A}}$.
• Figure 5: A plot of $\int_0^x \tilde{R}_{\delta}(\tilde{\nu}) \cot (\pi \tilde{\nu}) d \tilde{\nu}$ for $x=0..1$, $\ell=0.4$, $v=1/3$, $\delta=[0.01,0.02,0.04,0.08, 0.1]$. For $x=1$, the integral yields zero for all nonzero values of $\delta$.