Conformal Field Theory Couplings for Intersecting D-branes on Orientifolds

TL;DR

This work applies open-string conformal field theory to compute exact four-point and three-point correlation functions of bosonic twist fields arising at D-brane intersections on toroidal orientifolds, incorporating both quantum fluctuations and disk worldsheet instantons. The authors develop a detailed method to separate the amplitude into a quantum piece $Z_{qu}(x)$ and a classical lattice sum, enabling a complete expression for the Yukawa couplings of chiral fermions to Higgs fields on $T^6$ and the corresponding four-point couplings. They derive canonical forms valid for general wrapping numbers and complex structures, fix overall normalization via unitarity, and extend the analysis to multiple intersections, two independent angles, and orientifold/orbifold projections. The resulting Yukawa couplings are expressed as products over the three $T^2$ factors with exponential disk-instanton contributions depending on triangle areas $A_j(m)$, providing a direct bridge between worldsheet dynamics and low-energy effective couplings. This framework offers a principled, calculable route to phenomenologically relevant couplings in intersecting D-brane models and can be extended to more intricate orientifold/orbifold setups.

Abstract

We present a conformal field theory calculation of four-point and three-point correlation functions for the bosonic twist fields arising at the intersections of D-branes wrapping (supersymmetric) homology cycles of Type II orientifold compactifications. Both the quantum contribution from local excitations at the intersections and the world-sheet disk instanton corrections are computed. As a consequence we obtain the complete expression for the Yukawa couplings of chiral fermions with the Higgs fields. The four-point correlation functions in turn lead to the determination of the four-point couplings in the effective theory, and may be of phenomenological interest.

Figures (5)

• Figure 1: Target space: the intersection of two parallel branes separated by respective distances $d_1$ and $d_2$ and intersecting at angles $\pi \nu$ (Figure 1a). World-sheet: a disk diagram of the four twist fields located at $x_{1,2,3,4}$ (Figure 1b). The calculation involves a map from the world-sheet to target space.
• Figure 2: Target space: the intersection of two branes intersecting respectively with the two parallel branes at angles $\pi \nu$ and $\pi \lambda$, respectively (Figure 2a). World-sheet: a disk diagram of the four twist fields located at $x_{1,2,3,4}$ (Figure 2b).The calculation involves a map from the world-sheet to target space, allowing for a factorization on three-point function.
• Figure 3: World sheet contours. The contours $C_1$ and $C_2$ (Figure 3a) are the two topologically inequivalent contours leading to two independent conditions. The contours in Figure 3b define the global monodromy conditions used in Section III.
• Figure 4: The fundamental domain for a brane with wrapping numbers (3,1) (solid line) and a brane with wrapping numbers (1,2) (broken line). There are five intersection points labelled in increasing order starting from 0 along the solid brane. Starting from 0 and moving along the second brane (broken line) one meets first fixed point 2. This is the integer k that generates the automorphism (\ref{['auto']}) in this example.
• Figure 5: The two configurations for a twist-antitwist pair at intersection i and a twist-antitwist pair at intersection j. Both configurations must be included in the string amplitude.