String-scale gauge coupling relations in the supersymmetric Pati-Salam models from intersecting D6-branes

We have constructed all the three-family 𝒩 = 1 supersymmetric Pati-Salam models from intersecting D6-branes, and obtained 33 independent models in total. But how to realize the string-scale gauge coupling relations in these models is a big challenge. We first discuss how to decouple the exotic particles in these models. In addition, we consider the adjoint chiral mulitplets for SU(4) C and SU(2) L gauge symmetries, the Standard Model (SM) vector-like particles from D6-brane intersections, as well as the vector-like particles from the 𝒩 = 2 subsector. We show that the gauge coupling relations at string scale can be achieved via two-loop renormalization group equation running for all these supersymmetric Pati-Salam models. Therefore, we propose a concrete way to obtain the string-scale gauge coupling relations for the generic intersecting D-brane models.