The four-fermi coupling of the supersymmetric non-linear sigma-model on G/S\otimesU(1)^k

TL;DR

This paper develops a geometric and algebraic framework to evaluate four-fermi couplings in supersymmetric non-linear sigma-models on reducible K"ahler coset spaces, with explicit results for models related to E_7/SU(5)×U(1)^3.

Contribution

It introduces a group-structure-based formula for Riemann curvature and applies it to compute four-fermi couplings in specific supersymmetric models.

Findings

Derived a formula linking Riemann curvature to Killing vectors.

Established an algebraic method for low-energy four-fermi coupling evaluation.

Explicitly calculated couplings for E_7/SU(5)×U(1)^3 model.

Abstract

The reducible K\"ahler coset space G/S\otimesU(1)^k is discussed in a geometrical approach. We derive the formula which expresses the Riemann curvature of the reducible K\"ahler coset space in terms of its Killing vectors. The formula manifests the group structure of G. On the basis of this formula we establish an algebraic method to evaluate the four-fermi coupling of the supersymmetric non-linear sigma-model on G/S\otimesU(1)^k at the low-energy limit. As an application we consider the supersymmetric non-linear sigma-model on E_7/SU(5)\otimesU(1)^3 which contains the three families of {10} + {5^*} + {1} of SU(5) as the pseudo NG fermions. The four-fermi coupling constants among diffferent families of the fermions are explicitly given at the low-energy limit.