Computing the scalar field couplings in 6D supergravity

Using non-chiral supersymmetry in 6D space-time, we compute the explicit expression of the metric the scalar manifold SO(1,1)×SO(4,20)SO(4)×SO(20) of the ten-dimensional type IIA superstring on generic K3. We consider as well the scalar field self-couplings in the general case where the non-chiral 6D supergravity multiplet is coupled to generic n vector supermultiplets with moduli space SO(1,1)×SO(4,n)SO(4)×SO(n). We also work out a dictionary giving a correspondence between hyper-Kähler geometry and the Kähler geometry of the Coulomb branch of 10D type IIA on Calabi-Yau threefolds. Others features are also discussed.