The unitary connections on the complex Grassmann manifold

Abstract

In the complex Grassmann manifold ℱ(m,n), the space of complexn-planes passes through the origin of C^m+n; the local coordinate of the space can be arranged into anm ×n matrixZ. It is proved that

$$K = K(Z,dZ) = (I + ZZ^\dag )^{ - \frac{1}{2}\overline \partial } (I + ZZ^\dag )^{\frac{1}{2}} - \partial (I + ZZ^\dag )^{\frac{1}{2}} + (I + ZZ^\dag )^{ - \frac{1}{2}} $$

is a U(m)-connection of ℱ(m,n) and its curvature form

$$\Omega _1 = dK + K\Lambda K$$

satisfies the Yang-Mills equation. Moreover,

$$B = B(Z,{\bf{ }}dZ) = K(Z,{\bf{ }}dZ) - \frac{{tr(K(Z,{\bf{ }}dZ))}}{m}I^{(m)} $$

is an (Sum)-connection and its curvature form

$$\Omega _2 = dB + B{\bf{ }}\Lambda {\bf{ }}B$$

satisfies the Yang-Mills equation.