[For Part I see the preceding review Zbl 0656.58043.] The properties of analytic fields on a Riemann surface represented by a branch covering of \({\mathbb{C}}{\mathbb{P}}^ 1\) are investigated in detail. Branch points are shown to correspond to the vertex operators with simple conformal properties. As applications we compute determinants of \({\bar \partial}_ j\) operators for \(Z_ n\)-symmetric surfaces and obtain various representations for the two-loop measure in the bosonic string theory together with various identities for theta-functions of hyperelliptic surfaces. We also present an integral representation for the quantum part of the twist field correlation functions, which describe propagation of the string on the orbifold background. We also calculate the quantum part of the structure constants of the twist-field operator algebra, generalizing the results of Dixon, Friedan, Martinec, and Shenker.