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We show that the quantum field theoretical formulation of the $\tau$-functiontheory has a geometrical interpretation within the classical transformationtheory of conjugate nets. In particular, we prove that i) the partial chargetransformations preserving the neutral sector are Laplace transformations, ii)the basic vertex operators are Levy and adjoint Levy transformations and iii)the diagonal soliton vertex operators generate fundamental transformations. Wealso show that the bilinear identity for the multicomponentKadomtsev-Petviashvili hierarchy becomes, through a generalized Miwa map, abilinear identity for the multidimensional quadrilateral lattice equations.Comment: 28 pages, 3 Postscript figure

Charged free fermions, vertex operators and the classical theory of conjugate nets