Multiloop amplitudes in the theory of quantum strings and complex geometry

The evaluation of multiloop amplitudes in the theory of closed oriented bosonic strings is reduced to the problem of finding the measure on the moduli space of Riemann surfaces. It is shown that the measure is equal to the product of the square of the modulus of a holomorphic function and the determinant of the imaginary part of the period matrix, raised to the power 13. A consequence of this theorem is that the measure can be expressed in terms of theta-functions. A variant of the holomorphy theorem, in the form of Quillen's theorem, is used to evaluate the dependence of the determinants of the Laplace operator on a Riemann surface on the boundary conditions. When the Riemann surface is represented by a branched covering of a plane, the measure is expressed in terms of the coordinates of the branch points, and to each branch point there corresponds a vertex operator. The measure is the correlation function of these operators, and this can be used to represent the sum over all the higher loops as the partition function of a certain two-dimensional conformal field theory.