Multiloop Amplitudes in The Theory of Quantum Strings and Complex Geometry

Description This classic paper was first published in Soviet Scientific Reviews in 1989 and is a review of the main results in the area of multiloop calculations in the theory of strings. The evaluation of multiloop amplitudes in the theory of closed oriented Bosnic strings reduces to finding the measure on the moduli space of Riemann surfaces. It is shown that the measure is the products of the squared modulus of a holomorphic function with the determinant of the imaginary part of the period matrix to the power-13. With the help of this theorem, the measure can be expressed in terms of theta functions. A version of the holomorphity theorem (Quillen’s theorem) is used for evaluating the dependence of the determinants of the Laplace operators on the boundary conditions on the Riemann surface. In the case where the Riemann surfaces are presented as ramified covering of CPI the measure is expressible in terms of the coordinates of the ramification points, in such a way that a vertex operator corresponds to each ramification point. The measure is the correlation of the operators, so that the sum over all higher loops can be written as the statistical sum of a two-dimensional conformal field theory.