Complex geometry and quantum string theory

Summation over closed oriented surfaces of genus p ≥ 2 (p - loop vacuum amplitudes in boson string theory) in a critical dimensions D=26 is reduced to integration over Mp space of complex structures of Riemann surfaces of genus p. The analytic properties of the integration measure as a function of the complex coordinates on Mp are studied. It is shown that the measure multiplied by (det Im τ-circumflex)13(τ-circumflex is the surface period matrix) is the square of the modulus of a function which is holomorphic on Mp and does not vanish anywhere. The function has a second order pole at infinity of compactified space of moduli Mp. These properties define the measure uniquely up to a constant multiple and this permits one to set up explicitformulae for p=2,3 in terms of the theta-constants. Power and logarithmic divergences connected with renormalization of the tachyon wave function and of the slope respectively are involved in the theory. Quantum geometry of critical strings turns out to be a complex geometry