KODAIRA-SPENCER THEORY OF GRAVITY AND EXACT RESULTS FOR QUANTUM STRING AMPLITUDES

We develop techniques to compute higher loop string amplitudes for twi sted N = 2 theories with c = 3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly eq uation. In a particular realization of the N = 2 theories, the resulti ng string field theory is equivalent to a topological theory in six di mensions, the Kodaira-Spencer theory, which may be viewed as the close d string analog of the Chern-Simons theory. Using the mirror map this leads to computation of the 'number' of holomorphic curves of higher g enus curves in Calabi-Yau manifolds. It is shown that topological ampl itudes can also be reinterpreted as computing corrections to superpote ntial terms appearing in the effective 4d theory resulting from compac tification of standard 10d superstrings on the corresponding N = 2 the ory. Relations with c = 1 strings are also pointed out.