The paper by Bershadsky, Cecotti, Ooguri, and Vafa develops techniques to compute higher-loop string amplitudes for twisted $N = 2$ theories with $\hat{c} = 3$ (the critical case). An important ingredient is the discovery of an anomaly in the decoupling of BRST trivial states at every genus, captured by a master anomaly equation. In a specific realization of the $N = 2$ theories, the resulting string field theory is equivalent to a topological theory in six dimensions, known as the Kodaira–Spencer theory, which can be viewed as the closed string analog of the Chern–Simons theory. Using the mirror map, this leads to the computation of the number of holomorphic curves of higher genus in Calabi–Yau manifolds. The topological amplitudes are also shown to compute corrections to superpotential terms in the effective 4d theory resulting from compactifying standard 10d superstrings on the corresponding $N = 2$ theory. Relations with $c = 1$ strings are also discussed. The paper includes a detailed review of twisted $N = 2$ theories, the holomorphic anomaly, and the open string version of these theories. It also explores the physical implications of topological amplitudes, including their interpretation in type II and open superstring theories, and their relevance to threshold corrections for heterotic strings.The paper by Bershadsky, Cecotti, Ooguri, and Vafa develops techniques to compute higher-loop string amplitudes for twisted $N = 2$ theories with $\hat{c} = 3$ (the critical case). An important ingredient is the discovery of an anomaly in the decoupling of BRST trivial states at every genus, captured by a master anomaly equation. In a specific realization of the $N = 2$ theories, the resulting string field theory is equivalent to a topological theory in six dimensions, known as the Kodaira–Spencer theory, which can be viewed as the closed string analog of the Chern–Simons theory. Using the mirror map, this leads to the computation of the number of holomorphic curves of higher genus in Calabi–Yau manifolds. The topological amplitudes are also shown to compute corrections to superpotential terms in the effective 4d theory resulting from compactifying standard 10d superstrings on the corresponding $N = 2$ theory. Relations with $c = 1$ strings are also discussed. The paper includes a detailed review of twisted $N = 2$ theories, the holomorphic anomaly, and the open string version of these theories. It also explores the physical implications of topological amplitudes, including their interpretation in type II and open superstring theories, and their relevance to threshold corrections for heterotic strings.