Type-A and type-B topological sigma models are two topological field theories in two dimensions. Although they contain a lot of information in genus 0, they turn out to be trivial for g > 1. This is essentially due to the fact that, in order to define these theories, it is necessary to consider a fixed metric in the Riemann surface. In order to obtain a non-trivial theory in higher genus the degrees of freedom of the two-dimensional metric must be introduced. This means that the topological field theories must be coupled to two-dimensional gravity. The coupling to gravity is done by using the fact that the structure of the twisted theory is tantalizingly close to that of the bosonic string. Topological sigma models may be defined not only on closed Riemann surfaces and closed topological strings, but also on the open case.
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