Open Strings and 3d Topological Field Theory

In the perturbative formulation of string theory the partition function in the background of D-branes can be expanded as Z string = g,b g −2+2g+b s Z g,b. (1) Here Z g,b is the value of the diagram given by a Riemann surface of genus g with b boundary components. One way to compute the numbers Z g,b is to work in conformal gauge. In this gauge one obtains a conformal field theory (CFT) living on the string world sheet, and the Z g,b result from integrating correlators of this CFT over the moduli space of the Riemann surface. In order for this approach to make sense, the CFT has to be well defined on surfaces of arbitrary genus and with an arbitrary number of boundaries. For type I strings also non-orientable surfaces have to be included in the expansion (1). In this note we restrict ourselves to type II strings and orientable world sheets. With this motivation in mind we set ourselves the aim to construct a CFT consistent on all surfaces relevant in (1). This can be treated entirely as a problem in CFT. We will not worry about whether a particular CFT actually appears in the conformal gauge description of some string background.