Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1205We have also encountered a drawback of our construction. Even thoughthe theories we have found can give us some interesting ‘semi–topological’information about the target spaces, one would like to be able to definegeneral nonzero n−point functions at genus g instead of just the partitionfunction at genus one and the particular correlation functions we calculatedat genus zero.It turns out that these two remarks are intimately related. In this sectionwe will go from topological field theory to topological string theory byintroducing integrals over all metrics, and in doing so we will find interestingnonzero correlation functions at any genus (see [Vonk (2005)]).Coupling to Topological GravityIn coupling an ordinary field theory to gravity, one has to perform thefollowing three steps.• First of all, one rewrites the Lagrangian of the theory in a covariantway by replacing all the flat metrics by the dynamical ones, introducingcovariant derivatives and multiplying the measure by a factor of √ det h.• Secondly, one introduces an Einstein–Hilbert term as the ‘kinetic’ termfor the metric field, plus possibly extra terms and fields to preserve thesymmetries of the original Lagrangian.• Finally, one has to integrate the resulting theory over the space of allmetrics.Here we will not discuss the first two steps in this procedure. As wehave seen in our discussion of topological field theories, the precise form ofthe Lagrangian only plays a comparatively minor role in determining theproperties of the theory, and we can derive many results without actuallyconsidering a Lagrangian. Therefore, let us just state that it is possible tocarry out the analog of the first two steps mentioned above, and construct aLagrangian with a ‘dynamical’ metric which still possesses the topologicalQ−symmetry we have constructed. The reader who is interested in thedetails of this construction is referred to the paper [Witten (1990)] and tothe lecture notes [Dijkgraaf et. al. (1991)].The third step, integrating over the space of all metrics, is the one wewill be most interested in here. Naively, by the metric independence of ourtheories, integrating their partition functions over the space of all metrics,and then dividing the results by the volume of the topological ‘gauge group’,