Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1213exactly the same formula, but with G replaced by the conformal ghost b.This ghost is the BRST–partner of the SEM–tensor in exactly the sameway as G is the Q−partner of T . Secondly, one can make the not unrelatedobservation that since {Q, G} = T, we can still use the standard argumentsto show independence of the theory of the parameters in a Lagrangian ofthe form L = {Q, V }. The only difference is that now we also have tocommute Q through G to make it act on the vacuum, but since T αβ itself isthe derivative of the action with respect to the metric h αβ , the terms we getin this way amount to integrating a total derivative over the moduli space.Therefore, apart from possible boundary terms these contributions vanish.Note that this reasoning also gives us an argument for using G zz insteadof T zz (which is more or less the only other reasonable option) in (6.325):if we had chosen T zz then all path integrals would have been over totalderivatives on the moduli space, and apart from boundary contributionsthe whole theory would have become trivial.If we consider the vector and axial charges of the full path integralmeasure, including the new path integral over the world–sheet metric h,we find a surprising result. Since the world–sheet metric does not transformunder R−symmetry, naively one might expect that its measure doesnot either. However, this is clearly not correct since one should also takeinto account the explicit G−insertions in (6.325) that do transform underR−symmetry. From the N = 2 superconformal algebra (or, more down–to–earth, from expressing the operators in terms of the fields), it followsthat the product of G and Ḡ has vector charge zero and axial charge 2.Therefore, the total vector charge of the measure remains zero, and theaxial charge gets an extra contribution of 6(g − 1), so we find a total axialR−charge of 6(g − 1) − 2m(g − 1). From this, we see that the case ofcomplex target space dimension 3 is very special: here, the axial charge ofthe measure vanishes for any g, and hence the partition function is nonzeroat every genus. If m > 3 and g > 1, the total axial charge of the measureis negative, and we have seen that we cannot cancel such a charge withlocal operators. Therefore, for these theories only the partition function atg = 1 and a specific set of correlation functions at genus zero give nonzeroresults. Moreover, for m = 2 and m = 1, the results can be shown to betrivial by other arguments. Therefore, a Calabi–Yau threefold is by far themost interesting target space for a topological string theory. It is a ‘happycoincidence’ (see [Vonk (2005)]) that this is exactly the dimension we aremost interested in from the string theory perspective.Finally, let us come back to the special cases of genus 0 and 1. At genus