Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1211is always a flat metric on a surface which is topologically a sphere (justconsider the sphere as a plane with a point added at infinity), and afterhaving chosen this metric there are no remaining parameters such as τ inthe torus case. For g = 1, the virtual dimension is m g = 0, but as we haveseen the actual dimension is 1.We can explain these discrepancies using the fact that, after we haveused the conformal invariance to fix the metric to be flat, the sphere and thetorus have leftover symmetries. In the case of the sphere, it is well known instring theory that one can use these extra symmetries to fix the positionsof three labelled points. In the case of the torus, after fixing the metricto be flat we still have rigid translations of the torus left, which we canuse to fix the position of a single labelled point. To see how this leads to adifference between the virtual and the actual dimensions, let us for exampleconsider tori with n labelled points on them. Since the virtual dimension ofthe moduli space of tori without labelled points is 0, the virtual dimensionof the moduli space of tori with n labelled points is n. One may expectthat at some point (and in fact, this happens already when n = 1), onereaches a sufficiently generic situation where the virtual dimension reallyis the actual dimension. However, even in this case we can fix one of thepositions using the remaining conformal (translational) symmetry, so thepositions of the points only represent n − 1 moduli. Hence, there must bean nth modulus of a different kind, which is exactly the shape parameterτ that we have encountered above. In the limiting case where n = 0, thisparameter survives, thus causing the difference between the virtual and thereal dimension of the moduli space.For the sphere, the reasoning is somewhat more formal: we analogouslyexpect to have three ‘extra’ moduli when n = 0. In fact, three extraparameters are present, but they do not show up as moduli. They must beviewed as the three parameters which need to be added to the problem tofind a 0D moduli space.Since the cases g = 0, 1 are thus somewhat special, let us begin bystudying the theory on a Riemann surface with g > 1. To arrive at thetopological string correlation functions, after gauge fixing we have to integrateover the remaining moduli space of complex dimension 3(g − 1). Todo this, we need to fix a measure on this moduli space. That is, given aset of 6(g − 1) tangent vectors to the moduli space, we want to produce anumber which represents the size of the volume element spanned by thesevectors, see Figure 6.27. We should do this in a way which is invariant undercoordinate redefinitions of both the moduli space and the world–sheet.