Ivancevic_Applied-Diff-Geom

1220 Applied Differential Geometry: A Modern Introductionwhere F g = ln Z g is the free energy at genus g, and the reason F g appears inthe above equation instead of Z g is, as usual in quantum field theory, thatthe expectation values in the r.h.s. are normalized such that 〈1〉 = 1, andso the l.h.s. should be normalized accordingly and equal Zg−1 ∂āZ g = ∂āF g[Vonk (2005)].Thus, as we have claimed before, we are integrating a total derivativeover the moduli space of genus g surfaces. If the moduli space did nothave a boundary, this would indeed give zero, but in fact the moduli spacedoes have a boundary. It consists of the moduli which make the genus gsurface degenerate. This can happen in two ways: an internal cycle of thegenus g surface can be pinched, leaving a single surface of genus g − 1, asin Figure 6.28 (a), or the surface can split up into two surfaces of genus g 1and g 2 = g −g 1 , as depicted in Figure 6.28 (b). By carefully considering theboundary contributions to the integral for these two types of boundaries,it was shown in [Bershadsky et. al. (1994)] that()∂F g= 1∂tā 2 c ∑g−1ā¯b¯ce 2K G¯bd G¯ce D d D e F g−1 + D d F r D e F g−r ,where G is the so–called Zamolodchikov metric on the space parameterizedby the coupling constants t a , tā; K is its Kähler potential, and the D a arecovariant derivatives on this space. The coefficients cā¯b¯c are the 3–pointfunctions on the sphere of the operators Ō(0) a . We will not derive the aboveformula in detail, but the reader should notice that the contributions fromthe two types of boundary are quite clear.r=1Fig. 6.28 At the boundary of the moduli space of genus g surfaces, the surfaces degeneratebecause certain cycles are pinched. This either lowers the genus of the surface (a)or breaks the surface into two lower genus ones (b) (see text for explanation).Using this formula, one can inductively determine the tā dependenceon the partition functions if the holomorphic t a −dependence is known.