Ivancevic_Applied-Diff-Geom

1206 Applied Differential Geometry: A Modern Introductionwould be equivalent to multiplication by a factor of 1,?Z[h 0 ] = 1 ∫D[h] Z[h], (6.320)G topfor any arbitrary background metric h 0 . There are several reasons why thisnaive reasoning might go wrong:• There may be metric configurations which cannot be reached from agiven metric by continuous changes.• There may be anomalies in the topological symmetry at the quantumlevel preventing the conclusion that all gauge fixed configurations areequivalent.• The volume of G top is infinite, so even if we could rigorously definea path integral the above multiplication and division would not bemathematically well–defined.For these reasons, we should really be more careful and precisely definewhat we mean by the ‘integral over the space of all metrics’. Let us notethe important fact that just like in ordinary string theory (and even beforetwisting), the 2D sigma models become conformal field theories when weinclude the metric in the Lagrangian. This means that we can borrow thetechnology from string theory to integrate over all conformally equivalentmetrics. As is well known, and as we will discuss in more detail later, theconformal symmetry group is a huge group, and integrating over conformallyequivalent metrics leaves only a nD integral over a set of world–sheetmoduli. Therefore, our strategy will be to use the analogy to ordinary stringtheory to first do this integral over all conformally equivalent metrics, andthen perform the integral over the remaining nD moduli space.In integrating over conformally equivalent metrics, one usually has toworry about conformal anomalies. However, here a very important factbecomes our help. To understand this fact, it is useful to rewrite ourtwisting procedure in a somewhat different language (see [Vonk (2005)]).Let us consider the SEM–tensor T αβ , which is the conserved Noethercurrent with respect to global translations on C. From conformal fieldtheory, it is known that T z¯z = T¯zz = 0, and the fact that T is a conservedcurrent, ∂ α T α β = 0, means that T zz ≡ T (z) and T¯z¯z ≡ ¯T (¯z) are (anti–)holomorphic in z. One can now expand T (z) in Laurent modes,T (z) = ∑ L m z −m−2 . (6.321)