Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1191over it. This is exactly what we will do once we start considering topologicalstring theories.Note that the word ‘topological’ in the definition may be somewhat ofa misnomer [Vonk (2005)]. The reason is that the above definition doesnot strictly imply that the observables depend only on the topology of M –there may be other background choices hidden in b on which they dependas well. For example, in the case of a complex manifold M, correlationfunctions will in general not only depend on the topology of M and itsmetric, but also on our specific way of combining the 2D real coordinateson M into d complex ones. This choice, a complex structure, is part ofthe background of the quantum field theory, and correlation functions in atopological field theory will in general still depend on it.If our quantum field theory has general coordinate invariance, as we willusually assume to be the case, then the above definition has an interestingconsequence. The reason is that in such a case we can do an arbitrarygeneral coordinate transformation, changing both the coordinates on Mand its metric, under which the correlation functions should be invariant.Then, using the topological invariance, we can transform back the metricto its old value. The combined effect is that we have only changed the x i in(6.302). That is, in a generally coordinate invariant topological field theory,the observables do not depend on the insertion points of the operators.Chern ClassesInspired by the identification of a connection with a gauge field, let usconsider the analogue of the non–Abelian field strength,F = dA − A ∧ A,where A ∧ A is a shorthand for A I Ji AJ Kj dxi ∧ dx j (note that A ∧ A ≠ 0).A short calculation shows that on the overlap of two patches of M (orequivalently, under a gauge transformation), this quantity transforms asF (b) = Λ (ba) F (a) Λ −1(ba) ,from which we see that F can be viewed as a section of a true vector bundleof Lie algebra valued 2–forms. In particular, we can take its trace and geta genuine 2–form:c 1 =i Tr(F ),2π