Ivancevic_Applied-Diff-Geom

1190 Applied Differential Geometry: A Modern Introductionto conformal field theory is given in the lecture notes [Schellekens (1996)].The basics of topological string theory were laid out in a series of beautifulpapers by E. Witten in 1990s [Witten (1988b); Witten (1988a); Witten(1988d); Witten (1989); Witten (1990); Witten (1991); Witten (1992);Witten (1995a); Witten (1991)], and more or less completed in a seminalpaper [Bershadsky et. al. (1994)]. Reviews about topological string theory,usually also contain a discussion of topological field theory. The firstone of these is the review [Dijkgraaf et. al. (1991)] from the same period.However, if we want to dig even deeper, there is the 900–page book [Horiet. al. (2003)], which discusses topological string theory from the point ofview of mirror symmetry.In particular, there exists a mathematically rigorous, axiomatic definitionof topological field theories due to [Atiyah (1988a)]. Instead of givingthis definition, we will define topological field theory in a more physically–intuitive way, but as a result somewhat less rigorous way.Recall (from subsection 6.5.1 above) that the output of a QFT is givenby its observables: correlation functions of products of operators,〈O 1 (x 1 ) · · · O n (x n )〉 b . (6.302)Here, the O i (x) are physical operators of the theory. What one calls ‘physical’is part of the definition of the theory, but it is important to realize thatin general not all combinations of fields are viewed as physical operators.For example, in a gauge theory, we usually require the observables to arisefrom gauge–invariant operators. That is, Tr F would be one of the O i , butTr A or A itself would not.The subscript b in the above formula serves as a reminder that thecorrelation function is usually calculated in a certain background. That is,the definition of the theory may involve a choice of a Riemannian manifoldM on which the theory lives, it may involve choosing a metric on M, itmay involve choosing certain coupling constants, and so on.The definition of a topological field theory is now as follows. Supposethat we have a quantum field theory where the background choices involvea choice of manifold M and a choice of metric h on M. Then the theoryis called a topological field theory if the observables (6.302) do not dependon the choice of metric h. Let us stress that it is part of the definitionthat h is a background field – in particular, we do not integrate over hin the path integral. One may wonder what happens if, once we have atopological field theory, we do make the metric h dynamical and integrate