Ivancevic_Applied-Diff-Geom

1098 Applied Differential Geometry: A Modern IntroductionTQFT originated in 1982, when Witten rewrote classical Morse theory (seesection 3.10.5.1 above, as well as section 3.13.5.2 below) in Dick Feynman’slanguage of quantum field theory [Witten (1982)]. Witten’s argumentsmade use of Feynman’s path integrals and consequently, at first, they wereregarded as mathematically non–rigorous. However, a few years later, A.Floer reformulated a rigorous Morse–Witten theory [Floer (1987)] (thatwon a Fields medal for Witten). This trend in which some mathematicalstructure is first constructed by quantum field theory methods and thenreformulated in a rigorous mathematical ground constitutes one of the tendenciesin modern physics.In TQFT our basic topological space is an nD Riemannian manifoldM with a metric g µν . Let us consider on it a set of fields {φ i }, and letS[φ i ] be a real functional of these fields which is regarded as the action ofthe theory. We consider ‘operators’, O α (φ i ), which are in general arbitraryfunctionals of the fields. In TQFT these functionals are real functionalslabelled by some set of indices α carrying topological or group–theoreticaldata. The vacuum expectation value (VEV) of a product of these operatorsis defined as∫〈O α1 O α2 · · · O αp 〉 = [Dφ i ]O α1 (φ i )O α2 (φ i ) · · · O αp (φ i ) exp (−S[φ i ]) .A quantum field theory is considered topological if the following relation issatisfied:δδg µν 〈O α 1O α2 · · · O αp 〉 = 0, (6.153)i.e., if the VEV of some set of selected operators is independent of the metricg µν on M. If such is the case those operators are called ‘observables’.There are two ways to guarantee, at least formally, that condition(6.153) is satisfied. The first one corresponds to the situation in whichboth, the action S[φ i ], as well as the operators O αi are metric independent.These TQFTs are called of Schwarz type. The most important representativeis Chern–Simons gauge theory. The second one corresponds to thecase in which there exist a symmetry, whose infinitesimal form is denotedby δ, satisfying the following properties:δO αi = 0, T µν = δG µν , (6.154)