Ivancevic_Applied-Diff-Geom

1194 Applied Differential Geometry: A Modern IntroductionFrom this, we see that if we define the action asS = k ∫L,4πwith k an integer, the action changes by 2πkm under gauge transformations,and the quantum measure e iS is invariant.So from this discussion we seem to arrive at the conclusion that thepartition function∫Z = D[A] e iS[A]for a line bundle of a fixed topology E is a topological invariant of M,as are the correlation functions of gauge–invariant operators such as Tr F .However, there is one more detail we have to worry about: there may be ananomaly in the quantum theory. That is, it may not be possible to definethe path integral measure D[A] in a gauge–invariant way.One way to see what problems can arise is to note that to actuallycalculate the path integral, one has to pick a gauge condition on A. Thatis, we have to pick one representative of A in each equivalence class undergauge transformations. To make such a choice will in general require achoice of metric. For example, from electromagnetism (where E is a 1Dcomplex line bundle and G = U(1)) we know that a useful gauge is theFeynman gauge, in which the equation of motion for A becomesM∆A = 0.As we have seen before, the Laplacian ∆ is an operator which, throughthe Hodge star, depends on the metric, and hence the results we find willa priori be metric dependent. To show that the results are truly metricindependent, one needs to show that the quantum results do not dependon our arbitrary choice of gauge.We will not go into the details of this, but state that one can show thatChern–Simons theory on a compact 3–manifold is anomaly–free, so ournaive argument above was correct, and one can indeed calculate topologicalinvariants of M in this way.Let us briefly discuss the kind of topological invariants that Chern–Simons theory can lead to. Recall that one can construct a Lie groupelement g from a Lie algebra element A as g = e A . Now suppose we have apath γ(t) inside M. Suppose that we chop up γ into very small line elementsgiven by tangent vectors ˙γδt. Then we can insert this tangent vector into