Ivancevic_Applied-Diff-Geom

Geometrical Path Integrals and Their Applications 1193that [Vonk (2005)]F = −dgdg −1 − gdg −1 gdg −1 = −dgdg −1 + dgdg −1 = 0.Thus, for a trivial bundle with this connection, c 0 = 1 and c n = 0 for alln > 0.Chern–Simons TheoryThe easiest way to construct a topological field theory is to construct atheory where both the action S (or, quantum measure e iS ) and the fieldsdo not include the metric at all. Such topological field theories are called‘Schwarz–type’ topological field theories. This may sound like a trivialsolution to the problem, but nevertheless it can lead to quite interestingresults. To see this, let us consider familiar example: Chern–Simons gaugetheory on a 3D manifold M – now from a physical point of view.Recall from subsection 5.11.8 that Chern–Simons theory is a gauge theory– that is, it is constructed from a vector bundle E over the base spaceM, with a structure group (gauge group) G and a connection (gauge field)A. The Lagrangian of Chern–Simons theory is then given byL = Tr(A ∧ dA − 2 3 A ∧ A ∧ A).It is a straightforward exercise to check how this Lagrangian changes underthe gauge transformationand one finds˜L ≡ Tr(à ∧ dà − 2 3à ∧ à ∧ Ã)à = gAg −1 − gdg −1 ,= Tr(A ∧ dA − 2 3 A ∧ A ∧ A) − d Tr(gA ∧ dg−1 ) + 1 3 Tr(gdg−1 ∧ dg ∧ dg −1 ).The second term is a total derivative, so if M does not have a boundary,the action, being the integral of L over M, does not get a contribution fromthis term. The last term is not a total derivative, but its integral turns outto be a topological invariant of the map g(x), which is quantized as124π 2 ∫MTr(gdg −1 ∧ dg ∧ dg −1 ) = m ∈ Z.