Ivancevic_Applied-Diff-Geom

1192 Applied Differential Geometry: A Modern Introductionwhere the prefactor is convention. It can be seen that this 2–form is closed:d(Tr(F )) = Tr(dF ) = − Tr(d(A ∧ A)) = − Tr(dA ∧ A − A ∧ dA) = 0.Therefore, we can take its cohomology class, for which we would like toargue that it is a topological invariant. Note that this construction is independentof the choice of coordinates on M. Moreover, it is independent ofgauge transformations. However, on a general vector bundle there may beconnections which cannot be reached in this way from a given connection.Changing to such a connection is called a ‘large gauge transformation’, andfrom what we have said it is not clear a priori that the Chern classes do notdepend on this choice of equivalence class of connections. However, withsome work we can also prove this fact. The invariant [c 1 ] is called the firstChern class. In fact, it might be better to call it a ‘relative topologicalinvariant’: given a base manifold M of fixed topology, we can topologicallydistinguish vector bundles over it by calculating the above cohomologyclass.By taking the trace of F , we loose a lot of information. There turns outto be a lot more topological information in F , and it can be extracted byconsidering the expression(c(F ) = det 1 + iF ),2πwhere 1 is the identity matrix of the same size as the elements of the Liealgebra of G. Again, it can be checked that this expression is invariantunder a change of coordinates for M and under a change of connection.Since the matrix components inside the determinant consist of the 0–form1 and the 2–form F , expanding the determinant will lead to an expressionconsisting of forms of all even degrees. One writes this asc(F ) = c 0 (F ) + c 1 (F ) + c 2 (F ) + . . .The sum terminates either at the highest degree encountered in expandingthe determinant, or at the highest allowed even form on M. Note thatc 0 (F ) = 1, and [c 1 (F )] is exactly the first Chern class we defined above.The cohomology class of c n is called the nth Chern class.As an almost trivial example, let us consider the case of a productbundle M × W . In this case there is a global section g(x) of the principalbundle P , and we can use this to construct a connection A = −gdg −1 , so