P(M,G)

E(M,V)

An invariant polynomial of order r s.t.

• is multilinear for
• P is symmetric in all entries.
• P is gauge invariant, forall .

Example:

• 
• 

invariant polynomials form an algebra.

Can avoid the need of symmetricity by using where . Want to extract coefficients of .

In practice use , then . Then is a 2r form, most of which are traces.

An invariant polynomial determines a collection of classes of M that depends on the bundle but not on the choice of connection.

source psfile jl@crush.caltech.edu index invariant_polynomial_algebra characteristic_class Chern-Weil_theorem transgression