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\begin{document}

P(M,G)

E(M,V)

An invariant polynomial of order r s.t.

\begin{itemize}

\item \( P(F_{1},...,F_{r}) \) is multilinear for \( F_{i}\in G \)

\item P is symmetric in all entries.

\item P is gauge invariant, \( P(F_{i})=P(t^{-1}Ft) \) forall \( t\in G \).

\end{itemize}

Example: 

\begin{itemize}

\item \( P=tr(F_{1}F_{2}) \)

\item \( P=\frac{1}{r!}\sum _{p\in S_{r}}tr(F_{p(1)}F_{p(2)}..F_{p(r)}) \)

\end{itemize}

invariant polynomials form an algebra.

Can avoid the need of symmetricity by using \( P(F,..,F)=P_{r}(F) \) where \( F=t_{i}F_{i} \). Want to extract
coefficients of \( t_{i} \).

In practice use \( F=\frac{1}{2}F_{uv}dx^{u}\wedge dx^{v} \), then \( Tr(F^{2})=Tr(F\wedge F)=\frac{1}{4}tr(F_{uv}F_{pl}dx^{u}\wedge dx^{v}\wedge dx^{p}\wedge dx^{l} \). Then \( P_{r}(F) \) is a 2r form, most of which are
traces.

An invariant polynomial determines a collection of classes of M that
depends on the bundle \( E(M,V_{p}) \) but not on the choice of connection.



\end{document}
