# What is the meaning of Yang-Mills action evaluated on Levi-Civita connection?

On a Riemannian manifold $M$ with riemann curvature tensor $R_{\mu\nu\rho\sigma}$ written as (endomorphism valued) curvature two-tensor of the Levi-Civita connection $R=R_{\mu\nu}dx^\mu\wedge dx^\nu$, consider the following term. $$\int_M tr(R\wedge *R)=2\int_M tr(R_{\mu\nu}R^{\mu\nu})dvol_g=2\int_M tr(R_{\mu\nu}{}_{\rho\sigma}R^{\mu\nu\sigma\rho})dvol_g$$ This is Yang-Mills action evaluated on Levi-Civita. My questions are

What is its meaning? Is it a topological term?

Does it have any physical applications, especially in GR?

• If you google "L2 norm of curvature", you'll find some papers regarding this functional, especially in dimension 4. – Deane Yang Mar 28 '14 at 21:52