Euler characteristic of an Einstein-Cartan manifold

Question

In the Einstien-Cartan theory, the Euler characteristic of the manifold $M$ is given by $$\chi(M) = \int F_{ab}\wedge\star F^{ab}$$ where $F_{ab} = d\omega_{ab}+(\omega\wedge\omega)_{ab}$, $\star F_{ab} = \epsilon^{abcd}F_{cd}$ and $\omega$ is the spin connection. Now, how does one compte the topology of a manifold using this formula? I have only seen people discussing metric on a manifold and using the metric to compute various properties of the manifold without having to refer to connections on it. How can one discuss a manifold purely based on some connection defined on it without refering to its metric?

Answer 1

I believe this is a special case of the Chern-Gauss-Bonnet theorem.

My understanding is that Einstein-Cartan gravity is a metric-affine theory rather than a purely affine theory, in that the Riemannian metric $g$ and the connection $\nabla$ are both dynamical variables. The connection is still metric, in that $\nabla g=0$, but the torsion is allowed to vary independently of $g$.

While it doesn't apply to general affine connections, the CGB theorem does apply to all metric connections, and it appears that the integrand $F_{ab}\wedge\star F^{ab}$ is indeed equal to $\operatorname{Pf}(F)$. Your formula seems to differ by a factor of $4\pi^2$, and there is some subtlety if you allow nonorientable spacetimes, but I suspect these minor discrepancies are due to unstated conventions that I'm not familiar with.