(I will edit this later to elaborate and make it more clear... just want to get my thought down immediately.)

**Motivation:** There have been the *instanton* (anti-self dual connection) solutions to the Yang-Mills equation $\int_M|F_A|^2$, leading to the Donaldson invariants and even a Floer homology. There have been the *monopole* (connection + spinor) solutions to the Seiberg-Witten equations $D_A\psi=0$ and $F_A^+=\psi\otimes\psi^\ast-\frac{1}{2}|\psi|^2$, leading to the Seiberg-Witten invariants and a nice Floer homology. These utilize the fundamental particles in the Standard-Model of physics... but not of General Relativity, where the *gravitons* arise.

So I would be interested in a Floer homology and/or invariants arising from *gravitational instantons* (Riemannian metrics), i.e. solutions to the Einstein Field Equations $R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R-g_{\mu\nu}\Lambda=0$ in vacuum (no stress-energy term $T$). Here $\Lambda$ is the "cosmological constant", which we may or may not want to assume is zero. Surely these have been studied extensively.

**Should I expect something to arise? Are there immediate obstacles?** Otherwise this would have been done by now, right?

**Downfall?**: Perhaps the moduli space is too big, or boring, or unknown.  
**Progress?**: Witten has even shown that (2+1)-dimensional gravity (with no cosmological constant) on $M=\Sigma\times \mathbb{R}$ (compact surface $\Sigma$) is an ISO(2,1) Chern-Simons theory, i.e. the equations of motion of the CS-action are precisely the field equations. And if there is a cosmological constant $\Lambda\ne 0$, then the same result holds when we replace the gauge group ISO(2,1) by SO(3,1) or SO(2,2) depending on the sign of $\Lambda$.