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

**Quick motivation:** There have been the *instanton* (anti-self dual connection) solutions to the Yang-Mills equation, leading to the Donaldson invariants and even a Floer homology. There have been the *monopole* (connection + spinor) solutions to the Seiberg-Witten equations, 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 (or just invariants) arising from *gravitational instantons* (I believe Riemannian metrics?), i.e. solutions to the Einstein Field Equations. Surely these have been studied extensively.  **Should I expect something like this to arise?** It would've been done by now though...  
Perhaps the moduli space is too big, or boring, or unknown. We even have a Chern-Simons action functional that can be used for gravity [insert reference here].  **Are there immediate obstacles?**