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(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 $d_A^\ast F_A=0$ which extremize the YM energy $\int_M|F_A|^2$, 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 $F_A^+=\psi\otimes\psi^\ast-\frac{1}{2}|\psi|^2$ which extremize the Chern-Simons-Dirac functional, leading to the SW 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 $\Lambda$.

Motivation: There have been the instanton (anti-self dual connection) solutions to the Yang-Mills equation $d_A^\ast F_A=0$ which extremize the YM energy $\int_M|F_A|^2$, 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 $F_A^+=\psi\otimes\psi^\ast-\frac{1}{2}|\psi|^2$ which extremize the Chern-Simons-Dirac functional, leading to the SW 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. (This term 'gravitational instanton' is used first (I think) in Stephen Hawking's seminal 1977 paper "Gravitational Instantons", and basic examples are the Schwarzschild and Taub-NUT metrics.)

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 $\Lambda$.
More: There is something to be said from Witten's recent paper "Analytic Continuation Of Chern-Simons Theory", but I am not ready to understand it.

(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 $\Lambda$.

(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 $d_A^\ast F_A=0$ which extremize the YM energy $\int_M|F_A|^2$, 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 $F_A^+=\psi\otimes\psi^\ast-\frac{1}{2}|\psi|^2$ which extremize the Chern-Simons-Dirac functional, leading to the SW 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 $\Lambda$.

(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?

(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 $\Lambda$.