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.

  • $\begingroup$ Typically in a Floer theory the relevant PDEs are elliptic, with a Fredholm linearization, at least after restricting to a slice for the gauge group action. Is that the case here? Are solutions to the Einstein field equations gradient flowlines (or maybe you meant them to be critical points?) for the Chern-Simons functional? $\endgroup$
    – Mike Usher
    Aug 14, 2012 at 3:45
  • $\begingroup$ True, I do not know whether we have elliptic/Fredholm conditions. And as clarified in my edit, the solutions in 3-dimensional gravity are the critical points for the CS-action. $\endgroup$ Aug 14, 2012 at 7:50
  • $\begingroup$ Your definition of "gravitational instantons" does not capture the gravitational concept analogous to Yang-Mills instantons. Exotic four-manifolds capture this more appropriately. See for example the answer bit.ly/MZwcdP $\endgroup$ Aug 14, 2012 at 8:31

2 Answers 2


Let me clarify a couple of issues from the previous answers/comments:

1) The linearization of $Rc-\tfrac{1}{2}Rg$ has mixed signs, and for this reason the equation $\partial_t g=-(Rc-\tfrac{1}{2}Rg)$ is bad, a kind of coupled backwards/forwards heat equation, i.e. no short time existence.

2) The linearization of $Rc$ is elliptic however (after fixing the gauge), and the Ricci flow $\partial_tg=-2Rc$ indeed is a good equation. In terms of functionals, it's better to consider the Perelman functional (whose gradient is $Rc$ up to gauge) instead of the Einstein-Hilbert functional (whose gradient is $Rc-\tfrac{1}{2}Rg$).

3) There was the issue of 1st order vs. 2nd order: While the YM-equation ($D^*F=0$) is second order in $A$, there is also the 1st order equation $F^+=0$ (antiselfdual connections). Solutions of this 1st order equation are special solutions of the YM-equation (as follows immediately from the Bianchi identity). There is a similar story for Ricci-flat metrics: Here the "special" solutions are the ones with special holonomy. Having holonomy $SU,Sp,G_2$ or $Spin_7$ implies that the metric is Ricci-flat. E.g. by solving the first order system $d\psi=0,d*\psi=0$ for a 3-form $\psi$ on a 7-manifold you get Ricci-flat metrics with holonomy $G_2$. Indeed, there is a proposal by Simon Donaldson for a higher order gauge theory based on $G_2$ (though, instead of just counting $G_2$-structures, the idea there is actually to "count" the number of associative submanifolds...)

  • $\begingroup$ Thanks! I had forgotten about the special holonomy equations. They do indeed appear to be the analogue of self-dual connections. $\endgroup$
    – Deane Yang
    Aug 23, 2012 at 16:14
  • $\begingroup$ Could you elaborate a little on your point#2? $\endgroup$ Nov 9, 2012 at 3:04
  • $\begingroup$ @Chris: Ricci-flat metrics are critical points but never extrema of the Einstein-Hilbert functional. You can see this from the formula for the second variation, which has a different sign (at the level of the symbol!) in conformal and in TT directions, i.e. there are always infinitely many positive and infinitely many negative directions. In Perelman's lambda-functional there is a minimization over all densities (this roughly corresponds to the conformal directions), which kills all the positive directions (except possibly finitely many), so you get extrema and the gradient flow makes sense. $\endgroup$ Nov 11, 2012 at 23:24
  • $\begingroup$ @RobertHaslhofer Donaldson's proposal, I understand it, is not to count have a gauge theory counting associative submanifolds but to have a gauge theory counting G$_2$ instantons, arrising from a certain G$_2$ functional. But it turns out that G$_2$ instantons can bubble out of associative $3$-folds and so such a gauge theory needs to keep track of associative $3$-folds. $\endgroup$
    – user100272
    Dec 7, 2017 at 20:53

Here is one problem. The instantons or the monopoles are critical points of certain energy functionals and thus they satisfy Euler-Lagrange equations. These are second-order p.d.e.'s. However the Yang-Mills equations or the Seiberg-Witten equations are first-order p.d.e.-s. The reason for this fortunate accident is that the instantons and the monopoles are not arbitrary critical points, they are absolute minima of their corresponding energy functionals.

The Floer theories arise by considering these equations on cylinders $\mathbb{R}\times M^3$. In this context these equations will continue to be first order and as such they describe gradient flow lines of a different energy functional. For an equation to be a gradient flow line equation one necessary condition is to be a first-order equation in the time parameter. To hope for a Floer theory one needs that the gradient-flow equation be elliptic. This forces the equation to be first-order in all the variables, temporal or spatial for else the principal symbol of the linearization won't be elliptic.

The Einstein's equation are second order equations so they are not appropriate for a Floer theory. A more appropriate strategy would be too look for absolute minima of the Einstein's functional, if such absolute minima exist and satisfy a first order p.d.e.

  • $\begingroup$ Just a clarification. The original Yang-Mills equations, i.e. the Euler-Lagrange equations for the curvature functional, are in fact second order. But what happens is that there are first order equations, curvature is self-dual, that imply the Yang-Mills equations, and these equations are a lot easier to study than the second order ones. But you're right that there seems to be no analogous first order system for Einstein's equations. $\endgroup$
    – Deane Yang
    Aug 23, 2012 at 15:48
  • $\begingroup$ @Deane You are right. $\endgroup$ Aug 23, 2012 at 17:45
  • 1
    $\begingroup$ one more clarification: the suggestion "absolute minima of the Einstein-Hilbert functional" doesn't work, since the second variation of this functional has opposite signs in conformal and TT directions (this is of course very much related to the fact that the linearization of the Einstein tensor is not elliptic). However, for the Perelman functional there are extrema, e.g. (with Perelman's sign convention) a flat metric on a torus or a Ricci-flat metric on a K3 surface are absolute maxima. $\endgroup$ Aug 23, 2012 at 18:49
  • $\begingroup$ The Hilbert-Palatini formulation of GR is first order. So, if one really wants to take this path, which I think is likely the wrong road to go down, one should study the Hilbert-Palatini formulation. However, I think the real problem is that GR instantons, as Witten mentions in "Global gravitational anomalies", should be thought of as exotic four-manifolds not as global minima of the Hilbert-Palatini action. $\endgroup$ Aug 23, 2012 at 19:52

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