Floer homology and Invariants for Einstein Field Equations? - MathOverflow

Floer homology and Invariants for Einstein Field Equations?

2012-08-13T20:49:56Z

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.

Answer by Liviu Nicolaescu for Floer homology and Invariants for Einstein Field Equations?

2012-08-23T08:32:03Z

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.

Answer by Robert Haslhofer for Floer homology and Invariants for Einstein Field Equations?

2012-08-23T16:03:54Z

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...)