Ricci solitons in two dimensions

27/01/2009

Today I have read recent changes to DispersiveWiki. This is a beautiful site about differential equations that is maintained at University of Toronto by Jim Colliander and has notable contributors as the Fields medallist Terence Tao. Terry introduced a new page about Liouville’s equation as he got involved with it in a way you can read here. Physicists working on quantum gravity has been aware of this equation since eighties as it is the equation of two-dimensional quantum gravity and comes out quite naturally in string theory. A beautiful paper about quantum field theory of Liouville equation is due to Roman Jackiw and one of his collaborators Eric D’Hoker (see here). But what people could have overlooked is that Liouville’s equation is the equation of the Ricci soliton in two dimensions. The reason is that in this case a set of isothermal coordinates can always be found and the metric is always conformal, that is

g=e^{\phi}g_0

being g_0 the Euclidean metric. The Ricci tensor takes here a quite simple form

R_{ik}=-e^{-\phi}(\partial^2_x+\partial^2_y)\phi\epsilon_{ik}

being \epsilon_{ik}=diag(1,1) . Then the Ricci flow is

\frac{\partial\phi}{\partial t}=e^{-2\phi}(\partial^2_x+\partial^2_y)\phi

and finally for the Ricci soliton one has

(\partial^2_x+\partial^2_y)\phi = H e^{2\phi}

being H a constant. After a simple rescaling we are left with the Euclidean Liouville’s equation

(\partial^2_x+\partial^2_y)u = \Lambda e^{u}.

Turning back to the Jackiw and D’Hoker paper, we can see that a 2D gravity theory emerges naturally as the equilibrium (Ricci soliton) solution of a Fokker-Planck (Ricci flow) equation. This scenario seems a beautiful starting point to build an understanding of quantum gravity. I am still thinking about a lot and I will put all this on a paper one day.


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