I have an Ising model on the upper half plane with a boundary condition that all the spins left of the origin are up and the spins right of the origin are down. Then there is some curve starting from the origin and going into the upper half plane (may or may not come back to the boundary) that describes the domain wall between up and down.
But by Riemann mapping
where D is the upper half plane or Poincare disk take your pick (I'm going to go with disk).
Loewner says we get a differential equation for g.
where the driving function
describes where the domain wall goes when we make it part of the boundary as we unzip with g.
The domain wall really should be random. The right driving force is Brownian motion. So Brownian motion on the circle gives via the Loewner equation a description of the boundary wall in the 2D Ising model. Brownian motion in 1D relates to conformal invariance in 2D. This still works even when the domain wall is not simple. Just take the unbounded component of the complement instead of simply the complement of the curve. We still get a description of the domain wall in terms of brownian motion on the circle.
The parameter of how fast the Brownian motion goes controls the situation in the CFT description of the Ising model.
There are many special values. The case we were talking about seems to be 3. Other values will give other kinds of random walks which correspond to different CFT's
The central charge is determined by this via
If you plug in the Ising case you get 1/2, and all is right with the world.
Another cool duality is that:
One is below 4 and the other above 4. The case of 4 being self dual as it probably should if it is the free field.
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