User:Sylvain Ribault/Bootstat2021/Galvani

Not directly related to the bootstrap. But thought-provoking result on CFT.

Study critical systems with boundaries. Focus on systems with fixed boundary conditions.

Uniformization hypothesis

Put bulk and boundary on the same footing:

• Metric with negative curvature sets boundary infinitely far away.
• Constant curvature so that system is homogeneous.
• Other condition.

Pick metric

$g_{ij}={\frac {\delta _{ij}}{\gamma (x)^{2}}}$

Require curvature to be constant and negative. This condition amounts to a Yamabe equation for the scale factor $\gamma (x)$. Existence of solution is known. For a ball, we have explicit simple solution. For a slab, it is $x(\gamma )$ which is a known hypergeometric function.

2d: allowed to use integer Yamabe equation, which coincides with fractional Yamabe equation, which is also equivalent to Liouville equation. So nothing new: we recover results from local conformal symmetry. Higher dimensions: conformal symmetry is much less powerful, the hypothesis makes up for this somehow?

Correlation functions

Main conjecture: correlation functions are expressed through the function $\gamma (x)$. One-point functions of primary field is a power of that function. Two-point function also depends on distance of two points, measured using our metric.

Can modify the exponent of the scale factor to account for scaling behaviour, leading to fractional Laplacian. The fractional Laplacian has many definitions on flat space, but they are not equivalent in bounded domains.

Applications

Percolation critical exponents