Cutting γ-Liouville quantum gravity by Schramm-Loewner evolution for κ∉{γ^2, 16/γ^2}
arxiv(2023)
摘要
There are many deep and useful theorems relating Schramm-Loewner evolution
(SLE_κ) and Liouville quantum gravity (γ-LQG) in the case when
the parameters satisfy κ∈{γ^2, 16/γ^2}. Roughly
speaking, these theorems say that the SLE_κ curve cuts the γ-LQG
surface into two or more independent γ-LQG surfaces. We extend these
theorems to the case when κ∉{γ^2, 16/γ^2}. Roughly
speaking we show that if we have an appropriate variant of SLE_κ and an
independent γ-LQG disk, then the SLE curve cuts the LQG disk into two or
more γ-LQG surfaces which are conditionally independent given the values
along the SLE curve of a certain collection of auxiliary imaginary geometry
fields, viewed modulo conformal coordinate change. These fields are sampled
independently from the SLE and the LQG and have the property that that the sum
of the central charges associated with the SLE_κ curve, the γ-LQG
surface, and the auxiliary fields is 26. This condition on the central charge
is natural from the perspective of bosonic string theory. We also prove
analogous statements when the SLE curve is replaced by, e.g., an LQG metric
ball or a Brownian motion path. Statements of this type were conjectured by
Sheffield and are continuum analogs of certain Markov properties of random
planar maps decorated by two or more statistical physics models. We include a
substantial list of open problems.
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