<p>Two-pointed quantum disks with a weight parameter <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$W>0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>W</mi> <mo>></mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> is a canonical family of finite-volume random surfaces in Liouville quantum gravity. We prove that the conformal welding of the forested variant of this disk gives a two-pointed quantum disk with an independent SLE<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$_\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mi>κ</mi> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\kappa \in (4,8)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. Furthermore, we show that the conformal welding of multiple forested quantum disks gives a surface arising in Liouville conformal field theory decorated by multiple SLE<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$_\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mi>κ</mi> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\kappa \in (4,8)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, such that the random conformal modulus contains the SLE partition function as a multiplicative factor. In particular, this gives a construction of the multiple SLE<InlineEquation ID="IEq6"> <EquationSource Format="TEX">$_\kappa $</EquationSource> <EquationSource Format="MATHML"><math> <mmultiscripts> <mrow /> <mi>κ</mi> <mrow /> </mmultiscripts> </math></EquationSource> </InlineEquation> associated with any given link pattern. As a corollary, for <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\kappa \in (4,8)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>4</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, we prove the existence of the multiple SLE partition functions, which are smooth functions satisfying a system of PDEs and conformal covariance. This was open for <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\kappa \in (6,8)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>κ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>6</mn> <mo>,</mo> <mn>8</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$N\ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation> prior to our work.</p>

Conformal welding of quantum disks and multiple SLE: the non-simple case

期刊论文
发表日期：

2025年9月27日

查看全文

Morris Ang,
Nina Holden,
Xin Sun,
Pu Yu

摘要

Two-pointed quantum disks with a weight parameter $W>0$ $W > 0$ is a canonical family of finite-volume random surfaces in Liouville quantum gravity. We prove that the conformal welding of the forested variant of this disk gives a two-pointed quantum disk with an independent SLE $_\kappa $ $_{κ}$ for $\kappa \in (4,8)$ $κ \in (4, 8)$ . Furthermore, we show that the conformal welding of multiple forested quantum disks gives a surface arising in Liouville conformal field theory decorated by multiple SLE $_\kappa $ $_{κ}$ for $\kappa \in (4,8)$ $κ \in (4, 8)$ , such that the random conformal modulus contains the SLE partition function as a multiplicative factor. In particular, this gives a construction of the multiple SLE $_\kappa $ $_{κ}$ associated with any given link pattern. As a corollary, for $\kappa \in (4,8)$ $κ \in (4, 8)$ , we prove the existence of the multiple SLE partition functions, which are smooth functions satisfying a system of PDEs and conformal covariance. This was open for $\kappa \in (6,8)$ $κ \in (6, 8)$ and $N\ge 3$ $N \geq 3$ prior to our work.