# Conformal welding for critical Liouville quantum gravity

## TL;DR

This paper proves the existence and uniqueness of conformal welding for critical Liouville quantum gravity surfaces with boundary, resulting in a surface decorated by an SLE$_4$, extending subcritical results to the critical case.

## Contribution

It establishes the well-defined conformal welding for critical LQG surfaces, a significant extension of Sheffield's subcritical quantum gravity zipper theorem.

## Key findings

- Existence of conformal welding for critical LQG surfaces.
- Uniqueness of the conformal welding operation.
- Critical LQG surfaces decorated by SLE$_4$ obtained through welding.

## Abstract

Consider two critical Liouville quantum gravity surfaces (i.e., $\gamma$-LQG for $\gamma=2$), each with the topology of $\mathbb{H}$ and with infinite boundary length. We prove that there a.s. exists a conformal welding of the two surfaces, when the boundaries are identified according to quantum boundary length. This results in a critical LQG surface decorated by an independent SLE$_4$. Combined with the proof of uniqueness for such a welding, recently established by McEnteggart, Miller, and Qian (2018), this shows that the welding operation is well-defined. Our result is a critical analogue of Sheffield's quantum gravity zipper theorem (2016), which shows that a similar conformal welding for subcritical LQG (i.e., $\gamma$-LQG for $\gamma\in(0,2)$) is well-defined.