Conformal welding for critical Liouville quantum gravity

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