Conformal welding for critical Liouville quantum gravity

Abstract

Consider two critical Liouville quantum gravity surfaces (i.e., γ -LQG for γ = 2), each with the topology of 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., γ -LQG for γ ∈ (0, 2)) is well-defined.Résumé. Considérons deux surfaces de gravité quantique de Liouville critiques (c'est-à-dire des surfaces γ -LQG avec γ = 2), chacune ayant la topologie de H et avec une longueur de bord infinie. Nous montrons qu'il existe p.s. une soudure conforme des deux surfaces quand les bords sont identifiés selon leur longueur de bord quantique. Ceci résulte en une surface critique LQG décorée d'un SLE 4 indépendant. En combinant ceci avec le résultat d'unicité d'une telle soudure, démontré récemment par McEnteggart, Miller, et Qian (2018), nous montrons que l'opération de soudure est bien définie. Notre résultat est un analogue dans le cas critique du théorème de la « fermeture éclair quantique » de Sheffield (2016), qui démontre le résultat similaire que l'opération de soudure conforme pour la LQG sous-critique (c'est-à-dire avec γ ∈ (0, 2)) est bien définie.

The critical Liouville quantum gravity metric induces the Euclidean topology

“…We refer [Pow20] for a survey of results on the critical LQG area measure. Critical LQG is also connected to Schramm-Loewner evolution (SLE κ ) at the critical value κ = 4 [HP18].…”

The trunks of CLE(4) explorations

“…An alternative strategy to prove our main results would have been to build on the results from [MSW20a] on asymmetric explorations of CLE κ 's for κ > 4, i.e., to take the limit κ ↓ 4 instead of κ ↑ 4. Yet another option would have been to try to directly derive results about CLE κ decorations on LQG surfaces with parameter γ = √ κ for κ = 4 and γ = 2, building on the LQG theory [HP18] in that case. But it seems that given the current literature, the approach chosen in the present paper is the shortest one to derive our results and we believe that understanding how to approximate CLE 4 explorations by other CLE κ explorations is also interesting on its own right.…”

Liouville conformal field theory and the quantum zipper

“…SLE κ is a simple curve when κ ≤ 4, self-intersecting when κ ∈ (4, 8), and space-filling when κ ≥ 8. When κ ∈ (0, 4] there is an infinite-volume γ-LQG surface which, when decorated by an independent SLE κ curve, is invariant in law under the operation of conformally welding the two boundary arcs according to their random length measures; this is called the quantum zipper [She16,HP21,KMS22]. Similar stories hold for other ranges of κ when the boundary of the LQG surface is modified to have non-trivial topology [DMS21].…”

“…The pair (η s , ĝs ) is unique modulo conformal automorphisms of H, so specifying the hydrodynamic normalization lim z→∞ ĝs (z) − z = 0 uniquely defines (η s , ĝs ). The existence and uniqueness of (η s , ĝs ) was shown in [She16] for γ < 2; for γ = 2 existence was established by [HP21] and uniqueness by [KMS22] (see also [MMQ18]). Thus, for φ 0 ∼ LF…”