Metric gluing of Brownian and 8/3....-Liouville quantum gravity surfaces.

In a recent series of works, Miller and Sheffield constructed a metric on 8/3....-Liouville quantum gravity (LQG) under which 8/3.... -LQG surfaces (e.g., the LQG sphere, wedge, cone and disk) are isometric to their Brownian surface counterparts (e.g., the Brownian map, half-plane, plane and disk). We identify the metric gluings of certain collections of independent 8/3.... -LQG surfaces with boundaries identified together according to LQG length along their boundaries. Our results imply in particular that the metric gluing of two independent instances of the Brownian half-plane along their positive boundaries is isometric to a certain LQG wedge decorated by an independent chordal SLE8/3 curve. If one identifies the two sides of the boundary of a single Brownian half-plane, one obtains a certain LQG cone decorated by an independent whole-plane SLE8/3. If one identifies the entire boundaries of two Brownian half-planes, one obtains a different LQG cone and the interface between them is a two-sided variant of whole-plane SLE8/3 . Combined with another work of the authors, the present work identifies the scaling limit of self-avoiding walk on random quadrangulations with SLE8/3 on 8/3....-LQG.