  In a recent series of works, Miller and Sheffield constructed a metric on
$\sqrt{8/3}$-Liouville quantum gravity (LQG) under which $\sqrt{8/3}$-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
$\sqrt{8/3}$-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 SLE$_{8/3}$ 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 SLE$_{8/3}$. 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
SLE$_{8/3}$.
  Combined with another work of the authors, the present work identifies the
scaling limit of self-avoiding walk on random quadrangulations with SLE$_{8/3}$
on $\sqrt{8/3}$-LQG.
