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2D Liouville quantum gravity (LQG) is used as a toy model for 4D quantumgravity and is the theory of world-sheet in string theory. Recently there hasbeen growing interest in studying LQG in the realm of probability theory:David, Kupiainen, Rhodes, Vargas (2014) and Duplantier, Miller, Sheffield(2014) both provide a probabilistic perspective of the LQG on the 2D sphere. Inparticular, in each of them one may find a definition of the so-called unitarea quantum sphere. We examine these two perspectives and prove theirequivalence by showing that the respective unit area quantum spheres are thesame. This is done by considering a unified limiting procedure for definingboth objects.

Two perspectives of the 2D unit area quantum sphere and their equivalence