Previous works in this series have shown that an instance of a \(\sqrt{8/3}\)-Liouville quantum gravity (LQG) sphere has a well-defined distance function, and that the resulting metric measure space (mm-space) agrees in law with the Brownian map (TBM). In this work, we show that given just the mm-space structure, one can a.s. recover the LQG sphere. This implies that there is a canonical way to parameterize an instance of TBM by the Euclidean sphere (up to Möbius transformation). In other words, an instance of TBM has a canonical conformal structure. The conclusion is that TBM and the \(\sqrt{8/3}\)-LQG sphere are equivalent. They ultimately encode the same structure (a topological sphere with a measure, a metric, and a conformal structure) and have the same law. From this point of view, the fact that the conformal structure a.s. determines the metric and vice-versa can be understood as a property of this unified law. The results of this work also imply that the analogous facts hold for Brownian and \(\sqrt{8/3}\)-LQG surfaces with other topologies.

本系列的先前工作表明，\（\ sqrt {8/3} \）- Liouville量子引力（LQG）球的实例具有定义明确的距离函数，并且所得到的度量单位空间（mm-space） ）在法律上同意布朗地图（TBM）。在这项工作中，我们表明，仅给出毫米空间结构，就可以恢复LQG球体。这意味着有一种规范的方法可以通过欧几里得球（直到莫比乌斯变换）对一个TBM实例进行参数化。换句话说，TBM实例具有规范的保形结构。结论是，TBM和\（\ sqrt {8/3} \）- LQG球面是等效的。它们最终编码相同的结构（带有度量，度量和保形结构）并具有相同的定律。从这个角度来看，保形结构决定度量，反之亦然的事实可以理解为该统一法则的属性。这项工作的结果也暗示类似事实适用于具有其他拓扑的Brownian和\（\ sqrt {8/3} \）- LQG曲面。