The Quantum Mirror to the Quartic del Pezzo Surface

A log Calabi--Yau surface $(X,D)$ is given by a smooth projective surface $X$, together with an anti-canonical cycle of rational curves $D \subset X$. The homogeneous coordinate ring of the mirror to such a surface, or to the complement $X\setminus D$, is constructed in the work of Gross-Hacking-Keel, following previous work of Gross-Siebert, using wall structures, and it is generated by theta functions. In our work with Mark Gross we had provided a recipe to concretely compute these theta functions from a combinatorially constructed wall structure in arbitrary dimensions. In this paper, we first apply this recipe to obtain theta functions and equations for the mirror to the quartic del Pezzo surface, denoted by $dP_4$, together with an anti-canonical cycle of $4$ rational curves. We then describe the deformation quantization of this coordinate ring, following the work of Bousseau. This gives a non-commutative algebra, generated by quantum theta functions. There is a totally different approach, due to Chekhov-Mazzocco-Rubtsov, to construct the deformation quantization using the realization of the mirror as the monodromy manifold of the Painlev\'e IV equation. We show that these two approaches agree.