The Quantum Mirror to the Quartic del Pezzo Surface

TL;DR

This paper constructs the mirror to the quartic del Pezzo surface using theta functions, and explores its deformation quantization, showing agreement between two different approaches to this quantization.

Contribution

It applies a combinatorial recipe to explicitly compute the mirror's theta functions and equations for the quartic del Pezzo surface, and demonstrates the equivalence of two quantization methods.

Findings

Explicit theta functions and equations for the mirror of the quartic del Pezzo surface.

Deformation quantization results in a non-commutative algebra generated by quantum theta functions.

Confirmation that two different approaches to deformation quantization agree.

Abstract

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.