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As announced in Gross and Siebert (in Algebraic geometry: Salt Lake City 2015, Proceedings of Symposia in Pure Mathematics, vol 97, no 2. AMS, Providence, pp 199–230, 2018) in 2016, we construct and prove consistency of the canonical wall structure. This construction starts with a log Calabi–Yau pair (X, D) and produces a wall structure, as defined in Gross et al. (Mem. Amer. Math. Soc. 278(1376), 1376, 1–103, 2022). Roughly put, the canonical wall structure is a data structure which encodes an algebro-geometric analogue of counts of Maslov index zero disks. These enumerative invariants are defined in terms of the punctured invariants of Abramovich et al. (Punctured Gromov–Witten invariants, 2020. arXiv:2009.07720v2 [math.AG]). There are then two main theorems of the paper. First, we prove consistency of the canonical wall structure, so that, using the setup of Gross et al. (Mem. Amer. Math. Soc. 278(1376), 1376, 1–103, 2022), the canonical wall structure gives rise to a mirror family. Second, we prove that this mirror family coincides with the intrinsic mirror constructed in Gross and Siebert (Intrinsic mirror symmetry, 2019. arXiv:1909.07649v2 [math.AG]). While the setup of this paper is narrower than that of Gross and Siebert (Intrinsic mirror symmetry, 2019. arXiv:1909.07649v2 [math.AG]), it gives a more detailed description of the mirror. [ABSTRACT FROM AUTHOR]

We study the higher genus equivariant Gromov--Witten theory of the Hilbert scheme of n points of C². Since the equivariant quantum cohomology, computed by Okounkov and Pandharipande [Invent. Math. 179 (2010), 523-557], is semisimple, the higher genus theory is determined by an R-matrix via the Givental--Teleman classification of Cohomological Field Theories (CohFTs). We uniquely specify the required R-matrix by explicit data in degree 0. As a consequence, we lift the basic triangle of equivalences relating the equivariant quantum cohomology of the Hilbert scheme Hilbn"(C²) and the Gromov--Witten/Donaldson--Thomas correspondence for 3-fold theories of local curves to a triangle of equivalences in all higher genera. The proof uses the analytic continuation of the fundamental solution of the QDE of the Hilbert scheme of points determined by Okounkov and Pandharipande [Transform. Groups 15 (2010), 965-982]. The GW/DT edge of the triangle in higher genus concerns new CohFTs defined by varying the 3-fold local curve in the moduli space of stable curves. The equivariant orbifold Gromov--Witten theory of the symmetric product Symn"(C²) is also shown to be equivalent to the theories of the triangle in all genera. The result establishes a complete case of the crepant resolution conjecture [Bryan and Graber, Algebraic Geometry--Seattle 2005, Part 1, Proceedings of Symposia in Pure Mathematics, 80 (American Mathematical Society, Providence, RI, 2009), 23-42; Coates et al., Geom. Topol. 13 (2009), 2675-2744; Coates & Ruan, Ann. Inst. Fourier (Grenoble) 63 (2013), 431-478]. [ABSTRACT FROM AUTHOR]