We prove the genus-one restriction of the all-genus Landau–Ginzburg/Calabi–Yau conjecture of Chiodo and Ruan, stated in terms of the geometric quantization of an explicit symplectomorphism determined by genus-zero invariants. This gives the first evidence supporting the higher-genus Landau–Ginzburg/Calabi–Yau correspondence for the quintic $3$-fold, and exhibits the first instance of the ‘genus zero controls higher genus’ principle, in the sense of Givental’s quantization formalism, for non-semisimple cohomological field theories.

We study a class of flat bundles, of finite rank $N$, which arise naturally from the Donaldson–Thomas theory of a Calabi–Yau threefold $X$ via the notion of a variation of BPS structure. We prove that in a large $N$ limit their flat sections converge to the solutions to certain infinite-dimensional Riemann–Hilbert problems recently found by Bridgeland. In particular this implies an expression for the positive degree, genus 0 Gopakumar–Vafa contribution to the Gromov–Witten partition function of $X$ in terms of solutions to confluent hypergeometric differential equations.

Gromov–Witten invariants have been constructed to be deformation invariant, but their behavior under other transformations is subtle. We show that logarithmic Gromov–Witten invariants are also invariant under appropriately defined logarithmic modifications.

We prove that the tautological ring of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$, the moduli space of $n$-pointed genus two curves of compact type, does not have Poincaré duality for any $n\geqslant 8$. This result is obtained via a more general study of the cohomology groups of ${\mathcal{M}}_{2,n}^{\mathsf{ct}}$. We explain how the cohomology can be decomposed into pieces corresponding to different local systems and how the tautological cohomology can be identified within this decomposition. Our results allow the computation of $H^{k}({\mathcal{M}}_{2,n}^{\mathsf{ct}})$ for any $k$ and $n$ considered both as $\mathbb{S}_{n}$-representation and as mixed Hodge structure/$\ell$-adic Galois representation considered up to semi-simplification. A consequence of our results is also that all even cohomology of $\overline{{\mathcal{M}}}_{2,n}$ is tautological for $n<20$, and that the tautological ring of $\overline{{\mathcal{M}}}_{2,n}$ fails to have Poincaré duality for all $n\geqslant 20$. This improves and simplifies results of the author and Orsola Tommasi.

We prove a Givental-style mirror theorem for toric Deligne–Mumford stacks ${\mathcal{X}}$. This determines the genus-zero Gromov–Witten invariants of ${\mathcal{X}}$ in terms of an explicit hypergeometric function, called the $I$-function, that takes values in the Chen–Ruan orbifold cohomology of ${\mathcal{X}}$.

We explicitly calculate some Gromov–Witten correspondences determined by maps of labelled curves of genus 0 to the moduli spaces of labelled curves of genus 0. We consider these calculations as the first step towards studying the self-referential nature of motivic quantum cohomology.