Mirror symmetry for open $r$-spin invariants

We show that a generating function for open $r$-spin enumerative invariants produces a universal unfolding of the polynomial $x^r$. Further, the coordinates parametrizing this universal unfolding are flat coordinates on the Frobenius manifold associated to the Landau–Ginzburg model $(\mathbb{C}, x^r)$ via Saito–Givental theory. This result provides evidence for the same phenomenon to occur in higher dimension, proven in the sequel $\href{https://arxiv.org/abs/2203.02435}{[\textrm{GKT}22]}$.

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The first author acknowledges support from the EPSRC under Grants EP/N03189X/1, a Royal Society Wolfson Research Merit Award, and the ERC Advanced Grant MSAG.

The second author acknowledges that this paper is based upon work supported by the UKRI and EPSRC under fellowships MR/T01783X/1 and EP/N004922/2.

The third author, incumbent of the Lillian and George Lyttle Career Development Chair, acknowledges support provided by the ISF grant No. 335/19 and by a research grant from the Center for New Scientists of Weizmann Institute.

Received 13 April 2022

Received revised 28 September 2023

Accepted 10 December 2023

Published 3 April 2024