Givental-type reconstruction at a non-semisimple point

TL;DR

This paper develops a Givental-type reconstruction method for the Gromov-Witten theory of a specific orbifold curve, employing mirror symmetry and FJRW theory, and demonstrates its application to CY/LG correspondence.

Contribution

It introduces a systematic Givental-type reconstruction approach for orbifold curves using mirror symmetry and FJRW theory, applicable to non-semisimple cases.

Findings

Reconstruction of Gromov-Witten theory via Givental's action.

Establishment of CY/LG correspondence for the orbifold curve.

Recovery of FJRW theory from point theories using the developed methods.

Abstract

In this paper we consider the orbifold curve, which is a quotient of an elliptic curve $\mathcal{E}$ by a cyclic group of order 4. We develop a systematic way to obtain a Givental-type reconstruction of Gromov-Witten theory of the orbifold curve via the product of the Gromov-Witten theories of a point. This is done by employing mirror symmetry and certain results in FJRW theory. In particular, we present the particular Givental's action giving the CY/LG correspondence between the Gromov-Witten theory of the orbifold curve $\mathcal{E} / \mathbb{Z}_4$ and FJRW theory of the pair defined by the polynomial $x^4+y^4+z^2$ and the maximal group of diagonal symmetries. The methods we have developed can easily be applied to other finite quotients of an elliptic curve. Using Givental's action we also recover this FJRW theory via the product of the Gromov-Witten theories of a point. Combined with the CY/LG action we get a result in "pure" Gromov-Witten theory with the help of modern mirror symmetry conjectures.