Givental-type reconstruction at a non-semisimple point

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.