## The Michigan Mathematical Journal

### Givental-Type Reconstruction at a Nonsemisimple Point

#### Abstract

We consider the orbifold curve that 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.

#### Article information

Source
Michigan Math. J., Volume 67, Issue 2 (2018), 333-369.

Dates
Revised: 5 September 2017
First available in Project Euclid: 13 April 2018

https://projecteuclid.org/euclid.mmj/1523584849

Digital Object Identifier
doi:10.1307/mmj/1523584849

Mathematical Reviews number (MathSciNet)
MR3802257

Zentralblatt MATH identifier
06914766

#### Citation

Basalaev, Alexey; Priddis, Nathan. Givental-Type Reconstruction at a Nonsemisimple Point. Michigan Math. J. 67 (2018), no. 2, 333--369. doi:10.1307/mmj/1523584849. https://projecteuclid.org/euclid.mmj/1523584849

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