The Michigan Mathematical Journal

Givental-Type Reconstruction at a Nonsemisimple Point

Alexey Basalaev and Nathan Priddis

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We consider the orbifold curve that is a quotient of an elliptic curve 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 E/Z4 and FJRW theory of the pair defined by the polynomial x4+y4+z2 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

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

Received: 19 September 2016
Revised: 5 September 2017
First available in Project Euclid: 13 April 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J33: Mirror symmetry [See also 11G42, 53D37] 14N35: Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants [See also 53D45]
Secondary: 14B05: Singularities [See also 14E15, 14H20, 14J17, 32Sxx, 58Kxx]


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.

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