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Moscow Mathematical Journal

Volume 23, Issue 3, July–September 2023  pp. 309–317.

A Formula For the Gromov--Witten Potential of an Elliptic Curve

Authors:  Alexandr Buryak
Author institution:Faculty of Mathematics, National Research University Higher School of Economics, 6 Usacheva str., Moscow, 119048, Russian Federation;
Center for Advanced Studies, Skolkovo Institute of Science and Technology, 1 Nobel str., Moscow, 143026, Russian Federation;
P.G. Demidov Yaroslavl State University, 14 Sovetskaya str., Yaroslavl, 150003, Russian Federation


An algorithm to determine all the Gromov–Witten invariants of any smooth projective curve was obtained by Okounkov and Pandharipande in 2006. They identified stationary invariants with certain Hurwitz numbers and then presented Virasoro type constraints that allow to determine all the other Gromov–Witten invariants in terms of the stationary ones. In the case of an elliptic curve, we show that these Virasoro type constraints can be explicitly solved leading to a very explicit formula for the full Gromov–Witten potential in terms of the stationary invariants.

2020 Math. Subj. Class. 14H10, 14H70.

Keywords: Elliptic curve, Gromov–Witten invariant, generating series.

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