The Chekanov torus in $S^2 \times S^2$ is not real

We prove that the count of Maslov index $2$ $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold with minimal Maslov number at least two in a closed spherically monotone symplectic manifold must be even. As a corollary, we exhibit a genuine real symplectic phenomenon in terms of involutions, namely that the Chekanov torus $T_{\operatorname{Chek}}$ in $S^2 \times S^2$, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus $T_{\operatorname{Clif}}$, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.

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This work was supported by the Samsung Science and Technology Foundation under Project Number SSTFBA1901-01.

Received 6 October 2019

Accepted 7 May 2020

Published 26 March 2021