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

TL;DR

The paper demonstrates that the Chekanov torus in S^2×S^2 cannot be realized as a fixed point set of an antisymplectic involution, revealing a new real symplectic phenomenon related to Maslov index 2 holomorphic discs.

Contribution

It establishes that the Chekanov torus is not the fixed point set of any antisymplectic involution, highlighting a novel real symplectic property involving Maslov index 2 discs.

Findings

Maslov index 2 disc counts are even for real Lagrangians

Chekanov torus cannot be realized as an antisymplectic involution fixed point set

Reveals a new phenomenon in real symplectic topology

Abstract

We prove that the count of Maslov index 2 $J$-holomorphic discs passing through a generic point of a real Lagrangian submanifold 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 $\mathbb{T}_{\text{Chek}}$ in $S^2\times S^2$, which is a monotone Lagrangian torus not Hamiltonian isotopic to the Clifford torus $\mathbb{T}_{\text{Clif}}$, can be seen as the fixed point set of a smooth involution, but not of an antisymplectic involution.