Real Lagrangians in Calabi–Yau threefolds

We compute the mod 2 cohomology groups of real Lagrangians in torus fibrations on Calabi–Yau threefolds constructed by Gross. To do this we study a long exact sequence introduced by Castaño‐Bernard–Matessi, which relates the cohomology of the Lagrangians to the cohomology of the Calabi–Yau. We show that the connecting homomorphism in this sequence is given by the map squaring divisor classes in the mirror Calabi–Yau.