On real Calabi-Yau threefolds twisted by a section

TL;DR

This paper investigates the mod 2 cohomology of real Calabi-Yau threefolds with real structures twisted by a Lagrangian section, extending previous results and connecting cohomology with mirror symmetry through a twisted divisor squaring operation.

Contribution

It extends the cohomological analysis of real Calabi-Yau threefolds to twisted structures and links these to mirror symmetry via a novel divisor squaring map.

Findings

Exact sequences linking real and complex Calabi-Yau cohomology.

Identification of the twisting homomorphism via a twisted divisor squaring.

Example of a connected (M-2)-real quintic threefold.

Abstract

We study the mod $2$ cohomology of real Calabi-Yau threefolds given by real structures which preserve the torus fibrations constructed by Gross. We extend the results of Casta\~no-Bernard-Matessi and Arguz-Prince to the case of real structures twisted by a Lagrangian section. In particular we find exact sequences linking the cohomology of the real Calabi-Yau with the cohomology of the complex one. Applying SYZ mirror symmetry, we show that the connecting homomorphism is determined by a ``twisted squaring of divisors'' in the mirror Calabi-Yau, i.e. by $D \mapsto D^2 + DL$ where $D$ is a divisor in the mirror and $L$ is the divisor mirror to the twisting section. We use this to find an example of a connected $(M-2)$-real quintic threefold.