Unwinding toric degenerations and mirror symmetry for Grassmannians

TL;DR

This paper extends mirror symmetry concepts from Fermat hypersurfaces to Calabi-Yau hypersurfaces in Grassmannians, exploring group actions, toric degenerations, and GIT to relate mirror pairs.

Contribution

It generalizes mirror symmetry and toric degeneration techniques to Grassmannians with group actions, providing a new compactification of the mirror.

Findings

Established relations between Calabi-Yau hypersurfaces in Grassmannians and their quotients.

Described a compactification of the mirror inside a blow-up of the quotient Grassmannian.

Connected GIT variation and toric degenerations in the context of Grassmannian mirror symmetry.

Abstract

The most fundamental example of mirror symmetry compares the Fermat hypersurfaces in P^n and P^n/G, where G is a finite group that acts on P^n and preserves the Fermat hypersurface. We generalise this to hypersurfaces in Grassmannians, where the picture is richer and more complex. There is a finite group G that acts on the Grassmannian Gr(n,r) and preserves an appropriate Calabi-Yau hypersurface. We establish how mirror symmetry, toric degenerations, blow-ups and variation of GIT relate the Calabi-Yau hypersurfaces inside Gr(n,r) and Gr(n,r)/G. This allows us to describe a compactification of the Eguchi-Hori-Xiong mirror to the Grassmannian, inside a blow-up of the quotient of the Grassmannian by G.