The many symmetries of Calabi-Yau compactifications

TL;DR

This paper reviews the mathematical structures involved in Calabi-Yau compactifications of supergravity, emphasizing symplectic symmetry and its topological origins, to aid both beginners and experts in understanding these complex geometries.

Contribution

It provides a detailed, pedagogical review of the symplectic structure and topological aspects of Calabi-Yau compactifications in supergravity, filling a gap in existing literature.

Findings

Reproduces the full calculation of dimensional reduction

Highlights the role of symplectic symmetry from topology

Clarifies the structure of hypermultiplets in supergravity

Abstract

We review the major mathematical concepts involved in the dimensional reduction of D=11 N=1 supergravity theory over a Calabi-Yau manifold with non-trivial complex structure moduli resulting in ungauged D=5 N=2 supergravity theory with hypermultiplets. This last has a particularly rich structure with many underlying geometries. We reproduce the entire calculation and particularly emphasize its symplectic symmetry and how that arises from the topology of the underlying subspace. The review is intended to fill in a specific gap in the literature with the hope that it would be useful to both the beginner and the expert alike.