Compactification on Calabi-Yau threefolds: Consistent truncation to pure supergravity

TL;DR

This paper constructs and verifies consistent truncations of higher-dimensional supergravity theories on Calabi-Yau threefolds to lower-dimensional pure supergravity, enabling solutions like black holes to be embedded in higher dimensions.

Contribution

It provides explicit, universal truncations to pure supergravity from Calabi-Yau compactifications, independent of specific geometries, and demonstrates their consistency to all orders in fermions.

Findings

Constructed consistent truncations to pure supergravity in 4 and 5 dimensions.

Embedded 4D extremal black holes into higher-dimensional theories.

Proposed universal truncation Ansätze for supergravities with matter.

Abstract

We study compactifications of eleven- and ten-dimensional maximal supergravity on Calabi-Yau threefolds. We explicitly construct truncations to pure supergravity with eight supercharges in five and four dimensions and show that they are consistent, i.e. that every solution of the lower-dimensional equations of motion fully solves the higher-dimensional ones. We furthermore match the supersymmetry transformations and demonstrate the consistency to full non-linear order in fermions. Our construction is independent of the choice of Calabi-Yau threefold and only involves the universal structures such as the K\"ahler form and the holomorphic three-form, in agreement with implicit constructions in the generalised geometry literature. As an immediate application, we embed four-dimensional extremal black holes in the higher-dimensional supergravities. We furthermore propose Ans\"atze for consistent truncations on all universal structures, leading to supergravities with additional matter multiples. An extensive list of equations of motion and supersymmetry transformations for various supergravity theories is provided in the appendix.