Topological Einstein gravity as Kodaira-Spencer gravity

TL;DR

This paper demonstrates how three-dimensional Einstein gravity with negative cosmological constant can be embedded into Kodaira-Spencer gravity, linking 3D gravity, topological string theory, and holography.

Contribution

It establishes a classical uplift of 3D Einstein gravity to Kodaira-Spencer gravity on Calabi-Yau three-folds, connecting gravity solutions to complex structure deformations.

Findings

Explicit uplift of Ba ilde{n}ados solutions.

Embedding of 3D gravity into topological string theory.

Connection to AdS$_3$/CFT$_2$ duality and twisted holography.

Abstract

As a contribution towards quantizing three-dimensional gravity, we show at the classical level that Euclidean three-dimensional Einstein gravity with a negative cosmological constant is uplifted to the $SU(2)$-invariant sector of Kodaira-Spencer gravity on a Calabi-Yau three-fold. Kodaira-Spencer gravity appears in the target space description of the B-model topological string theory and describes deformations of a complex structure. We prove that given a reference solution of Einstein gravity in the first-order formulation, a second off-shell configuration uplifts to a unique complex structure deformation in six dimensions. If the configuration satisfies Einstein's equations, the complex structure deformation is integrable, i.e. a solution of Kodaira-Spencer gravity. We demonstrate the uplift explicitly for Ba\~nados solutions. Our construction embeds three-dimensional gravity into topological string theory and AdS$_3$/CFT$_2$ duality into twisted holography.