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arXiv · 2512.03142

Topological Einstein gravity as Kodaira-Spencer gravity

Johanna Erdmenger·Jonathan Karl·Jani Kastikainen·René Meyer·Henri Scheppach
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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.

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Johanna Erdmenger, Jonathan Karl, Jani Kastikainen, René Meyer, Henri Scheppach. 2025-12-02. Topological Einstein gravity as Kodaira-Spencer gravity. https://doi.org/10.1007/jhep05(2026)060

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