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Topological Einstein gravity as Kodaira-Spencer gravity

Johanna Erdmenger, Jonathan Karl, Jani Kastikainen, René Meyer, Henri Scheppach

TL;DR

The paper establishes a classical, explicit map between Euclidean 3D Einstein gravity with negative cosmological constant and the SU(2)-invariant sector of Kodaira–Spencer gravity on a six-dimensional SU(2) principal bundle. By uplifting 3D CS data to a six-dimensional flat connection, it identifies deformations of the complex structure with KS fields and shows that Einstein solutions correspond to integrable KS deformations, embedding 3D gravity into topological string theory. The construction yields a Calabi–Yau background naturally from a reference Einstein solution and furnishes explicit uplift of Bañados geometries, linking Schwarzian boundary data to KS moduli. The work lays groundwork for quantum extensions and holographic interpretations within twisted holography, potentially illuminating AdS3/CFT2 and topological string dualities.

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 Bañados solutions. Our construction embeds three-dimensional gravity into topological string theory and AdS$_3$/CFT$_2$ duality into twisted holography.

Topological Einstein gravity as Kodaira-Spencer gravity

TL;DR

The paper establishes a classical, explicit map between Euclidean 3D Einstein gravity with negative cosmological constant and the SU(2)-invariant sector of Kodaira–Spencer gravity on a six-dimensional SU(2) principal bundle. By uplifting 3D CS data to a six-dimensional flat connection, it identifies deformations of the complex structure with KS fields and shows that Einstein solutions correspond to integrable KS deformations, embedding 3D gravity into topological string theory. The construction yields a Calabi–Yau background naturally from a reference Einstein solution and furnishes explicit uplift of Bañados geometries, linking Schwarzian boundary data to KS moduli. The work lays groundwork for quantum extensions and holographic interpretations within twisted holography, potentially illuminating AdS3/CFT2 and topological string dualities.

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 -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 Bañados solutions. Our construction embeds three-dimensional gravity into topological string theory and AdS/CFT duality into twisted holography.

Paper Structure

This paper contains 61 sections, 272 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Summary of results
  3. Complex structures on principal bundles.
  4. Relation between 3D Einstein and KS gravities.
  5. Matching of symmetries.
  6. Uplift of Bañados solutions.
  7. Review of three-dimensional Einstein gravity
  8. Metric and Chern--Simons formulations
  9. First-order formulation.
  10. Chern--Simons formulation.
  11. Local symmetries.
  12. Solutions of three-dimensional Einstein gravity
  13. Metric formulation.
  14. Chern--Simons formulation.
  15. Review of Kodaira--Spencer gravity
  16. ...and 46 more sections

Figures (1)

  • Figure 1: Principal bundle geometry $P_6$ used in the uplift of geometric data from $B_3$. $y$ denotes a set of coordinates of $B_3$ and $\theta$ a set of coordinates of the compact $S^3$ directions.