Gravity in the 3+1-Split Formalism II: Self-Duality and the Emergence of the Gravitational Chern-Simons in the Boundary

TL;DR

The paper analyzes self-duality in gravity with a cosmological constant within the 3+1-split formalism and uncovers a holographic link to the three-dimensional gravitational Chern–Simons theory. In Lorentzian signature, bulk self-dual (conformally flat) geometries are determined by boundary CS data (Weyl vacua), while in Euclidean signature self-dual configurations are governed by CS boundary data, with the boundary stress tensor related to the Cotton tensor. The gravitational CS thus acts as an exact generating functional (or leading effective action) for the boundary theory in AdS$_4$/CFT$_3$, establishing a precise holographic description of self-dual gravity through 3D CS dynamics. The work provides concrete formulations, including FG expansions and explicit self-dual solutions (e.g., Eguchi–Hanson and Fubini–Study metrics), bridging 4D self-dual gravity and 3D CS dynamics and highlighting pathways to CDM-like holographic structures in related theories.

Abstract

We study self-duality in the context of the 3+1-split formalism of gravity with non-zero cosmological constant. Lorentzian self-dual configurations are conformally flat spacetimes and have boundary data determined by classical solutions of the three-dimensional gravitational Chern-Simons. For Euclidean self-dual configurations, the relationship between their boundary initial positions and initial velocity is also determined by the three-dimensional gravitational Chern-Simons. Our results imply that bulk self-dual configurations are holographically described by the gravitational Chern-Simons theory which can either viewed as a boundary generating functional or as a boundary effective action.