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Weyl-Ambient Geometries

Weizhen Jia, Manthos Karydas, Robert G. Leigh

TL;DR

The paper develops a Weyl-generalized ambient construction by introducing a Weyl-ambient space $( ilde{M}, ilde{g})$ that extends the Fefferman–Graham ambient metric to Weyl manifolds. From a top-down viewpoint, it is shown that the Weyl-ambient space induces a codimension-2 Weyl manifold $(M,[g,a])$, where the boundary data include an induced metric $ ilde{ m g}^{(0)}_{ij}= ilde{ m g}_{ij}|_{ ho=0,t=1}={ m g}^{(0)}_{ij}$ and a Weyl connection $a^{(0)}_i=a_i|_{ ho=0}$, with ambient Weyl diffeomorphisms corresponding to Weyl transformations on $M$. A bottom-up construction then treats a given Weyl manifold $(M,[g,a])$ as the initial surface for a unique Weyl-ambient space in Weyl-normal form, proving the existence and uniqueness of $ ilde{g}$ under a Ricci-flatness condition and an initial data set $(g_{ij},a_i, ext{acceleration})$, thereby enabling a perturbative expansion with Weyl-obstruction tensors emerging as poles in the ambient expansion. The work defines extended Weyl-obstruction tensors $\nhat{oldsymbol{ m m oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{oldsymbol{ extit{}}}}}}}}}}}}}}}}}$ from ambient curvature and proves their Weyl covariance, establishing a precise link between ambient geometry and Weyl-covariant data on $M$. Applications to holography, corner symmetries, and Carroll structures are discussed as potential extensions. The paper thus provides a robust ambient framework that unifies Weyl geometry with conformal ambient techniques and offers a rigorous route to Weyl anomalies via extended Weyl-obstruction tensors.

Abstract

Weyl geometry is a natural extension of conformal geometry with Weyl covariance mediated by a Weyl connection. We generalize the Fefferman-Graham (FG) ambient construction for conformal manifolds to a corresponding construction for Weyl manifolds. We first introduce the Weyl-ambient metric motivated by the Weyl-Fefferman-Graham (WFG) gauge. From a top-down perspective, we show that the Weyl-ambient space as a pseudo-Riemannian geometry induces a codimension-2 Weyl geometry. Then, from a bottom-up perspective, we start from promoting a conformal manifold into a Weyl manifold by assigning a Weyl connection to the principal $\mathbb{R}_+$-bundle realizing a Weyl structure. We show that the Weyl structure admits a well-defined initial value problem, which determines the Weyl-ambient metric. Through the Weyl-ambient construction, we also investigate Weyl-covariant tensors on the Weyl manifold and define extended Weyl-obstruction tensors explicitly.

Weyl-Ambient Geometries

TL;DR

The paper develops a Weyl-generalized ambient construction by introducing a Weyl-ambient space that extends the Fefferman–Graham ambient metric to Weyl manifolds. From a top-down viewpoint, it is shown that the Weyl-ambient space induces a codimension-2 Weyl manifold , where the boundary data include an induced metric and a Weyl connection , with ambient Weyl diffeomorphisms corresponding to Weyl transformations on . A bottom-up construction then treats a given Weyl manifold as the initial surface for a unique Weyl-ambient space in Weyl-normal form, proving the existence and uniqueness of under a Ricci-flatness condition and an initial data set , thereby enabling a perturbative expansion with Weyl-obstruction tensors emerging as poles in the ambient expansion. The work defines extended Weyl-obstruction tensors from ambient curvature and proves their Weyl covariance, establishing a precise link between ambient geometry and Weyl-covariant data on . Applications to holography, corner symmetries, and Carroll structures are discussed as potential extensions. The paper thus provides a robust ambient framework that unifies Weyl geometry with conformal ambient techniques and offers a rigorous route to Weyl anomalies via extended Weyl-obstruction tensors.

Abstract

Weyl geometry is a natural extension of conformal geometry with Weyl covariance mediated by a Weyl connection. We generalize the Fefferman-Graham (FG) ambient construction for conformal manifolds to a corresponding construction for Weyl manifolds. We first introduce the Weyl-ambient metric motivated by the Weyl-Fefferman-Graham (WFG) gauge. From a top-down perspective, we show that the Weyl-ambient space as a pseudo-Riemannian geometry induces a codimension-2 Weyl geometry. Then, from a bottom-up perspective, we start from promoting a conformal manifold into a Weyl manifold by assigning a Weyl connection to the principal -bundle realizing a Weyl structure. We show that the Weyl structure admits a well-defined initial value problem, which determines the Weyl-ambient metric. Through the Weyl-ambient construction, we also investigate Weyl-covariant tensors on the Weyl manifold and define extended Weyl-obstruction tensors explicitly.
Paper Structure (18 sections, 12 theorems, 150 equations, 1 figure, 1 table)

This paper contains 18 sections, 12 theorems, 150 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Ambient Metrics
  3. Flat Ambient Metrics
  4. Ricci-Flat Ambient Metrics
  5. Weyl-Ambient Metric
  6. Weyl-Ambient Space: Top-Down Perspective
  7. Induced Weyl Geometry
  8. Weyl-Obstruction Tensors: First Order Formalism
  9. Weyl-Ambient Space: Bottom-Up Perspective
  10. Formal Theory
  11. Weyl-Obstruction Tensors: Second Order Formalism
  12. Conclusions
  13. Coordinate Systems of the Flat Ambient Space
  14. Details of Null Frame Calculations
  15. Proofs
  16. ...and 3 more sections

Key Result

Proposition 3.1

Let $IJKLM_{1}\dots M_{r}$ be a list of indices, $s_{+}$ of which are $+$, $s_{M}$ of which correspond to $x^i$, and $s_{-}$ of which are $-$, then under the ambient Weyl diffeomorphism eq:Weyldiff, we have

Figures (1)

  • Figure 1: Sketch of a constant-$\rho$ surface (red) and a constant-$t$ surface (green) of the flat ambient metric \ref{['Flat_Ambient_2']} in the Lorentzian coordinate system $\{X^{0}, X^{i}\}$. Constant-$t$ surfaces are past directed light cones. Changing $t$ moves the apex $P$ of the cone along the $X^{0}$-axes. Constant-$\rho$ surfaces are future directed timelike cones. When $\rho\to 0^{-}$ the constant $\rho$ surface becomes the light cone ${\cal N}^{+}$ (blue).

Theorems & Definitions (31)

  • Proposition 3.1
  • proof
  • Definition 1
  • Proposition 3.2
  • proof
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 4.1
  • ...and 21 more