Topics in Weyl Geometry and Quantum Anomalies

TL;DR

The thesis extends holographic and anomaly formalisms by embedding Weyl geometry into ambient-space methods, defining Weyl-obstruction tensors as Weyl-covariant building blocks for the holographic Weyl anomaly up to eight dimensions. It introduces the Weyl-ambient construction that yields a codimension-2 Weyl manifold on the boundary and demonstrates a consistent, Weyl-covariant holographic dictionary via the Weyl-Fefferman–Graham gauge. In Part II, the BRST formalism is geometrized with Atiyah and general Lie algebroids, revealing that BRST cohomology is naturally captured by Lie algebroid cohomology, unifying consistent and covariant anomalies through a common geometric framework. The results illuminate how Weyl structure and ambient geometry organize higher-dimensional anomalies and suggest broader applications to corner symmetries, celestial holography, and holographic hydrodynamics. Overall, the work offers a coherent ambient-geometric perspective that links holographic Weyl anomalies with a Lie-algebroid view of quantum anomalies, providing new tools for understanding symmetry breaking in curved backgrounds and gauge theories.

Abstract

The first part of this thesis focuses on the Weyl-covariant nature of holography. We generalize the Fefferman-Graham ambient construction for conformal geometry to a corresponding construction for Weyl geometry. Through the Weyl-ambient construction, we investigate Weyl-covariant quantities on the Weyl manifold and define Weyl-obstruction tensors. We show that Weyl-obstruction tensors appear as poles in the Fefferman-Graham expansion of the AlAdS bulk metric for even boundary dimensions. Under holographic renormalization in the Weyl-Fefferman-Graham gauge, we compute the Weyl anomaly of the boundary theory in multiple dimensions and demonstrate that Weyl-obstruction tensors can be used as the building blocks for the Weyl anomaly of the dual quantum field theory. The holographic calculation with a background Weyl geometry also suggests an underlying geometric interpretation of the Weyl anomaly. The second part of this thesis is devoted to understanding the geometric nature of the BRST formalism and quantum anomalies. Using the language of Lie algebroids, the BRST complex can be encoded in the exterior algebra of an Atiyah Lie algebroid derived from the principal bundle of the gauge theory. We showed that the cohomology of an Atiyah Lie algebroid in a trivialization gives rise to the BRST cohomology. We then apply the Lie algebroid cohomology in studying quantum anomalies and demonstrate the computation for chiral and Lorentz-Weyl anomalies. In particular, we pay close attention to the fact that the geometric intuition afforded by the Lie algebroid (which was absent in the traditional BRST complex) provides hints of a deeper picture that simultaneously geometrizes the consistent and covariant forms of the anomaly. In the algebroid construction, the difference between the consistent and covariant anomalies is simply a different choice of basis.

Figures (3)

• 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) Jia:2023gmk.
• Figure 2: A connection on $A$ gives a global split $A=H\oplus V$, which locally can be viewed as determined by a gauge field $b$ defined with respect to "axes" corresponding to sub-bundles $TM$ and $L$Jia:2023tki.
• Figure 3: The three-legged stool of the Weyl anomaly.

Theorems & Definitions (42)

• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Theorem 3.1
• Theorem 3.2
• Lemma 3.3
• proof
• Lemma 3.4
• proof