Topics in Weyl Geometry and Quantum Anomalies

TL;DR

This thesis explores the geometric structures underlying Weyl geometry and quantum anomalies, extending holographic techniques and Lie algebroid theory to better understand anomalies and their geometric origins.

Contribution

It generalizes the Fefferman-Graham construction to Weyl geometry and applies Lie algebroid cohomology to analyze quantum anomalies in gauge theories.

Findings

Weyl-obstruction tensors appear as poles in holographic expansions.

Weyl anomalies can be constructed from Weyl-obstruction tensors.

Lie algebroid cohomology encodes BRST and anomaly structures.

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.