University of Illinois Urbana-Champaign

Topics in Weyl geometry and quantum anomalies

Jia, Weizhen

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JIA-DISSERTATION-2024.pdf

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https://hdl.handle.net/2142/125555

Description

Title
Topics in Weyl geometry and quantum anomalies
Author(s)
Jia, Weizhen
Issue Date
2024-07-03
Director of Research (if dissertation) or Advisor (if thesis)
Leigh, Robert G
Doctoral Committee Chair(s)
Faulkner, Thomas
Committee Member(s)
  • Hughes, Taylor L
  • Noronha, Jorge
Department of Study
Physics
Discipline
Physics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
  • ambient space
  • ambient metric
  • Weyl-ambient metric
  • Weyl-ambient geometry
  • obstruction tensor
  • Weyl-obstruction tensor
  • AdS/CFT
  • holography
  • conformal geometry
  • Weyl geometry
  • Weyl manifold
  • Weyl connection
  • Weyl anomaly
  • holographic Weyl anomaly
  • Weyl-Fefferman-Graham gauge
  • Weyl class
  • BRST
  • Lie algebroid
  • anomaly
  • cohomology
  • gauge theory
  • ghost
  • consistent anomaly
  • covariant anomaly
  • topology
  • BRST complex
  • Characteristic class
  • Lorentz-Weyl structure
  • chiral anomaly
  • descent equations
  • Wess-Zumino consistency condition
  • de Rham cohomology
  • BRST cohomology
  • Lie algebroid cohomology
  • principal bundle
  • Atiyah Lie algebroid
Abstract
The interplay between geometry, symmetry, and physics reveals fundamental insights of Nature. In this thesis we explore several facets of these topics, including Weyl geometry and its applications in holographic duality, and the geometric structure of gauge theory and quantum anomalies in the language of Lie algebroids. The first part of this thesis focuses on the Weyl-covariant nature of holography. The conformal boundary of an asymptotically locally AdS (ALAdS) spacetime carries a conformal geometry. The commonly used Fefferman-Graham (FG) gauge explicitly breaks the Weyl symmetry of the boundary theory. This can be resolved by applying the Weyl-Fefferman-Graham (WFG) gauge, in which the boundary carries a Weyl geometry, which is a natural extension of conformal geometry with the Weyl covariance mediated by a Weyl connection. Based on this idea, we generalize the Fefferman-Graham ambient construction for conformal geometry to a corresponding construction for Weyl geometry. We modify the FG ambient metric into a Weyl-ambient metric by implementing the WFG gauge, then we show that the Weyl-ambient space as a pseudo-Riemannian geometry at codimension-2 a Weyl manifold. Conversely, we also show that the Weylambient metric can be uniquely reconstructed from a codimension-2 Weyl manifold provided the initial data of the metric and Weyl connection. 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 WFG 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 (QFT). Furthermore, the holographic calculation with a background Weyl geometry also suggests an underlying geometric interpretation of the Weyl anomaly, which motivates the second part of this thesis. The second part of this thesis is devoted to understanding the geometric nature of the Becchi-Rouet-Stora-Tyutin (BRST) formalism and quantum anomalies. Conventionally, the geometric interpretation for anomalies is studied through the Wess-Zumino consistency condition and descent equations, where the anomaly lives in the ghost number one sector of the BRST cohomology. 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 develop the correspondence of the BRST cohomology and the Lie algebroid cohomology. We showed explicitly that the cohomology of an Atiyah Lie algebroid in a trivialization gives rise to the BRST cohomology. In addition, in the framework of Lie algebroid, the gauge transformations and diffeomorphisms are implemented on an equal footing. We then apply the Lie algebroid cohomology in studying quantum anomalies and demonstrate the computation for chiral and Lorentz-Weyl (LW) 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. This indicates that the Lie algebroid cohomology is indeed a suitable formulation for geometrizing quantum anomalies. The two parts of this thesis are structured to be self-contained and can be read independently. While each part delves into distinct topics, they converge on the subject of the Weyl anomaly. Collectively, they contribute to advancing our understanding of the Weyl anomaly from various perspectives.
Graduation Semester
2024-08
Type of Resource
Thesis
Handle URL
https://hdl.handle.net/2142/125555
Copyright and License Information
Copyright 2024 Weizhen Jia

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