  The first part of this thesis focuses on the Weyl-covariant nature of
holography. We generalize the Fefferman-Graham (FG) 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 (QFT). 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.
