# BRST Cohomology is Lie Algebroid Cohomology

Weizhen<EMAIL_ADDRESS>Marc S<EMAIL_ADDRESS>and
Robert G<EMAIL_ADDRESS>

_ Illinois Center for Advanced Studies of the Universe & Department of
Physics, University of Illinois, 1110 West Green St., Urbana IL 61801, U.S.A.
_

###### Abstract

In this paper we demonstrate that the exterior algebra of an Atiyah Lie
algebroid generalizes the familiar notions of the physicist’s BRST complex. To
reach this conclusion, we develop a general picture of Lie algebroid
isomorphisms as commutative diagrams between algebroids preserving the
geometric structure encoded in their brackets. We illustrate that a necessary
and sufficient condition for such a diagram to define a morphism of Lie
algebroid brackets is that the two algebroids possess gauge-equivalent
connections. This observation indicates that the aforementioned set of Lie
algebroid isomorphisms should be regarded as equivalent to the set of local
diffeomorphisms and gauge transformations. Moreover, a Lie algebroid
isomorphism being a chain map in the exterior algebra sense ensures that
isomorphic algebroids are cohomologically equivalent. The Atiyah Lie
algebroids derived from principal bundles with common base manifolds and
structure groups may therefore be divided into equivalence classes of
isomorphic algebroids. Each equivalence class possesses a local representative
which we refer to as the trivialized Lie algebroid, and we show that the
exterior algebra of the trivialized algebroid gives rise to the BRST complex.
We conclude by illustrating the usefulness of Lie algebroid cohomology in
computing quantum anomalies, including applications to the 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
naive 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.

###### Contents

  1. 1 Introduction
  2. 2 Background on Atiyah Lie Algebroids
    1. 2.1 The Lie Algebroid Derived from a Principal Bundle
    2. 2.2 The Exterior Algebra of an Atiyah Lie Algebroid
  3. 3 Lie Algebroid Isomorphisms
    1. 3.1 A Commutative Diagram
    2. 3.2 Local Trivialization of an Atiyah Lie Algebroid
    3. 3.3 The Cohomology of Trivialized Lie Algebroids
  4. 4 Anomalies from Lie Algebroid Cohomology
    1. 4.1 Characteristic Classes and Lie Algebroid Cohomology
    2. 4.2 Descent Equations and the Consistent Anomaly
    3. 4.3 The Horizontal-Vertical Splitting and the Covariant Anomaly
    4. 4.4 Examples
      1. 4.4.1 Chiral Anomaly in $2d$
      2. 4.4.2 Lorentz-Weyl Anomaly in $2d$
  5. 5 Conclusions
  6. A Chain Maps
  7. B Free Variation and the Covariant Anomaly

##  1 Introduction

The geometric analysis of gauge theories is a rich area of physics which is
deeply interconnected with mathematics [1, 2, 3, 4, 5, 6]. The historical
approach to quantifying topological behavior in gauge theories runs through
the BRST formalism, which was originally introduced to facilitate the
covariant quantization of gauge theories [7, 8, 9]. It was subsequently
realized that the BRST formalism gives rise to an exterior algebra, later
dubbed the BRST complex[10, 11, 12, 13, 14, 15, 16], which can be used to
calculate cohomology classes relevant to quantum anomalies [17, 18, 19, 20,
21, 22, 23, 24, 25]. Starting from a principal bundle $P(M,G)$, the basic
objective of the BRST complex is to design an exterior algebra that combines
the de Rham cohomology of the base manifold $M$ with the cohomology of the
local gauge algebra associated with the structure group $G$. The BRST complex
accomplishes this task in a series of steps. First, it takes a local section
of $P(M,G)$ to define the gauge field $A$, which descends from a bona-fide
principal connection. In this way, it forgets about the vertical sub-bundle of
$TP$, and restricts its attention only to the de Rham cohomology of the base
manifold. Next, the vacuum left behind by the vertical sub-bundle is filled by
introducing a graded algebra generated by a set of Grassmann valued fields
$c^{A}(x)$ called ghosts. In this way, one obtains the BRST complex as an
exterior bi-algebra consisting of $p$-forms on $M$ contracted with $q$ factors
of the ghost field, where the number $q$ is referred to as the ghost number.

A priori, the ghost fields have no geometric interpretation, rather being
interpreted as a computational device. However, it has been argued that a
geometric interpretation for the ghost fields exists as the “vertical
components” of an extended gauge field [26, 27, 28, 29, 30, 31, 32, 33, 34,
35, 36, 37, 38, 39]. The basic idea behind this interpretation is to contract
the ghost fields with the set of Lie algebra generators
$c=c^{A}\otimes\underline{t}_{A}$ and define the extended “connection” form
$\hat{A}=A+c$ by appending the ghost field to the gauge field. Viewing
$\hat{A}$ as a connection, it is natural to define an associated curvature
$\hat{F}={\rm d}_{\rm BRST}\hat{A}+\frac{1}{2}[\hat{A},\hat{A}]$, where the
coboundary operator of the BRST complex is identified as ${\rm d}_{\rm
BRST}={\rm d}+{\rm s}$, which is simply the combination of the de Rham
differential ${\rm d}$ and the BRST operator ${\rm s}$. Enforcing the extra
condition that the curvature should have extent only in the de Rham part of
the BRST complex, one arrives at a pair of equations defining the action of
the BRST operator which can be identified with the Chevalley-Eilenberg
differential appearing in Lie algebra cohomology [40, 41, 42]. In addition,
the action of ${\rm s}$ on the gauge field $A$ can be interpreted as that of
an infinitesimal gauge transformation generated by $c(x)$.

With the “connection” $\hat{A}$, “curvature” $\hat{F}$, and coboundary
operator ${\rm d}_{\rm BRST}$ in hand, one can construct “characteristic
classes” in the BRST complex by naively following the Chern-Weil theorem [43,
44]. Due to the fact that $\hat{F}$ was manufactured to have zero ghost
number, the Chern-Simons form associated with a given characteristic class in
the BRST complex can be shown to satisfy a series of equations known as the
descent equations [41, 45, 46, 47]. One of the resulting equations is the
Wess-Zumino consistency condition[48], which ultimately determines the
algebraic form of candidates for quantum anomalies.

The success of the BRST approach is undeniable. However, it motivates a series
of questions. Why should the Grassmann valued fields $c^{A}(x)$, which started
their life in the BRST quantization procedure have an interpretation as the
generators of a local gauge transformation? Why is it reasonable to combine
the de Rham complex and the ghost algebra into a single exterior bi-algebra?
On a related note, why is it reasonable to consider the combination
$\hat{A}=A+c$ as a “connection”, and moreover what horizontal distribution
does it define? Why should the “curvature” $\hat{F}$ be taken to have ghost
number zero, and why does enforcing this constraint turn the BRST operator
${\rm s}$ into the Chevalley-Eilenberg operator for the Lie algebra of the
structure group? These are the questions that we will answer in this paper.
Quite serendipitously, we will show that there is not an answer to each of
these questions individually, but rather each of these individual questions
are resolved by the answer to a single question: What is the appropriate
geometric interpretation for the BRST complex? Indeed, our main objective will
be to demystify the BRST complex once and for all, and in doing so provide a
unified geometric picture of quantum anomalies. The mathematical language
which is up to this task is that of Lie algebroids [49, 50, 51, 52, 53, 54,
55], the existing uses of which in the context of gauge theories can be found
in, e.g., [56, 57, 58, 59, 60, 61, 62, 63] and the citations therein.

In [63] it was argued that the exterior algebra of an Atiyah Lie algebroid
derived from a principal $G$-bundle $P(M,G)$ is a geometrization of the
physicist’s BRST complex. In this note we will provide a novel perspective on
this correspondence by elaborating on the concept of the _Lie algebroid
trivialization_ , which pushes the discussion in [63] further. In Section 2 we
review the necessary background on Atiyah Lie algebroids, concentrating
especially on aspects of the exterior algebra defined therein. In Section 3 we
discuss the role of Lie algebroid isomorphisms in facilitating the study of
topological aspects of Atiyah Lie algebroids. We introduce an explicit form of
Lie algebroid isomorphism between Atiyah Lie algebroids modeled on a
commutative diagram, and demonstrate how this isomorphism may be interpreted
as implementing both gauge transformations and diffeomorphisms in physical
contexts. In Subsection 3.2 we study Lie algebroid isomorphisms as a tool for
trivializing an Atiyah Lie algebroid. We introduce the Lie algebroid atlas
which allows for the Lie algebroid trivialization to be carried into the
global context. In Subsection 3.3 we study trivializations of the exterior
algebra associated with an Atiyah Lie algebroid, and demonstrate that the
resulting cohomology is equivalent to that of the BRST complex. In Section 4
we apply the lessons from the previous sections to study quantum anomalies. We
place an emphasis on the fact that the exterior algebra of the Atiyah Lie
algebroid can directly quantify both the consistent and covariant anomaly
polynomials. This machinery is applied to the chiral anomaly and the Lorentz-
Weyl anomaly in Subsection 4.4. We conclude in Section 5 in which we provide
answers to the questions posed in this introduction, and address directions
for follow up work.

This paper is one in a series of ongoing projects intended to synthesize the
local properties of gauge theories using the mathematical language of Atiyah
Lie algebroids, in route towards a consistent approach to quantizing gauge
theories including gravity.

##  2 Background on Atiyah Lie Algebroids

In this section we provide an introduction to Atiyah Lie algebroids focusing
on their exterior algebras. We begin by reviewing the construction of an
Atiyah Lie algebroid derived from a principal bundle. We subsequently recall
the formulation of the exterior algebra of an arbitrary Atiyah Lie algebroid
and the coboundary operator $\hat{{\rm d}}$. Here, our intention is to include
enough detail relevant to the present paper; for more detailed discussions of
Lie algebroids, see [63] or [55].

###  2.1 The Lie Algebroid Derived from a Principal Bundle

Let $P(M,G)$ be a principal $G$-bundle over the base manifold $M$ with
structure group $G$. We will denote the Lie algebra of $G$ by $\mathfrak{g}$.
The principal bundle $P$ comes equipped with two canonical maps:

$\pi:P\rightarrow M\,,\qquad R:P\times G\rightarrow P\,,$ (1)

corresponding respectively to the projection and the free right action.

The Atiyah Lie algebroid derived from the principal bundle $P(M,G)$ is given
by the vector bundle $A\equiv P\times_{G}TP=TP/G$ over $M$. In particular, $A$
is obtained as the quotient of the tangent bundle $TP$ by the canonically
defined right action of $G$. We note that while $TP$ is a bundle over $P$,
$A=TP/G$ is importantly a vector bundle over $M$. Furthermore, $A$ is a Lie
algebroid because it inherits a bracket algebra from $TP$, denoted by
$[\cdot,\cdot]_{A}$, and possesses an anchor map $\rho$ in the form of the
pushforward by the projection, i.e., $\rho=\pi_{*}:A\rightarrow TM$. Moreover,
the map $\rho$ can easily be seen to be surjective, and hence the algebroid
$A$ is automatically transitive. This means that we have the following short
exact sequence of vector bundles over $M$:

${0}$${L}$${A}$${TM}$${0\,.}$$\scriptstyle{j}$$\scriptstyle{\rho}$ (2)

$L$ is the kernel of the anchor map $\rho$, called the isotropy bundle over
$M$. The short exact sequence (2) therefore dictates that a section of $A$ can
be identified (locally) with the direct sum of a local gauge transformation
generated by $\underline{\mu}\in\Gamma(L)$ and a diffeomorphism generated by
$\underline{X}\in\Gamma(TM)$.

The Atiyah Lie algebroid $A$ has a canonically defined _vertical sub-bundle_
$V\subset A$ given by the image of $L$ under the morphism $j$ as $V=j(L)$.
This predicates the notion of a Lie algebroid connection as the choice of a
_horizontal sub-bundle_ which is complimentary to $V$. In the context of the
Atiyah Lie algebroid, a connection is quantified by a pair of maps
$\omega:A\rightarrow L$ and $\sigma:TM\rightarrow A$ satisfying
$\text{ker}(\omega)=\text{im}(\sigma)$, defining a second short exact sequence
in the direction opposite to the first one:

${0}$${L}$${A}$${TM}$${0\,.}$$\scriptstyle{j}$$\scriptstyle{\rho}$$\scriptstyle{\omega}$$\scriptstyle{\sigma}$
(3)

The map $\omega$ is called the connection reform, and must also satisfy the
condition $\omega\circ j=-Id_{L}$. In terms of the connection, the horizontal
sub-bundle is given by $H=\text{ker}(\omega)=\text{im}(\sigma)$, and the
connection corresponds to a globally defined split of $A$, namely $A=H\oplus
V$.

###  2.2 The Exterior Algebra of an Atiyah Lie Algebroid

The main focus of this work is to analyze the exterior algebra of $A$, denoted
by $\Omega(A)=\oplus_{p=1}^{\text{rank}\,A}\Omega^{p}(A)$. Each
$\Omega^{p}(A)\equiv\wedge^{p}A^{*}$ consists of totally antisymmetric
$p$-linear maps from $A^{\otimes p}$ into $C^{\infty}(M)$. The exterior
algebra $\Omega(A)$ has a well-defined coboundary operator $\hat{{\rm
d}}:\Omega^{p}(A)\rightarrow\Omega^{p+1}(A)$ determined by the anchor map
$\rho$ and the bracket on $A$, via the Koszul formula[40, 64]:

$\displaystyle\hat{{\rm
d}}\eta(\underline{\mathfrak{X}}_{1},\ldots,\underline{\mathfrak{X}}_{p+1})={}$
$\displaystyle\sum_{i}(-1)^{i+1}\rho(\underline{\mathfrak{X}}_{i})\eta(\underline{\mathfrak{X}}_{1},\ldots,\widehat{\underline{\mathfrak{X}}_{i}},\ldots,\underline{\mathfrak{X}}_{p+1})$
$\displaystyle+\sum_{i<j}(-1)^{i+j}\eta([\underline{\mathfrak{X}}_{i},\underline{\mathfrak{X}}_{j}]_{A},\underline{\mathfrak{X}}_{1},\ldots,\widehat{\underline{\mathfrak{X}}_{i}},\ldots,\widehat{\underline{\mathfrak{X}}_{j}},\ldots,\underline{\mathfrak{X}}_{p+1})\,,$
(4)

where $\underline{\mathfrak{X}}_{1},\ldots,\underline{\mathfrak{X}}_{p+1}$ are
arbitrary sections on $A$, and $\eta$ a section of $\Omega^{p}(A)$, with
$\eta(\underline{\mathfrak{X}}_{1},\ldots,\underline{\mathfrak{X}}_{p})\in
C^{\infty}(M)$ the complete contraction of $\eta$ with sections of $A$.

The exterior algebra $\Omega(A)$ can be extended to $\Omega(A;E)$, namely the
exterior algebra on $A$ with values in the vector bundle $E$, by introducing a
suitable differentiation of sections of $E$. Such a notion comes in the form
of a Lie algebroid representation, which is a morphism
$\phi_{E}:A\rightarrow\text{Der}(E)$ compatible with the anchor. We note that
$\text{Der}(E)$ is itself a Lie algebroid, with isotropy bundle given by
$\text{End}(E)$ and bracket given via the composition of derivations. The
morphism condition simply means that $\phi_{E}$ has a vanishing curvature:

$R^{\phi_{E}}(\underline{\mathfrak{X}},\underline{\mathfrak{Y}}):=[\phi_{E}(\underline{\mathfrak{X}}),\phi_{E}(\underline{\mathfrak{Y}})]_{\text{Der}(E)}-\phi_{E}([\underline{\mathfrak{X}},\underline{\mathfrak{Y}}]_{A})=0\,,\qquad\forall\underline{\mathfrak{X}},\underline{\mathfrak{Y}}\in\Gamma(A)\,.$
(5)

The compatibility condition ensures that $\phi_{E}$ maps into a derivation by
enforcing the Leibniz-like identity

$\phi_{E}(\underline{\mathfrak{X}})(f\underline{\psi})=f\phi_{E}(\underline{\mathfrak{X}})(\underline{\psi})+\rho(\underline{\mathfrak{X}})(f)\underline{\psi}\,,\qquad\forall\underline{\mathfrak{X}}\in\Gamma(A)\,,\quad
f\in C^{\infty}(M)\,,\quad\underline{\psi}\in\Gamma(E)\,.$ (6)

Given such a representation, there is a corresponding Koszul formula
generalizing (2.2):

$\displaystyle\hat{{\rm
d}}^{E}\eta(\underline{\mathfrak{X}}_{1},\ldots,\underline{\mathfrak{X}}_{p+1})={}$
$\displaystyle\sum_{i}(-1)^{i+1}\phi_{E}(\underline{\mathfrak{X}}_{i})\eta(\underline{\mathfrak{X}}_{1},\ldots,\widehat{\underline{\mathfrak{X}}_{i}},\ldots,\underline{\mathfrak{X}}_{p+1})$
$\displaystyle+\sum_{i<j}(-1)^{i+j}\eta([\underline{\mathfrak{X}}_{i},\underline{\mathfrak{X}}_{j}]_{A},\underline{\mathfrak{X}}_{1},\ldots,\widehat{\underline{\mathfrak{X}}_{i}},\ldots,\widehat{\underline{\mathfrak{X}}_{j}},\ldots,\underline{\mathfrak{X}}_{p+1})\,.$
(7)

The operator $\hat{{\rm d}}^{E}$ can be seen to be nilpotent as a combination
of (5) and the fact that the bracket on $A$ satisfies the Jacobi identity. For
simplicity, we will later refer to the coboundary operator as simply
$\hat{{\rm d}}$, leaving the particular representation $E$ implicit.

A connection on $A$ specified by $\omega$ and $\sigma$ induces a Lie algebroid
representation on any vector bundle $E$ that furnishes a representation space
of $L$. Such a representation is determined through the combination of (1) a
covariant derivative operator on $E$, $\nabla^{E}:TM\rightarrow\text{Der}(E)$,
and (2) an endomorphism on $E$, $v_{E}:L\rightarrow\text{End}(E)$. In
particular, we take [63]

$\phi_{E}(\underline{\mathfrak{X}})(\underline{\psi})=\nabla^{E}_{\rho(\underline{\mathfrak{X}})}\underline{\psi}-v_{E}\circ\omega(\underline{\mathfrak{X}})\underline{\psi}\,.$
(8)

$\phi_{E}$ being a Lie algebroid representation through (8) implies two
things. Firstly, $v_{E}$ must be a morphism, or in other words a linear
representation of $L$. Secondly, the curvature of $\nabla^{E}$ viewed as a
connection on $TM$ is determined entirely by the curvature of the horizontal
distribution $H$:[63]

$R^{\nabla^{E}}(\underline{X},\underline{Y})=[\nabla^{E}_{\underline{X}},\nabla^{E}_{\underline{Y}}]_{\text{Der}(E)}-\nabla^{E}_{[\underline{X},\underline{Y}]_{TM}}=-v_{E}\circ\omega(R^{\sigma}(\underline{X},\underline{Y}))\,.$
(9)

Given the covariant derivative $\nabla^{E}$, the corresponding connection
coefficients are given by

$\nabla^{E}_{\rho(\underline{\mathfrak{X}})}\underline{e}_{a}={\cal
A}^{b}{}_{a}(\underline{\mathfrak{X}}_{H})\underline{e}_{b}\,,$ (10)

where $\underline{e}_{a}$ is a basis section of $E$. Hence, we can see that
the representation $\phi_{E}$ acts as

$\phi_{E}(\underline{\mathfrak{X}})(\underline{e}_{a})=\Big{(}{\cal
A}^{b}{}_{a}(\underline{\mathfrak{X}}_{H})-(v_{E}(\omega(\underline{\mathfrak{X}}_{V})))^{b}{}_{a}\Big{)}\underline{e}_{b}\,.$
(11)

##  3 Lie Algebroid Isomorphisms

Given that $\hat{{\rm d}}$ is nilpotent on $\Omega(A,E)$, it provides a well-
defined notion of cohomology, which we refer to as Lie algebroid cohomology.
In this section, our intention is to explain how this cohomology is related to
the usual notion of BRST cohomology. In [63], it was shown that the action of
$\hat{{\rm d}}$ can be thought of as containing within it the BRST
transformation. In this section, we will emphasize the role played by
isomorphisms of Lie algebroids. We will show that two Lie algebroids with
connection that are related by an isomorphism are different representatives of
a topological class, and the cohomology of the respective $\hat{{\rm d}}$
agree. In this sense, the $\hat{{\rm d}}$ cohomology is invariant under
isomorphism. In [63] the notion of a local trivialization of a Lie algebroid
was reviewed. This is a map $\tau_{U}:A\big{|}_{U}\to TU\oplus L\big{|}_{U}$,
with $U$ an open subset of the base manifold $M$, through which the connection
on $A$ can be expressed locally as a gauge field. We will show below that it
is in this description that the usual physics notation $\hat{{\rm
d}}_{\tau}\to{\rm d}+s$ makes sense. This isomorphism may then be used to
relate Lie algebroid cohomology to the usual physics notions of BRST
cohomology.

###  3.1 A Commutative Diagram

A Lie algebroid morphism is a map $\varphi:A_{1}\rightarrow A_{2}$ between two
Lie algebroids, which preserves the geometric structure of the Lie algebroids
as encoded in their brackets. That is, for all
$\underline{\mathfrak{X}},\underline{\mathfrak{Y}}\in\Gamma(A_{1})$,

$R^{\varphi}(\underline{\mathfrak{X}},\underline{\mathfrak{Y}}):=-\varphi([\underline{\mathfrak{X}},\underline{\mathfrak{Y}}]_{A_{1}})+[\varphi(\underline{\mathfrak{X}}),\varphi(\underline{\mathfrak{Y}})]_{A_{2}}=0\,.$
(12)

In this section we focus on a subclass of Lie algebroid morphisms which are,
in fact, isomorphisms of the underlying vector bundles. Consider a set of Lie
algebroids that share the same base manifold and structure group. In general,
two such algebroids may be topologically distinct. Our goal is to emphasize
that two algebroids in this set, $A_{1}$ and $A_{2}$, will be topologically
equivalent if there exists an isomorphism between them. To accomplish this
goal, we seek to understand the conditions under which the set of structure
maps of two Lie algebroids define a commutative diagram of the following form:

${A_{1}}$${0}$${L}$${TM}$${0\,.}$${A_{2}}$$\scriptstyle{\varphi}$$\scriptstyle{\omega_{1}}$$\scriptstyle{\rho_{1}}$$\scriptstyle{j_{1}}$$\scriptstyle{j_{2}}$$\scriptstyle{\sigma_{1}}$$\scriptstyle{\sigma_{2}}$$\scriptstyle{\omega_{2}}$$\scriptstyle{\rho_{2}}$$\scriptstyle{\overline{\varphi}}$
(13)

Note that $J\equiv\sigma_{2}\circ\rho_{1}$ is a map from $H_{1}$ to $H_{2}$,
while $K\equiv j_{2}\circ\omega_{1}$ is a map from $V_{1}$ to $V_{2}$.
Clearly, we can write $\varphi=J-K$. Our motivation for considering (13) is
that it respects the horizontal and vertical splittings of the two algebroids,
and will subsequently provide a useful physical picture for general Lie
algebroid isomorphisms.444Here, we are discussing isomorphisms using an active
language; in the corresponding passive description, an isomorphism would be
understood as a change of basis for the same algebroid.

By commutativity, the maps $\varphi$ and $\overline{\varphi}$ in (13)
apparently define isomorphisms of the vector bundles $A_{1}$ and $A_{2}$.
However, it is not immediately clear that these maps respect the algebras
defined by the brackets on these bundles. To this end, we will now demonstrate
that the map $\varphi$ will be a Lie algebroid morphism if and only if the
horizontal distributions of $A_{1}$ and $A_{2}$ as defined by their respective
connections $\omega_{1}$ and $\omega_{2}$ share the same curvature. Recall
that the curvature of a connection reform $\omega$ is the horizontal
$L$-valued form given by555We have introduced the graded Lie bracket between
$L$-valued differential forms. For $\alpha\in\Omega^{m}(A;L)$ and
$\beta\in\Omega^{n}(A;L)$, $[\alpha,\beta]_{L}$ is defined as
$[\alpha,\beta]_{L}(\underline{\mathfrak{X}}_{1},\ldots,\underline{\mathfrak{X}}_{m+n})=\sum_{\sigma}\text{sgn}(\sigma)[\alpha(\underline{\mathfrak{X}}_{\sigma(1)},\ldots,\underline{\mathfrak{X}}_{\sigma(m)}),\beta(\underline{\mathfrak{X}}_{\sigma(m+1)},\ldots,\underline{\mathfrak{X}}_{\sigma(m+n)})]_{L}\,,$
where $\underline{\mathfrak{X}}_{1},\ldots,\underline{\mathfrak{X}}_{m+n}$ are
arbitrary sections on $A$, $\sigma$ denotes the permutations of
$(1,\ldots,m+n)$, and $\text{sgn}(\sigma)=1$ for even permutations and
$\text{sgn}(\sigma)=-1$ for odd permutations.

$\displaystyle\Omega=\hat{{\rm d}}\omega+\frac{1}{2}[\omega,\omega]_{L}\,.$
(14)

Suppose the curvatures of $\omega_{1}$ and $\omega_{2}$ are $\Omega_{1}$ and
$\Omega_{2}$, respectively. We can compute that

$\displaystyle
R^{\varphi}(\underline{\mathfrak{X}}_{H},\underline{\mathfrak{Y}}_{H})$
$\displaystyle=R^{\sigma_{2}}(\rho_{1}(\underline{\mathfrak{X}}_{H}),\rho_{1}(\underline{\mathfrak{Y}}_{H}))+j_{2}(R^{-\omega_{1}}(\underline{\mathfrak{X}}_{H},\underline{\mathfrak{Y}}_{H})$
$\displaystyle=j_{2}(\Omega_{2}(\varphi(\underline{\mathfrak{X}}),\varphi(\underline{\mathfrak{Y}})))-j_{2}(\Omega_{1}(\underline{\mathfrak{X}},\underline{\mathfrak{Y}}))\,,$
(15)

where we used $\varphi=J-K$ and Eq. (43) of [63]:

$\displaystyle
R^{\sigma}(\rho(\underline{\mathfrak{X}}),\rho(\underline{\mathfrak{Y}}))=j(\Omega(\underline{\mathfrak{X}},\underline{\mathfrak{Y}}))=-j(R^{-\omega}(\underline{\mathfrak{X}}_{H},\underline{\mathfrak{Y}}_{H}))\,.$
(16)

In this way, we see that $\varphi$ will be a morphism of the brackets if and
only if

$\displaystyle\Omega_{1}(\underline{\mathfrak{X}},\underline{\mathfrak{Y}})=\Omega_{2}(\varphi(\underline{\mathfrak{X}}),\varphi(\underline{\mathfrak{Y}}))\,.$
(17)

Provided $\varphi$ is an isomorphism, it will induce a linear transformation
on bundles associated to $A_{1}$ and $A_{2}$ to preserve Lie algebroid
representations. Let $E_{1}$ and $E_{2}$ be isomorphic vector bundles over $M$
which are associated, respectively, to $A_{1}$ and $A_{2}$ by Lie algebroid
representations $\phi_{E_{j}}:A_{j}\to\text{Der}(E_{j})$, with $j=1,2$. Then,
accompanying the Lie algebroid isomorphism $\varphi$, there is a corresponding
map on the associated bundles, which can be written as

$g_{\varphi}:E_{1}\rightarrow E_{2}\,.$ (18)

By construction, we enforce that this map is compatible with the Lie algebroid
representations of $A_{1}$ and $A_{2}$ in the sense that

$\displaystyle\phi_{E_{2}}\circ\varphi(\underline{\mathfrak{X}})(g_{\varphi}(\underline{\psi}))=g_{\varphi}(\phi_{E_{1}}(\underline{\mathfrak{X}})(\underline{\psi}))\,,\qquad\forall\underline{\mathfrak{X}}\in\Gamma(A_{1})\,,\quad\underline{\psi}\in\Gamma(E_{1})\,.$
(19)

Let $\varphi^{*}:\Omega(A_{2};E_{2})\rightarrow\Omega(A_{1};E_{1})$ denote the
Lie algebroid pullback map induced by $\varphi$. Explicitly, given
$\eta\in\Omega^{r}(A_{2};E_{2})$ and
$\underline{\mathfrak{X}}_{1},\ldots,\underline{\mathfrak{X}}_{r}\in\Gamma(A_{1})$
we have

$(\varphi^{*}\eta)(\underline{\mathfrak{X}}_{1},\ldots,\underline{\mathfrak{X}}_{r})=g_{\varphi}^{-1}\big{(}\eta(\varphi(\underline{\mathfrak{X}}_{1}),\dots,\varphi(\underline{\mathfrak{X}}_{r}))\big{)}\,.$
(20)

Using this notation along with the properties (12) and (19) it is easy to
establish that

$\displaystyle\hat{{\rm d}}_{1}\circ\varphi^{*}=\varphi^{*}\circ\hat{{\rm
d}}_{2}\,,$ (21)

which means that $\varphi$ is a Lie algebroid chain map in the exterior
algebra sense. For an explicit demonstration, we refer the reader to Appendix
A.

Using (20) we can rewrite (17) as

$\displaystyle\Omega_{1}=\varphi^{*}\Omega_{2}\,.$ (22)

Eq. (22) indicates that a Lie algebroid isomorphism of the form (13) involves
a topological consideration about the algebroids in question. In Section 4 we
introduce a version of the Chern-Weil homomorphism which is applicable to Lie
algebroid cohomology. This will provide a recipe for constructing Atiyah Lie
algebroid cohomology classes in terms of characteristic polynomials in
curvature. Recall that a characteristic class satisfies a so-called
“naturality” condition, which essentially implies that the pullback commutes
through the characteristic class; i.e., if $\lambda(\Omega)$ is a
characteristic class of a curvature $\Omega$, then

$\displaystyle\lambda(\varphi^{*}\Omega)=\varphi^{*}\lambda(\Omega)\,.$ (23)

Hence, two Lie algebroids whose curvatures are related as (17) will possess an
isomorphism between their cohomologies. Eq. (21) similarly implies that
isomorphic Lie algebroids possess isomorphic cohomology classes. In light of
these observations, we can view the Lie algebroid isomorphism as a device for
organizing the set of Atiyah Lie algebroids with connection into topological
equivalence classes. Let $(A,\omega)$ denote an Atiyah Lie algebroid $A$ with
connection reform $\omega$. Then,

$[(A,\omega)]:=\\{(A^{\prime},\omega^{\prime})\;|\;\exists\varphi:A\rightarrow
A^{\prime}\text{ s.t. }\Omega=\varphi^{*}\Omega^{\prime}\\}$ (24)

can be regarded as the set of topologically equivalent Atiyah Lie algebroids
with connection.

From a physical perspective Eqs. (17) and (21) establish the fact that the
commutative diagram (13) encodes diffeomorphisms and gauge transformations
relating isomorphic Lie algebroids. In particular, it is straightforward to
establish that the connection coefficients [see Eq. (10)] satisfy

$\displaystyle({\cal A}_{1})_{\underline{\alpha}_{1}}{}^{a_{1}}{}_{b_{1}}$
$\displaystyle=J^{\underline{\alpha}_{2}}{}_{\underline{\alpha}_{1}}(g_{\varphi}^{-1})^{a_{1}}{}_{a_{2}}\Big{(}({\cal
A}_{2})_{\underline{\alpha}_{2}}{}^{a_{2}}{}_{b_{2}}+\delta^{a_{2}}{}_{b_{2}}\rho(\underline{E}_{\underline{\alpha}_{2}})\Big{)}g_{\varphi}^{b_{2}}{}_{b_{1}}\,,$
(25)
$\displaystyle(v_{E}(\omega_{1}))_{\underline{A}_{1}}{}^{a_{1}}{}_{b_{1}}$
$\displaystyle=K^{\underline{B}_{2}}{}_{\underline{A}_{1}}(g_{\varphi}^{-1})^{a_{1}}{}_{a_{2}}(v_{E}(\omega_{2}))_{\underline{B}_{2}}{}^{a_{2}}{}_{b_{2}}g_{\varphi}^{b_{2}}{}_{b_{1}}\,.$
(26)

That is, the components of ${\cal A}$ and $\omega$ transform like a gauge
field and a gauge ghost, respectively. Eq. (25) is compatible with (17);
recall that the curvatures of gauge fields related by a gauge transformation
are equivalent up to a conjugation. In this respect, we can also identify the
Lie algebroid isomorphism (13) as encoding the data of a gauge transformation.
In other words, the set $[(A,\omega)]$ can be regarded as an orbit of gauge
equivalent algebroids. In a separate work [65], we use this remark to
construct the configuration algebroid, which can be regarded as a concise
definition of the space of gauge orbits of connections that can be employed in
any gauge theory formulated in terms of Atiyah Lie algebroids.

###  3.2 Local Trivialization of an Atiyah Lie Algebroid

In the last section we have shown that there exists a Lie algebroid
isomorphism of the form (13) between Lie algebroids with connection whose
horizontal distributions have curvatures related by (17). It is perhaps worth
mentioning that this very same construction was used in constructing a
representation of a Lie algebroid $A$ by the Lie algebroid $\text{Der}(E)$,
for some associated vector bundle $E$. In fact, this is a slight
generalization of what we presented above, in that whereas the isomorphism in
question is $\phi_{E}:A\to\text{Der}(E)$, these two algebroids do not share
the same isotropy bundle, but instead there is a further isomorphism
$v_{E}:L\to\text{End}(E)$ between them. Locally this isomorphism can be
thought to give a matrix representation (on the fibres of $E$) of the Lie
algebra.

A local trivialization of a Lie algebroid can also be thought of as an example
of a Lie algebroid isomorphism, with the details presented in terms of local
data. Using the notation of [63], on open sets $U_{i}\subset M$, we
have666Here $A^{U_{i}}$ is the restriction of the Lie algebroid $A$ to the
local neighborhood $U_{i}\subset M$. In other words, $A^{U_{i}}$ is a vector
bundle over $U_{i}$.

$\displaystyle\tau_{i}:A^{U_{i}}\to TU_{i}\oplus L^{U_{i}}\,,$ (27)

and so local sections of $A$ can be expressed in terms of local bases for $TM$
and $L$

$\displaystyle\tau_{i}(\underline{\mathfrak{X}}_{H})=\mathfrak{X}_{i,H}^{\alpha}\tau_{i}{}^{\mu}{}_{\alpha}(\underline{\partial}^{U_{i}}_{\mu}+b_{i}{}_{\mu}^{A}\underline{t}^{U_{i}}_{A})\,,\qquad\tau_{i}(\underline{\mathfrak{X}}_{V})=\mathfrak{X}_{i,V}^{\underline{A}}\tau_{i}{}^{A}{}_{\underline{A}}\underline{t}^{U_{i}}_{A}\,.$
(28)

The coefficients $b_{i}{}_{\mu}^{A}$ are the components of a
$\mathfrak{g}$-valued 1-form on $M$, that transforms on overlapping open sets
as a gauge field by consequence of (25).

Figure 1: A visualization of a Lie algebroid. A connection 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
$L$.

As we have now established, in each open set $U_{i}\subset M$, we realize a
Lie algebroid isomorphism $\tau_{i}:A^{U_{i}}\rightarrow TU_{i}\oplus
L^{U_{i}}$.777Note that here we are using the notion of isomorphism in the
active sense, and hence we distinguish $A^{U_{i}}$ from $TU_{i}\oplus
L^{U_{i}}$. In what follows, the reader may find it profitable to think from a
passive perspective: indeed our use of $A^{U_{i}}$ versus $TU_{i}\oplus
L^{U_{i}}$ can be thought of as simply corresponding to a different choice of
basis, the first natural from the $H\oplus V$ split, the second natural from
the local $TU\oplus L$ split. Suppose $\\{U_{i}\\}$ form an open cover for the
base $M$, then we can sew together the aforementioned local charts to obtain a
Lie algebroid atlas. Sewing the charts $\tau_{i}$ together requires that we
also specify transition functions $t_{ij}:A^{U_{i}}\rightarrow A^{U_{j}}$,
which are Lie algebroid isomorphisms with support in the intersection
$U_{i}\cap U_{j}$ for each pair of $U_{i}$ and $U_{j}$. This is equivalent to
the perhaps more familiar notion that overlapping charts in a principal bundle
must agree up to a gauge transformation. The presence of non-trivial
transition functions in the algebroid context ensures that topological data is
preserved under trivialization. Together, the collection
$\\{U_{i},\tau_{i},t_{ij}\\}$ carries the intuition of the Lie algebroid
trivialization into a global context. In the following we will use the
abbreviated notation $\tau:A\rightarrow A_{\tau}$ to refer to the local Lie
algebroid isomorphism mapping $A$ into the trivialized Lie algebroid
$A_{\tau}\simeq TU\oplus L^{U}$ for some $U\subset M$. That is, the notation
$A_{\tau}$ serves to remind that $A_{\tau}$ involves restricting $A$ to an
open set. We leave the open subset $U$ unspecified with the understanding that
the Lie algebroid atlas allows for the algebroid $A$ to be trivialized when
restricted to any open neighborhood of the base.

To be precise about details, we will introduce explicit bases for the various
vector bundles; although we will not indicate so, these should be understood
to be valid locally on some open set of $M$. So we introduce the notation for
bases for the bundles $TM$ and $L$ and their dual bundles:

$\begin{split}TM&=\text{span}\\{\underline{\partial}_{\mu}\\}\,,\qquad
T^{*}M=\text{span}\\{{\rm d}x^{\mu}\\}\,,\qquad\mu=1,\ldots,\dim M\,,\\\
L&=\text{span}\\{\underline{t}_{A}\\}\,,\qquad
L^{*}=\text{span}\\{t^{A}\\}\,,\qquad A=1,\ldots,\dim G\,.\end{split}$ (29)

These bases are dual in the sense that

${\rm d}x^{\mu}(\underline{\partial}_{\nu})=\delta^{\mu}{}_{\nu},\qquad
t^{A}(\underline{t}_{B})=\delta^{A}{}_{B}\,,\qquad{\rm
d}x^{\mu}(\underline{t}_{A})=0\,,\qquad
t^{A}(\underline{\partial}_{\mu})=0\,.$ (30)

Given the above notation, we have a choice to make for a basis of sections of
the trivialized Lie algebroid $A_{\tau}$. We will refer to such choices as
“splittings”, and we will make reference to two natural choices which we refer
to as the _consistent splitting_ and the _covariant splitting_ , respectively.
The relevance of this nomenclature will become clear shortly. These two
splittings correspond in fact to the two sets of axes shown in Figure 1, and
they are distinguished precisely because of the non-trivial connection on
$(A_{\tau},\omega_{\tau})$.

By a covariant splitting, we mean a split basis as described in [63]. Consider
an algebroid $(A,\omega)$ for which we take a basis of sections
$\\{\underline{E}_{\underline{\alpha}},\underline{E}_{\underline{A}}\\}$
(where $\underline{\alpha}=1,\ldots,\dim M$, $\underline{A}=1,\ldots,\dim G$).
Such a basis has the virtue that
$\omega(\underline{E}_{\underline{\alpha}})=\underline{0}$ and
$\rho(\underline{E}_{\underline{A}})=\underline{0}$, namely they span $H$ and
$V$ respectively. Given the map $\tau$, it is natural to choose a basis
$\\{\tau(\underline{E}_{\underline{\alpha}}),\tau(\underline{E}_{\underline{A}})\\}$
for $A_{\tau}$. Since we will now deal directly with $A_{\tau}$, we will for
brevity denote such a basis by
$\\{\underline{\hat{E}}_{\underline{\alpha}},\underline{\hat{E}}_{\underline{A}}\\}$.
Thus a covariant splitting corresponds to a choice of basis sections that are
aligned with the global split $A_{\tau}=H_{\tau}\oplus V_{\tau}$. Locally,
these sections can be expressed in terms of the bases for $TM$ and $L$ as

$\displaystyle\underline{\hat{E}}_{\underline{\alpha}}=\rho_{\tau}^{\mu}{}_{\underline{\alpha}}(\underline{\partial}_{\mu}+b_{\mu}^{A}\underline{t}_{A})\,,\qquad\underline{\hat{E}}_{\underline{A}}=-\omega_{\tau}^{A}{}_{\underline{A}}\underline{t}_{A}\,,$
(31)

while the dual bases can be written as (also referred to as the “mixed local
basis” in [59])

$\displaystyle\hat{E}^{\underline{\alpha}}=\sigma_{\tau}^{\underline{\alpha}}{}_{\mu}{\rm
d}x^{\mu}\,,\qquad{\hat{E}}^{\underline{A}}=j_{\tau}^{\underline{A}}{}_{A}(t^{A}-b^{A}_{\mu}{\rm
d}x^{\mu})\,.$ (32)

Suppose $\underline{X}=X^{\mu}\underline{\partial}_{\mu}\in\Gamma(TM)$ and
$\underline{\mu}=\mu^{A}\underline{t}_{A}\in\Gamma(L)$, then we have a section
$\underline{\mathfrak{X}}$ of $A_{\tau}$ which can be expressed in this
covariant splitting as

$\displaystyle\underline{\mathfrak{X}}=X^{\mu}\sigma_{\tau}(\underline{\partial}_{\mu})-\mu^{A}j_{\tau}(\underline{t}_{A})=X^{\mu}\sigma_{\tau}^{\underline{\alpha}}{}_{\mu}\underline{\hat{E}}_{\underline{\alpha}}-\mu^{A}j_{\tau}^{\underline{A}}{}_{A}\underline{\hat{E}}_{\underline{A}}=X^{\mu}(\underline{\partial}_{\mu}+b_{\mu}^{A}\underline{t}_{A})+\mu^{A}\underline{t}_{A}\,.$
(33)

On the other hand, by a consistent splitting, we mean a choice of basis for
$A_{\tau}$ that is aligned with the bases for $TM$ and $L$. That is, in the
consistent splitting, we can write a section of $A_{\tau}$ as

$\displaystyle\underline{\mathfrak{X}}=\mathfrak{X}^{\mu}\underline{\partial}_{\mu}+\mathfrak{X}^{A}\underline{t}_{A}\,.$
(34)

By comparing to the covariant split (33), we see that

$\displaystyle\mathfrak{X}^{\mu}=X^{\mu}\,,\qquad\mathfrak{X}^{A}=\mu^{A}+X^{\mu}b_{\mu}^{A}\,,$
(35)

and thus in the consistent splitting, the gauge field is contained in an off-
block-diagonal piece of $\sigma_{\tau}$.

In the current set up, the connection reform $\omega_{\tau}$ which defines the
horizontal distribution through its kernel can be written in the consistent
splitting as

$\omega_{\tau}=\omega_{\tau}^{A}{}_{\underline{A}}\hat{E}^{\underline{A}}\otimes\underline{t}_{A}=\omega_{\tau}^{A}{}_{\underline{A}}j_{\tau}^{\underline{A}}{}_{B}(t^{B}-b^{B}_{\mu}{\rm
d}x^{\mu})\otimes\underline{t}_{A}=(b^{A}_{\mu}{\rm
d}x^{\mu}-t^{A})\otimes\underline{t}_{A}=b-\varpi\,.$ (36)

where we defined

$\displaystyle\varpi=\varpi^{A}\otimes\underline{t}_{A}=t^{A}\otimes\underline{t}_{A}\,,$
(37)

which can be interpreted as the Maurer-Cartan form on $L$. Recall that $L$ is
a bundle of Lie algebras, which means that the $\varpi$ given in (37) should
be interpreted as the Maurer-Cartan form for the group $G$ pointwise on the
base manifold $M$. In other words, $\varpi$ is a field of Maurer-Cartan forms,
with $\varpi(x)$ being the Maurer-Cartan form for each fiber of $L$ at $x\in
M$. The spatial dependence of $\varpi$ will play a significant role in
defining the exterior algebra in the consistent splitting.

Eq. (36) explicitly shows that the connection reform can be understood as the
sum of two pieces, the first related to the gauge field, and the second
related to the Maurer-Cartan form of the gauge algebra, if we interpret it in
the consistent splitting (i.e., in terms of the bases for $TM$ and $L$ and
their duals). This equation should be compared with the idea of an extended
“connection” in the BRST complex which is typically taken to be of the form
$\hat{A}=A+c$ where $A$ is a local gauge field and $c$ is the ghost field [26,
27, 28]. However, Eq. (36) has an advantage over the conventional extended
“connection” because it possesses a manifestly geometric interpretation as a
genuine connection in the algebroid context.

###  3.3 The Cohomology of Trivialized Lie Algebroids

We now turn our attention to the main focus of this section—understanding the
exterior algebra of the trivialized algebroid. The bracket on $A_{\tau}$ can
be written explicitly for the basis sections as

$\displaystyle[\underline{\hat{E}}_{\underline{\alpha}},\underline{\hat{E}}_{\underline{\beta}}]_{A_{\tau}}$
$\displaystyle=\sigma_{\tau}\left([\rho_{\tau}(\underline{\hat{E}}_{\underline{\alpha}}),\rho_{\tau}(\underline{\hat{E}}_{\underline{\beta}})]_{TM}\right)+j_{\tau}(\Omega_{\underline{\alpha}\underline{\beta}})\,,$
(38)
$\displaystyle[\underline{\hat{E}}_{\underline{\alpha}},\underline{\hat{E}}_{\underline{B}}]_{A_{\tau}}$
$\displaystyle=-j_{\tau}\left(R^{-\omega_{\tau}}(\underline{\hat{E}}_{\underline{\alpha}},\underline{\hat{E}}_{\underline{B}})\right)=j_{\tau}\left(\nabla^{L}_{\underline{\hat{E}}_{\underline{\alpha}}}(\omega_{\tau}^{A}{}_{\underline{B}}\underline{t}_{A})\right)=j_{\tau}\left(\phi_{L}(\underline{\hat{E}}_{\underline{\alpha}})(\omega_{\tau}^{A}{}_{\underline{B}}\underline{t}_{A})\right)\,,$
(39)
$\displaystyle[\underline{\hat{E}}_{\underline{A}},\underline{\hat{E}}_{\underline{B}}]_{A_{\tau}}$
$\displaystyle=j_{\tau}\left([\omega_{\tau}(\underline{\hat{E}}_{\underline{A}}),\omega_{\tau}(\underline{\hat{E}}_{\underline{B}})]_{L}\right)=-\omega_{\tau}^{A}{}_{\underline{A}}\omega_{\tau}^{B}{}_{\underline{B}}f_{AB}{}^{C}\underline{\hat{E}}_{\underline{C}}j_{\tau}^{\underline{C}}{}_{C}\,.$
(40)

The coboundary operator for the complex $\Omega(A_{\tau};E)$, denoted by
$\hat{{\rm d}}_{\tau}$, is defined precisely by the Koszul formula (2.2). In
terms of the isomorphism $\tau:A\to A_{\tau}$, we have, as in (21), $\hat{{\rm
d}}\circ\tau^{*}=\tau^{*}\circ\hat{{\rm d}}_{\tau}$. Working in $A_{\tau}$, we
now have two different ways of splitting $\Omega(A_{\tau};E)$ into a bi-
complex. Firstly, we can use the covariant splitting of $A_{\tau}$ to identify

$\Omega^{p}(A_{\tau};E)=\bigoplus_{r+s=p}\Omega^{(r,s)}(H_{\tau},V_{\tau};E)\,,$
(41)

where $\Omega^{(r,s)}(H_{\tau},V_{\tau};E)$ consists of bi-forms of degree $r$
in the algebra of $H_{\tau}$ and degree $s$ in the algebra of $V_{\tau}$. This
is certainly the most natural splitting of the exterior algebra, as it is
globally defined given a connection. We will show that this is equivalent to,
but not the same as, the usual splitting, where $r$ counts the de Rham form
degree and $s$ counts ghost number.

Alternatively, using the consistent splitting for $A_{\tau}$ we can identify

$\Omega^{p}(A_{\tau};E)=\bigoplus_{r+s=p}\Omega^{(r,s)}(TM,L;E)\,,$ (42)

where $\Omega^{p}(A_{\tau};E)$ now consists of bi-forms of degree $r$ in the
de Rham cohomology of $M$ and degree $s$ in the Chevalley-Eilenberg algebra of
$L$.

To understand precisely how this works, we consider the action of $\hat{{\rm
d}}_{\tau}$ on sections of various bundles. We will show that the action of
$\hat{{\rm d}}_{\tau}$ can be interpreted as acting as ${\rm d}+{\rm s}$ on
the components of sections, reproducing the usual physics notation [63] (apart
from the fact that the usual Grassmann quantities appear instead as forms).

As a first example, we consider an $E$-valued scalar
$\underline{\psi}=\psi^{a}\underline{e}_{a}\in\Gamma(E)$. Using the Koszul
formula, we have

$\displaystyle\hat{{\rm d}}_{\tau}\underline{\psi}$
$\displaystyle=\hat{E}^{\underline{M}}\otimes\phi_{E}(\underline{\hat{E}}_{\underline{M}})(\underline{\psi})$
$\displaystyle=\rho_{\tau}^{\mu}{}_{\underline{\alpha}}\left(\partial_{\mu}\psi^{a}+v_{E}(b_{\mu})^{a}{}_{b}\psi^{b}\right)\hat{E}^{\underline{\alpha}}\otimes\underline{e}_{a}-v_{E}(\omega_{\hat{A}})^{a}{}_{b}\psi^{b}\;E^{\hat{A}}\otimes\underline{e}_{a}$
$\displaystyle=\Big{(}{\rm
d}\psi^{a}+v_{E}(\underline{t}_{A})^{a}{}_{b}\varpi^{A}\psi^{b}\Big{)}\otimes\underline{e}_{a}\,,$
(43)

which we identify with888It should be noted that in [63] this was written as
$\hat{{\rm d}}\underline{\psi}=\nabla^{E}\underline{\psi}+{\rm
s}\underline{\psi}$. These results are consistent, given that $\hat{{\rm
d}}\underline{\psi}=\nabla^{E}\underline{\psi}+\psi^{a}{\rm
s}\underline{e}_{a}+{\rm s}\psi^{a}\otimes\underline{e}_{a}={\rm
d}\psi^{a}\otimes\underline{e}_{a}+{\rm s}\psi^{a}\otimes\underline{e}_{a}$.
This is a general feature: by extracting the basis elements, the gauge fields
in the covariant derivative are canceled by those coming from ${\rm
s}\underline{e}_{a}$. We will see this pattern repeated in additional
examples.

$\displaystyle\hat{{\rm d}}_{\tau}\underline{\psi}=({\rm d}+{\rm
s})\psi^{a}\otimes\underline{e}_{a}\,,$ (44)

if we interpret

$\displaystyle{\rm
s}\psi^{a}:=v_{E}(\underline{t}_{A})^{a}{}_{b}\varpi^{A}\psi^{b}\,.$ (45)

As a second example, consider a section $\beta\in\Gamma(A_{\tau}^{*}\times
E)$. Employing the Koszul formula (which is most easily employed by
translating $\beta$ into the covariant split basis), we find

$\displaystyle\hat{{\rm d}}_{\tau}\beta={}$
$\displaystyle\frac{1}{2}\hat{E}^{\underline{M}}\wedge\hat{E}^{\underline{N}}\otimes\Big{(}\phi_{E}(\underline{\hat{E}}_{\underline{M}})(\beta^{a}_{\underline{N}}\underline{e}_{a})-\phi_{E}(\underline{\hat{E}}_{\underline{N}})(\beta^{a}_{\underline{M}}\underline{e}_{a})-\beta([\underline{\hat{E}}_{\underline{M}},\underline{\hat{E}}_{\underline{N}}]_{A_{\tau}})\Big{)}$
$\displaystyle={}$ $\displaystyle\Big{[}\Big{(}{\rm
d}(\sigma_{\tau}^{\underline{\alpha}}{}_{\nu}\beta^{a}_{\underline{\alpha}}-j_{\tau}^{\underline{B}}{}_{B}\beta^{a}_{\underline{B}}b_{\nu}^{B})+v_{E}(\underline{t}_{A})^{a}{}_{b}t^{A}(\sigma_{\tau}^{\underline{\alpha}}{}_{\nu}\beta^{a}_{\underline{\alpha}}-j_{\tau}^{\underline{B}}{}_{B}\beta^{a}_{\underline{B}}b_{\nu}^{B})\Big{)}\wedge{\rm
d}x^{\nu}$ $\displaystyle+\Big{(}{\rm
d}(j_{\tau}^{\underline{B}}{}_{B}\beta^{a}_{\underline{B}})+v_{E}(\underline{t}_{A})^{a}{}_{b}t^{A}(j_{\tau}^{\underline{B}}{}_{B}\beta^{b}_{\underline{B}})-\frac{1}{2}f_{AB}{}^{C}(j_{\tau}^{\underline{B}}{}_{C}\beta^{b}_{\underline{B}})t^{A}\Big{)}\wedge
t^{B}\Big{]}\otimes\underline{e}_{a}\,.$ (46)

Recognizing
$\beta^{a}_{\nu}=\sigma_{\tau}^{\underline{\alpha}}{}_{\nu}\beta^{a}_{\underline{\alpha}}-j_{\tau}^{\underline{B}}{}_{B}\beta^{a}_{\underline{B}}b_{\nu}^{B}$
and $\beta^{a}_{A}=j_{\tau}^{\underline{B}}{}_{A}\beta^{a}_{\underline{B}}$,
we have

$\displaystyle\hat{{\rm d}}_{\tau}\beta=\Big{(}{\rm
d}\beta^{a}_{\nu}+v_{E}(\underline{t}_{A})^{a}{}_{b}t^{A}\beta^{a}_{\nu}\Big{)}\wedge{\rm
d}x^{\nu}\otimes\underline{e}_{a}+\Big{(}{\rm
d}\beta^{a}_{B}+v_{E}(\underline{t}_{A})^{a}{}_{b}t^{A}\beta^{b}_{B}-\frac{1}{2}f_{AB}{}^{C}\beta^{a}_{C}t^{A}\Big{)}\wedge
t^{B}\otimes\underline{e}_{a}\,,$ (47)

so we see that

$\displaystyle\hat{{\rm d}}_{\tau}\beta=({\rm d}+{\rm
s})\beta^{a}_{\mu}\wedge{\rm d}x^{\mu}\otimes\underline{e}_{a}+({\rm d}+{\rm
s})\beta^{a}_{A}\wedge t^{A}\otimes\underline{e}_{a}\,,$ (48)

if

$\displaystyle{\rm
s}\beta^{a}_{\nu}=v_{E}(\underline{t}_{A})^{a}{}_{b}\varpi^{A}\beta^{a}_{\nu}\,,\qquad{\rm
s}\beta^{a}_{B}=v_{E}(\underline{t}_{A})^{a}{}_{b}\varpi^{A}\beta^{b}_{B}-\frac{1}{2}f_{AB}{}^{C}\beta^{b}_{C}\varpi^{A}\,.$
(49)

We note that this is of a similar form to the previous example in (44).

As a final example, we consider the connection reform $\omega_{\tau}$, which
we regard as an element of $\Omega^{1}(A_{\tau},L)$. We have

$\displaystyle\hat{{\rm d}}_{\tau}\omega_{\tau}$ $\displaystyle=\hat{{\rm
d}}_{\tau}(b-\varpi)$
$\displaystyle=(\Omega_{\tau}^{A}-\frac{1}{2}f_{BC}{}^{A}\omega_{\tau}^{B}\wedge\omega_{\tau}^{C})\otimes\underline{t}_{A}$
(50) $\displaystyle=({\rm d}b^{A}+f_{BC}{}^{A}\varpi^{B}\wedge
b^{C}-\frac{1}{2}f_{BC}{}^{A}\varpi^{B}\wedge\varpi^{C})\otimes\underline{t}_{A}\,,$
(51)

where in the last line we made use of the result (36), writing
$\varpi=\varpi^{A}\otimes\underline{t}_{A}$.

We note that if we identify

$\displaystyle{\rm s}b^{A}={\rm d}\varpi^{A}+f_{BC}{}^{A}\varpi^{B}\wedge
b^{C}\,,\qquad{\rm
s}\varpi^{A}=\frac{1}{2}f_{BC}{}^{A}\varpi^{B}\wedge\varpi^{C}\,,$ (52)

then we obtain

$\displaystyle\hat{{\rm d}}_{\tau}\omega_{\tau}=({\rm d}+{\rm
s})\omega_{\tau}^{A}\otimes\underline{t}_{A}\,.$ (53)

To understand (53) one must establish an interpretation for the ${\rm
d}\varpi^{A}$ in (52). As we have alluded to below (37), $\varpi$ is not
spatially constant, and therefore has a nonzero derivative under de Rham ${\rm
d}$. Considering the following pair of facts:

$\displaystyle\hat{i}_{-j(\underline{\mu})}\varpi^{A}=-\mu^{A}\,,\qquad\hat{\cal
L}_{-j(\underline{\mu})}\varpi^{A}=0\,,\qquad\forall\underline{\mu}\in\Gamma(L)\,,$
(54)

and noticing that $\hat{\cal
L}_{\underline{\mathfrak{X}}}=\hat{i}_{\underline{\mathfrak{X}}}\hat{{\rm
d}}+\hat{{\rm d}}\hat{i}_{\underline{\mathfrak{X}}}$, we have

$\displaystyle\hat{i}_{-j(\underline{\mu})}{\rm d}\varpi^{A}={\rm
d}\mu^{A}\,.$ (55)

Then, the first equation in (52) is consistent with the standard variation of
the gauge field:

$\displaystyle\hat{i}_{-j(\underline{\mu})}{\rm s}b^{A}={\rm
d}\mu^{A}+[b,\underline{\mu}]^{A}\,.$ (56)

Therefore, starting from the formal definition (2.2) of the nilpotent
coboundary operator in the algebroid exterior algebra, we established the
relationship between $\hat{{\rm d}}_{\tau}$ and the BRST differentiation ${\rm
s}$. Again, we emphasize that this result is a natural consequence of the
geometric structure of the algebroid.

##  4 Anomalies from Lie Algebroid Cohomology

We have now demonstrated that the fundamental features of the BRST complex are
geometrically encoded in the Atiyah Lie algebroid. Working in the consistent
splitting, the exterior algebra of the trivialized algebroid is a bi-complex
consisting of differential forms on the base manifold $M$ and differential
forms in the exterior algebra associated to the local gauge group. This is the
state of affairs described in the BRST complex but only after making a series
of choices [8, 22, 24, 46, 66]. We have now shown why these choices are
reasonable. For example, the counterpart of the extended “connection”
$\hat{A}=A+c$ is identified with $\omega_{\tau}=b-\varpi$ in the algebroid
context; $b$ corresponds to the gauge field $A$, and $\varpi$ corresponds to
the ghost field $c$ (up to a sign difference). Significantly, $\omega_{\tau}$
is a genuine connection which defines a horizontal distribution on the
algebroid. Moreover, the appearance of the Maurer-Cartan form $\varpi$
justifies the interpretation of the ghost field $c$ in the BRST formalism as a
generator of gauge transformations.

As discussed in [63], the “Russian formula” central to the BRST analysis (see,
for example, [27, 21]) is also simply a geometric fact in the algebroid
context arising from the observation that the curvature of a Lie algebroid
connection is zero when contracted with vertical vector fields. Working in the
consistent splitting of the trivialized algebroid, this version of the Russian
formula can be stated in a more familiar form as:

$\Omega_{\tau}=\hat{{\rm
d}}_{\tau}\omega_{\tau}+\frac{1}{2}[\omega,\omega]_{L}=({\rm d}+{\rm
s})(b^{A}-\varpi^{A})\otimes\underline{t}_{A}+\frac{1}{2}[b-\varpi,b-\varpi]_{L}={\rm
d}b+\frac{1}{2}[b,b]_{L}=F\,,$ (57)

where $F\equiv{\rm d}b+\frac{1}{2}[b,b]_{L}$ is the gauge field strength of
the gauge field $b$. In words, the curvature $\Omega_{\tau}$ is automatically
“ghost free” without the need to apply any additional requirements.

In the BRST context, the Russian formula leads to the descent equations which
subsequently characterize anomalies from a topological point of view [22, 24,
46, 48]. This form of the anomaly is referred to as the _consistent anomaly_
as it satisfies the Wess-Zumino consistency condition. However, the consistent
form of the anomaly is not gauge covariant, and one can separately introduce
the corresponding covariantized version, called the _covariant anomaly_ [67].
In this final section we will demonstrate how this story carries over into the
algebroid language. Moreover, we will give an illustration of how the
algebroid may afford us with a more complete picture by demonstrating that it
is capable of geometrizing the consistent form of the anomaly as well as the
covariant form. The conventional analysis of the BRST complex can only cover
the former. Here we will be computing anomalies from a purely cohomological
perspective which is independent of any particular physical theory. In other
words, we simply mean that the consistent and covariant anomaly polynomials we
derive have the correct topological and algebraic properties to be the
anomalous divergences of the consistent and covariant currents that appear in
the familiar physical considerations.

###  4.1 Characteristic Classes and Lie Algebroid Cohomology

The cohomological formulation of anomalies begins by considering
characteristic classes and their associated Chern-Simons forms. In this
section we will work in the context of an arbitrary Atiyah Lie algebroid $A$,
with connection reform $\omega$. Recall that the curvature of the connection
reform is given by $\Omega=\hat{{\rm
d}}\omega+\frac{1}{2}[\omega,\omega]_{L}$.

We begin by computing

$\hat{{\rm d}}\Omega=-[\omega,\Omega]_{L}\,,$ (58)

which can be recognized as the Bianchi identity, given $\hat{{\rm d}}^{2}=0$.
The pair of equations

$\hat{{\rm d}}\omega=\Omega-\frac{1}{2}[\omega,\omega]_{L},\qquad\hat{{\rm
d}}\Omega=-[\omega,\Omega]_{L}$ (59)

imply that the ring of polynomials generated by $\omega$ and $\Omega$ form a
closed subalgebra of $\Omega(A)$. This is the basis of the Chern-Weil
homomorphism, which states that one can formulate cohomology classes in
$\Omega(A)$ using such polynomials [43, 44, 68, 69].

To be precise, let $Q:L^{\otimes l}\rightarrow\mathbbm{R}$ correspond to a
symmetric, order-$l$ polynomial function on $L$ which is invariant under Lie
algebroid morphisms. Such an object can be represented by a symmetric
$l$-linear map in the tensor algebra of $L$. In other words, given the dual
basis $\\{t^{A}\\}$ for $L^{*}$, with $A=1,\ldots,\text{dim}(G)$, we can write
$Q=Q_{A_{1}\ldots A_{l}}\bigotimes_{j=1}^{l}t^{A_{j}}$. In terms of such a
symmetric, invariant polynomial we can define the _characteristic class_

$\lambda_{Q}(\Omega)=Q(\underbrace{\Omega,\ldots,\Omega}_{l})=Q_{A_{1}\ldots
A_{l}}\wedge_{j=1}^{l}\Omega^{A_{j}}\in\Omega^{2l}(A)\,.$ (60)

The Chern-Weil theorem999Strictly speaking, the Chern-Weil theorem is proven
in the context of principal bundle cohomology. However, the basis of the proof
hinges on the fact that the principal connection and curvature satisfy the
same algebraic relations as the algebroid connection and curvature given in
(59). Hence, the proof carries over to this case as well. See [64] for a more
rigorous discussion. establishes that each $\lambda_{Q}(\Omega)$ defines an
element of the cohomology class of degree $2l$ in the exterior algebra
$\Omega(A)$. Specifically, it consists of the following two statements [70]:

  1. 1.

Characteristic classes are closed $2l$-forms in $\Omega(A)$:

$\hat{{\rm d}}\lambda_{Q}(\Omega)=l!Q(\hat{{\rm
d}}\Omega,\underbrace{\Omega,\ldots,\Omega}_{l-1})=l!Q(\hat{{\rm
d}}\Omega+[\omega,\Omega]_{L},\underbrace{\Omega,\ldots,\Omega}_{l-1})=0\,,$
(61)

which follows from the symmetry of $Q$ and the Bianchi identity.

  2. 2.

Given two different connections $\omega_{1}$ and $\omega_{2}$, with respective
curvatures $\Omega_{1}$ and $\Omega_{2}$, we have that
$\lambda_{Q}(\Omega_{2})-\lambda_{Q}(\Omega_{1})\in\Omega^{2l}(A)$ is
$\hat{{\rm d}}$-exact. The relevant $(2l-1)$-form potential is defined by
introducing a one parameter family of connections
$\omega_{t}=\omega_{1}+t(\omega_{2}-\omega_{1})$ which interpolates between
$\omega_{1}$ and $\omega_{2}$ as $t$ goes from $0$ to $1$. Then,

$\lambda_{Q}(\Omega_{2})-\lambda_{Q}(\Omega_{1})=\hat{{\rm
d}}\left[Q_{A_{1}\cdots A_{l}}\int_{0}^{1}{\rm
d}t\;(\omega_{2}-\omega_{1})^{A_{1}}\wedge_{j=2}^{l}\left(\hat{{\rm
d}}\omega_{t}+\frac{1}{2}[\omega_{t},\omega_{t}]_{L}\right)^{A_{j}}\right]\,.$
(62)

An immediate corollary of the Chern-Weil theorem is that the characteristic
class $\lambda_{Q}(\Omega)$ will be globally exact if there exists a one
parameter family of connections for which $\omega_{2}=\omega$ and $\omega_{1}$
is any connection that has zero curvature.101010Note that a connection having
zero curvature does not imply $\omega=0$, which would be inconsistent with
$\omega\circ j=-Id_{L}$. Rather, in the consistent splitting one can realize a
connection with zero curvature by ensuring that the gauge field vanishes,
i.e., $b=0$. This implies $\omega_{\tau}=-\varpi$, which is consistent with
the aforementioned identity. In physical contexts, this corresponds to the
case that the connection is “pure gauge”. This inspires the topological
interpretation of the characteristic class which will be cohomologically
trivial if and only if any connection $\omega$ can be homotopically connected
to the trivial connection. Nonetheless, it will always be true locally that
any characteristic class can be written as $\hat{{\rm d}}$ acting on a
$(2l-1)$-form defined using (62). That is,

$\lambda_{Q}(\Omega)=\hat{{\rm d}}\mathscr{C}_{Q}(\omega)\,,$ (63)

where

$\mathscr{C}_{Q}(\omega):=Q_{A_{1}\cdots A_{l}}\int_{0}^{1}{\rm
d}t\,\omega^{A_{1}}\wedge_{j=2}^{l}\left(t\hat{{\rm
d}}\omega+\frac{1}{2}t^{2}[\omega,\omega]_{L}\right)^{A_{j}}$ (64)

is the Chern-Simons form associated with the symmetric invariant polynomial
$Q$. Note that (63) indicates that there does not exist
$\gamma\in\Omega^{2l-2}(A)$ such that $\mathscr{C}_{Q}=\hat{\rm d}\gamma$, and
$\mathscr{C}_{Q}$ can only be determined up to a $\hat{\rm d}$ closed term. As
we will see, a characteristic class $\lambda_{Q}(\Omega)$ and its associated
Chern-Simons form $\mathscr{C}_{Q}(\omega)$ play central roles in the
cohomological analysis of anomalies.

###  4.2 Descent Equations and the Consistent Anomaly

Now, let us move into the trivialized algebroid and work in the consistent
splitting. As we have shown, in the consistent splitting
$\omega_{\tau}=b-\varpi$, and $\hat{{\rm d}}_{\tau}\to{\rm d}+{\rm s}$. It is
therefore natural to organize the Chern-Simons form order by order in the bi-
complex $\Omega(TM,L)$ as

$\mathscr{C}_{Q}(b-\varpi)=\sum_{r+s=2l-1}\alpha^{(r,s)}(b,\varpi)\,,$ (65)

where $\alpha^{(r,s)}(b,\varpi)\in\Omega^{(r,s)}(TM,L)$, and
$\alpha^{(2l-2,1)}(b,\varpi)=\mathscr{C}_{Q}(b)$.

Combining (57) and (63) yields

$\hat{{\rm
d}}_{\tau}\mathscr{C}_{Q}(b-\varpi)=\lambda_{Q}(\Omega)=\lambda_{Q}(F)={\rm
d}\mathscr{C}_{Q}(b)\,.$ (66)

From this point it is straightforward to derive the descent equations simply
by plugging (65) into (66), and enforcing the equality order by order in the
bi-complex $\Omega^{(r,s)}(TM,L)$. The descent equations can be expressed as

${\rm d}\alpha^{(r,s)}(b,\varpi)+{\rm
s}\alpha^{(r+1,s-1)}(b,\varpi)=0\,,\qquad r+s=2l-1\,,\quad r\neq 2l-1\,,$ (67)

In particular, the term with $r=2l-3$ yields the Wess-Zumino consistency
condition:

${\rm d}\alpha^{(2l-3,2)}(b,\varpi)+{\rm s}\alpha^{(2l-2,1)}(b,\varpi)=0\,.$
(68)

On the other hand, from the fact that $\mathscr{C}_{Q}(b-\varpi)$ is not
$\hat{\rm d}_{\tau}$ exact we also have

$\alpha^{(2l-2,1)}(b,\varpi)\neq{\rm d}\gamma^{(2l-3,1)}(b,\varpi)+{\rm
s}\gamma^{(2l-2,0)}(b,\varpi)\,.$ (69)

The term $\alpha^{(2l-2,1)}(b,\varpi)$ satisfying (68) and (69) is a candidate
to be the density of the consistent anomaly (see [24, 46, 71] for a
description from a physical and algebraic perspective). Thus, we have now
demonstrated that the consistent anomaly arises naturally in the algebroid
context:

$\mathfrak{a}_{\rm con}=\int_{M}\alpha^{(2l-2,1)}(b,\varpi)\,.$ (70)

###  4.3 The Horizontal-Vertical Splitting and the Covariant Anomaly

Strictly speaking, the results discussed in the previous subsection are merely
a reformulation of those obtained in the BRST analysis [72], although now they
come from a transparent formal and geometric foundation which makes their
origin and meaning clear. However, beyond simply improving our interpretation
of the BRST analysis, we would now like to demonstrate that the algebroid
approach has the potential to produce new results in the study of anomalies.

As we have stressed, the trivialized algebroid has two relevant splittings. By
analyzing the cohomology of the consistent splitting above we found the
consistent anomaly. This inspires the question of whether the covariant
splitting also has an interpretation related to an anomaly. Following the
previous subsection, we can instead organize the Chern-Simons form on
$A_{\tau}$ order by order in the bi-complex
$\Omega^{(r,s)}(H_{\tau},V_{\tau})$. The most transparent way of doing this is
by expanding the Chern-Simons form as a polynomial in the connection
$\omega\in\Omega^{1}(V;L)$ and its curvature $\Omega\in\Omega^{2}(H;L)$. Here
again we see the Russian formula playing a crucial role in dictating that the
curvature can generate a sub-algebra of $\Omega(H_{\tau})$. The expansion of
the Chern-Simons form can now be written as

$\mathscr{C}_{Q}(\omega)=\sum_{r+s=2l-1}\beta^{(r,s)}(\omega,\Omega)\,,$ (71)

where $\beta^{(r,s)}(\omega,\Omega)\in\Omega^{(r,s)}(H,V)$ contains $r/2$
factors of the curvature and $s$ factors of the connection.

We will now show that the covariant splitting directly produces the covariant
anomaly. As was established in [67, 73, 74] the covariant anomaly is obtained
from the free variation of the Chern-Simons form with respect to the
connection. Computing this variation in the algebroid context, one arrives at
the following formula (see Appendix B for details):

$\delta\mathscr{C}_{Q}(\omega)=l\beta^{(2l-2,1)}(\delta\omega,\Omega)+\hat{{\rm
d}}\Theta(\omega,\delta\omega)\,,$ (72)

where

$\beta^{(2l-2,1)}(\delta\omega,\Omega)=\frac{1}{l}Q(\underbrace{\Omega,\ldots,\Omega}_{l-1},\delta\omega)\,.$
(73)

Hence, the covariant anomaly can be read off from the first term in (72). We
therefore recognize that the covariant anomaly is intimately related to the
term of order one in the vertical part of the Lie algebroid exterior algebra
appearing in the expansion of the Chern-Simons form. This establishes a
pleasant symmetry between the covariant anomaly and the consistent anomaly,
since the consistent anomaly was proportional to the “ghost number” one term
in the expansion of the Chern-Simons form when viewed in the consistent
splitting. We should note that from this point of view, the consistent and
covariant anomalies do not coincide precisely because $V^{*}$ is not
canonical, depending on the connection.

The covariant anomaly does not come with a series of descent equations that
leads to a consistency condition. Instead, its defining property is that it is
covariant with respect to the gauge transformation. In fact, we can now
readily interpret the geometric difference between the consistent and
covariant anomalies in the algebroid formulation. The former, being written in
the consistent splitting of the algebroid, respects the nilpotency of the
coboundary operator $\hat{{\rm d}}$ in both factors of its associated bi-
complex but spoils the gauge covariance. Conversely, the latter, although it
does not admit two nilpotent differential operators, respects the covariant
splitting defined by the connection $\omega$ and thus is endowed with gauge
covariance. Such a conclusion was not possible from the perspective of the
BRST complex, precisely because it lacked a geometry for its connection to
define a covariant splitting.

###  4.4 Examples

We close this section by exploring a pair of illuminating examples, namely the
chiral anomaly and the (type A) Lorentz-Weyl anomaly in $2d$. In both cases
the covariant and consistent forms of the anomaly are deduced by analyzing an
appropriate characteristic class and its associated Chern-Simons form. The
analysis done here can easily be generalized to arbitrary even dimension.

####  4.4.1 Chiral Anomaly in $2d$

The analysis of the chiral anomaly arises in the context of an Atiyah Lie
algebroid $A$ derived from a principal bundle $P(M,G)$, where $G$ is a
semisimple Lie group. The characteristic class that is relevant to the chiral
anomaly in $2d$ is the second Chern class111111For simplicity, we have taken a
basis such that the second Killing form is given by $\delta_{AB}$.

$\text{ch}_{2}(\Omega)=\delta_{AB}\;\Omega^{A}\wedge\Omega^{B}\,.$ (74)

The Chern-Simons form associated with $\text{ch}_{2}(\Omega)$ can be deduced
by employing the transgression formula (62):

$\mathscr{C}_{2}(\omega)=\delta_{AB}\left(\omega^{A}\wedge\hat{{\rm
d}}\omega^{B}+\frac{1}{3}\omega^{A}\wedge[\omega,\omega]_{L}^{B}\right)\,.$
(75)

Using (75), we can easily determine the algebraic form of candidates for the
covariant and consistent forms of the anomaly. To begin, still working in the
algebroid $A$ we can decompose (75) order by order in the bi-complex
$\Omega(H,V)$ by re-expressing it as a polynomial in the curvature and
connection; that is, where there is a $\hat{{\rm d}}\omega$ we will replace it
by $\Omega-\frac{1}{2}[\omega,\omega]_{L}$. The resulting expression is

$\mathscr{C}_{2}(\omega,\Omega)=\delta_{AB}\left(\omega^{A}\wedge\Omega^{B}-\frac{1}{6}\omega^{A}\wedge[\omega,\omega]_{L}^{B}\right)\,.$
(76)

In other words, the various terms in (71) are given by

$\beta^{(2,1)}(\omega,\Omega)=\delta_{AB}\;\omega^{A}\wedge\Omega^{B}\,,\qquad\beta^{(0,3)}(\omega,\Omega)=-\frac{1}{6}\delta_{AB}\;\omega^{A}\wedge[\omega,\omega]_{L}^{B}\,,$
(77)

from which we can read off by applying (72) that the covariant anomaly
polynomial is given in terms of the curvature $2\delta_{AB}\Omega^{B}$, as
expected.

To obtain the consistent anomaly polynomial, we pass to the trivialized Lie
algebroid. That is, we specify a map $\tau:A\rightarrow A_{\tau}$ along with
its inverse map $\overline{\tau}:A_{\tau}\rightarrow A$. Recall from
Subsection 3.1 that such a morphism implies the following relationships
between the connections, curvatures, and coboundary operators of the two
algebroids:

$\overline{\tau}^{*}\omega=\omega_{\tau}=b-\varpi\,,\qquad\overline{\tau}^{*}\Omega=\Omega_{\tau}=F\,,\qquad\overline{\tau}^{*}\circ\hat{{\rm
d}}=\hat{{\rm d}}_{\tau}\circ\overline{\tau}^{*}\,.$ (78)

Trivializing the Chern-Simons form, it follows from (51) that

$\displaystyle\overline{\tau}^{*}\mathscr{C}_{2}(\omega)=\mathscr{C}_{2}(\omega_{\tau})=\mathscr{C}_{2}(b)+\delta_{AB}\left(-\varpi^{A}\wedge{\rm
d}b^{B}-\frac{1}{2}b^{A}\wedge[\varpi,\varpi]_{L}^{B}+\frac{1}{6}\varpi^{A}\wedge[\varpi,\varpi]_{L}^{B}\right)\,.$
(79)

Then, the expansion (65) gives

$\begin{split}\alpha^{(3,0)}(b,\varpi)&=\mathscr{C}_{2}(b)\,,\qquad\alpha^{(2,1)}(b,\varpi)=-\delta_{AB}\varpi^{A}\wedge{\rm
d}b^{B}\,,\\\
\alpha^{(1,2)}(b,\varpi)&=-\frac{1}{2}\delta_{AB}b^{A}\wedge[\varpi,\varpi]_{L}^{B}\,,\qquad\alpha^{(0,3)}(b,\varpi)=\frac{1}{6}\delta_{AB}\varpi^{A}\wedge[\varpi,\varpi]_{L}^{B}\,.\end{split}$
(80)

The consistent anomaly polynomial can therefore be read off from the ghost
number one contribution to (79), which is $-\delta_{AB}\varpi^{A}\wedge{\rm
d}b^{B}$. Recall that $-\varpi^{A}$ corresponds to the ghost field, the
consistent anomaly can be recognized $\delta_{AB}{\rm d}b^{B}$, which is again
in agreement with the known result.

As promised, the covariant anomaly, which is written in terms of $\Omega$, is
indeed covariant, while the consistent anomaly, which is written in terms of
${\rm d}b$, is not. Moreover, it is straightforward to show that the series of
terms $\alpha^{(r,s)}(b,\varpi)$ satisfy the descent equations as introduced
in (67).

####  4.4.2 Lorentz-Weyl Anomaly in $2d$

To analyze the Lorentz-Weyl (LW) anomaly, let us begin by introducing the
geometric framework and characteristic classes for a Lorentz-Weyl structure in
arbitrary even dimension $d=2l$. Consider an Atiyah Lie algebroid $A$ derived
from a principal $G$-structure with $G=SO(1,d-1)\times\mathbbm{R}_{+}\subset
GL(d,\mathbbm{R})$. Here $SO(1,d-1)$ is the local Lorentz group, while
$\mathbbm{R}_{+}$ corresponds to local Weyl rescaling. The corresponding Lie
algebra can be expressed as
$\mathfrak{g}=\mathfrak{so}(1,d-1)\oplus\mathfrak{r}_{+}$. The adjoint bundle
of the group $G$ is given by $L=P\times_{G}\mathfrak{g}=L_{L}\oplus L_{W}$,
where $L_{L}=P\times_{SO(1,d-1)}\mathfrak{so}(1,d-1)$ and
$L_{W}=P\times_{\mathbbm{R}_{+}}\mathfrak{r}_{+}d$ correspond to the Lorentz
and Weyl factors, respectively. The connection reform on $A$ will therefore
split as $\omega=\omega_{L}+\omega_{W}$ where $\omega_{L}$ and $\omega_{W}$
are the connection reform on the Lorentz and Weyl sub-algebroids,
respectively. The curvature of the connection reform $\omega$ will have two
pieces

$\Omega=\hat{{\rm
d}}\omega+\frac{1}{2}[\omega,\omega]_{L}=\Omega_{L}+\Omega_{W}\,,$ (81)

where $\Omega_{L}\in\Omega^{2}(H;L_{L})$ is related to the Riemann tensor and
$\Omega_{W}\in\Omega^{2}(H;L_{W})$ is the gauge field strength of the Weyl
connection. We can see that the curvature $\Omega$ remains horizontal.

There are two natural invariant structures associated with $L$. The Weyl
factor $L_{W}$ is an Abelian subalgebra of $L$. Thus, the map
$\text{tr}_{W}:L\rightarrow L_{W}$ which projects an element
$\underline{\mu}\in\Gamma(L)$ down to $L_{W}$ will be invariant under the
adjoint action of $L$ on itself. In a linear representation of $L$ given by
$v_{E}:L\rightarrow\text{End}(L)$, the generators of $L_{L}$ are represented
by traceless antisymmetric matrices. Hence, as the notation indicates, the map
$\text{tr}_{W}$ can also be understood by selecting a representation and
computing the ordinary trace. In other words, for any representation $E$ and
given $\text{tr}:\text{End}(E)\rightarrow C^{\infty}(M)$ we have

$\text{tr}_{W}(\underline{\mu})=\text{tr}\circ v_{E}(\underline{\mu})\,.$ (82)

Similarly, there is an invariant structure on $L_{L}$ which will correspond to
the Pfaffian. In particular we define

$\epsilon:L^{\otimes l}\rightarrow C^{\infty}(M)\,.$ (83)

One of the defining properties of the map $\epsilon$ is that
$\epsilon(\underline{\mu}_{1},\ldots,\underline{\mu}_{l})=0$ if
$\underline{\mu_{i}}\in\Gamma(L_{W})$ for any $i$. In other words, $\epsilon$
only sees the orthogonal factor of $G$, and is an invariant polynomial on this
factor. As was the case with the trace, $\epsilon$ can be computed by passing
to a linear representation. To be precise, we should take a $2l$-dimensional
representation space $E$ equipped with an inner product $g_{E}:E\times
E\rightarrow C^{\infty}(M)$ of appropriate signature. Then, we can define the
map $w_{E}:L\rightarrow\wedge^{2}E^{*}$ such that given
$\underline{\psi}_{1},\underline{\psi}_{2}\in\Gamma(E)$ we have

$w_{E}(\underline{\mu})(\underline{\psi}_{1},\underline{\psi}_{2})=g_{E}\left(\underline{\psi}_{1},v_{E}(\underline{\mu})(\underline{\psi}_{2})\right)\,.$
(84)

Notice that $w_{E}\circ\text{tr}_{W}=0$, since a Weyl rescaling cannot be
represented by an antisymmetric matrix. Given an oriented orthonormal basis
$\\{\underline{e}_{a}\\}$ for $E$ along with its dual basis $\\{e^{a}\\}$,
with $a=1,\ldots,2l$, we can define an $SO(1,d-1)$ invariant volume form on
$E$121212Note that we are not specifying a solder form, and so we have no way
to pull this volume form back to the base. Similarly the inner product on $E$
is not directly related to a metric on the base. These facts might be thought
of as being responsible for the topological nature of the characteristic
classes discussed below.

$\text{Vol}_{E}\equiv\epsilon_{a_{1}\cdots a_{d}}e^{a_{1}}\wedge\cdots\wedge
e^{a_{d}}\,.$ (85)

Thus, in this representation we can express:

$\epsilon(\underline{\mu}_{1},\ldots,\underline{\mu}_{l})=\epsilon_{a_{1}b_{1}\cdots
a_{l}b_{l}}w_{E}(\underline{\mu}_{1})^{a_{1}b_{1}}\cdots
w_{E}(\underline{\mu}_{l})^{a_{l}b_{l}}=\epsilon^{a_{1}}{}_{b_{1}}\cdots^{a_{l}}{}_{b_{l}}v_{E}(\underline{\mu}_{1})^{b_{1}}{}_{a_{1}}\cdots
v_{E}(\underline{\mu}_{l})^{b_{l}}{}_{a_{l}}\,.$ (86)

This construction satisfies the above-mentioned properties since
$w_{E}\circ\text{tr}_{W}(\underline{\mu})=0$ and

$\epsilon(\underline{\mu},\ldots,\underline{\mu})=\text{Pf}(\underline{\mu})\,.$
(87)

Note that this construction requires $d$ to be even, as the
$\epsilon^{a_{1}}{}_{b_{1}}\cdots^{a_{l}}{}_{b_{l}}$ has an equal number of up
and down indices (signifying its Weyl invariance).

We are now prepared to introduce the relevant characteristic class for the LW
anomaly. If we intend to derive the anomaly for a $d=2l$ dimensional theory,
we must construct a characteristic class of form degree $d+2=2(l+1)$. Hence,
we must construct a symmetric and invariant linear map
$Q^{LW,l+1}:L^{\otimes(l+1)}\rightarrow\mathbbm{R}$. As we have discussed, we
have at our disposal two invariant objects corresponding to the trace (82) and
the Pfaffian (83). We therefore obtain an $(l+1)$-order symmetric invariant
polynomial by taking the symmetrized product of these two maps:

$Q^{LW,l+1}(\underline{\mu}_{1},\ldots\underline{\mu}_{l+1})=\sum_{\pi}\epsilon(\underline{\mu}_{\pi(1)},\ldots,\underline{\mu}_{\pi(l)})\,\text{tr}_{W}(\underline{\mu}_{\pi(l+1)})\,,$
(88)

where $\pi$ denotes the permutations of $(1,\ldots,l+1)$. The characteristic
class associated with $Q^{LW,l+1}$ is therefore given by
$\lambda_{Q^{LW,l+1}}(\Omega)$ as dictated in (60). While
$\lambda_{Q^{LW,l+1}}$ is the appropriate characteristic class in the LW
context, in other situations (such as a simple or semi-simple group) one finds
an Euler class.131313Indeed in the literature [75, 76, 77, 78] there is an
analysis of Cartan geometry, in which the symmetry is enhanced to $SO(2,d)$,
and the type A conformal anomaly comes from the Euler class. Descending to the
subgroup $SO(1,d-1)\times\mathbbm{R}_{+}$ considered here, one obtains (88).

Let us now specialize to the case $d=2$ and show that $\lambda_{Q^{LW,2}}$
gives rise to the LW anomaly. The characteristic class of interest takes the
following form:

$\lambda_{Q^{LW,2}}(\Omega)=\frac{1}{2}\left(\epsilon(\Omega)\wedge\text{tr}_{W}(\Omega)+\text{tr}_{W}(\Omega)\wedge\epsilon(\Omega)\right)\,.$
(89)

In the $2d$ case, since the structure group $G=SO(1,1)\times\mathbbm{R}_{+}$
is Abelian, we can write $\Omega=\hat{{\rm d}}\omega$. Hence, the Chern-Simons
form can be obtained as

$\mathscr{C}_{LW,2}(\omega,\Omega)=\frac{1}{2}\left(\epsilon(\omega)\wedge\text{tr}_{W}(\Omega)+\text{tr}_{W}(\omega)\wedge\epsilon(\Omega)\right)\,.$
(90)

To read off the covariant form of the anomaly polynomial let us pass to a
representation on $E$. Then using (82) and (86) we can write the covariant
anomaly as (ignoring the constant factor)

$\Omega_{W}\epsilon^{a}{}_{b}+\text{Pf}(\Omega_{L})\delta^{a}{}_{b}\,.$ (91)

Noticing that $\epsilon(\omega)$ and $\text{tr}_{W}(\omega)$ picks out the
Lorentz and Weyl part of the connection, respectively, the first term in the
above result should be interpreted as the Lorentz anomaly, which vanishes when
the Weyl connection is turned off; the second term is the Weyl anomaly in
$2d$, which is proportional to the Ricci scalar of the spacetime. Therefore,
the LW anomaly is the mixed anomaly between the Lorentz and Weyl symmetry. In
fact, it is easy to see that by adding a total derivative term, one can remove
the Lorentz anomaly or Weyl anomaly but cannot remove both simultaneously.

To obtain the consistent form, we must employ a Lie algebroid trivialization.
Under the trivialization we find that

$\overline{\tau}^{*}\omega=b-\varpi_{L}+a-\varpi_{W}\,,\qquad\overline{\tau}^{*}\Omega=R+f\,,\qquad\overline{\tau}^{*}\circ\hat{{\rm
d}}=({\rm d}+{\rm s}_{L}+{\rm s}_{W})\circ\overline{\tau}^{*}\,,$ (92)

where $b$ and $a$ are the spin connection and Weyl connection on $M$, and $R$
and $f$ are their curvature 2-forms, respectively. The pairs $(\varpi_{L},{\rm
s}_{L})$ and $(\varpi_{W},{\rm s}_{W})$ are the Maurer-Cartan forms and BRST
operators for the $SO(1,1)$ and $\mathbbm{R}_{+}$ factors of $L$. Let $B=b+a$
and $\varpi=\varpi_{L}+\varpi_{W}$ denote the combined gauge field and Maurer-
Cartan forms. We subsequently identify the consistent LW anomaly from
$Q^{LW,2}(\varpi,{\rm d}B)$. Since in the index notation of the representation
we have

$({\rm d}B)^{a}{}_{b}=R\epsilon^{a}{}_{b}+f\delta^{a}{}_{b}\,,$ (93)

the consistent form of the LW anomaly is merely the pullback of the covariant
form by the trivialization $\overline{\tau}$, which reads

$f\epsilon^{a}{}_{b}+\text{Pf}(R)\delta^{a}{}_{b}\,,$ (94)

which has the same form as (91). This follows in this particular case from the
fact that $G$ is an Abelian group when $d=2$.

A simplified account of the LW anomaly in two dimensions appeared recently in
Appendix A of [79]. Note that here we have focussed on the type A Weyl
anomaly, and the type B Weyl anomaly remains an open question in general
dimension. A more elaborate discussion is required since obstruction tensors
are expected to make an appearance [80, 81, 82, 83, 84]. We expect to return
to this issue, as well as other $G$-structures, in a future publication.

##  5 Conclusions

In the introduction we raised a series of questions about the BRST formalism.
In the course of this paper we have provided answers to each of these
questions by geometrically formalizing the BRST complex in terms of the Atiyah
Lie algebroid. As we promised in the introduction, each answer follows
immediately from the geometry of the Atiyah Lie algebroid.

Q: Why should the Grassmann-valued fields $c^{A}(x)$, which started their life
in the BRST quantization procedure have an interpretation as the generators of
local gauge transformations? And why is it reasonable to combine the de Rham
complex and the ghost algebra into a single exterior bi-algebra?

A: In the algebroid context the Maurer-Cartan form $\varpi\in\Omega^{1}(L;L)$
plays the role of the gauge ghost, and is also a generator of local gauge
transformations. Working in the consistent splitting the exterior algebra of
the trivialized algebroid $A_{\tau}$ subsequently takes the form of a bi-
complex $\Omega^{(p,q)}(TM,L;E)$, where $p$ is the form degree with respect to
the de Rham cohomology of $M$, and $q$ is the “ghost number”. The coboundary
operator $\hat{{\rm d}}_{\tau}$ takes explicitly the form ${\rm d}+{\rm s}$ on
this exterior algebra, where ${\rm d}$ is the de Rham differential and ${\rm
s}$ is the BRST operator.

Q: Why is it reasonable to consider $\hat{A}=A+c$ as a “connection”, and
moreover what horizontal distribution does it define?

A: Still in the context of the trivialized Lie algebroid, one can introduce a
connection reform, $\omega_{\tau}:A_{\tau}\rightarrow L$, defining the
horizontal distribution $H_{\tau}=\text{ker}(\omega_{\tau})$ for which
$A_{\tau}=H_{\tau}\oplus V_{\tau}$. In the consistent splitting
$\omega_{\tau}=b-\varpi$, where $b:TM\rightarrow L$ is a local gauge field,
and $\varpi:L\rightarrow L$ is the Maurer-Cartan form on $L$. Hence, $\omega$
reproduces the “connection” $\hat{A}$ defined in the BRST complex, where again
we see the role of the gauge ghost being played by the Maurer-Cartan form.

Q: Why should the “curvature” $\hat{F}$ be taken to have ghost number zero?
And why does enforcing this requirement turn the BRST operator $s$ into the
Chevalley-Eilenberg operator for the Lie algebra of the structure group?

A: $\hat{F}$ in the context of the trivialized Lie algebroid is represented by
$\Omega_{\tau}=\hat{{\rm
d}}_{\tau}\omega_{\tau}+\frac{1}{2}[\omega_{\tau},\omega_{\tau}]_{L}$, namely
the curvature associated with $\omega_{\tau}$, which is fully horizontal as a
built-in geometric property of the algebroid. In the consistent splitting,
this reproduces the Russian formula and the BRST transformation as we
presented in (57).

The culmination of all of these facts gives rise to the descent equations (67)
and the Wess-Zumino consistency condition (68). Given a characteristic class
$\lambda_{Q}(\Omega)$ with associated Chern-Simons form
$\mathscr{C}_{Q}(\omega)$ we have

$\hat{{\rm d}}_{\tau}\mathscr{C}_{Q}(\omega)=({\rm d}+{\rm
s})\mathscr{C}_{Q}(b-\varpi)={\rm d}\mathscr{C}_{Q}(b)\,.$ (95)

From the above equation, one can immediately compute the _consistent_ anomaly
polynomial, which corresponds to the ghost number one contribution to
$\mathscr{C}_{Q}(b-\varpi)$, and can be shown to be an element of the first
cohomology of the BRST operator ${\rm s}$ once integrated over a space of
appropriate dimension. Furthermore, one can also obtain the _covariant_ form
of the anomaly by viewing the Chern-Simons form in the covariant splitting and
extracting the terms contributing with one exterior power in the vertical sub-
bundle of the associated exterior algebra (multiplied by the order $l$ of
$Q$). Although the formulae for finding the consistent and covariant anomalies
have been known [67], our approach to these anomalies provides a meaningful
explanation as to why the consistent anomaly is consistent and the covariant
anomaly is covariant. From the algebroid perspective, they just correspond to
different choices of splitting.

To understand the complete picture of the consistent and covariant anomalies
as well as the anomaly inflow mechanism that relates them, we will have to
further exploit the structure of the configuration space of Lie algebroid
connections. In this paper we established a powerful approach for studying Lie
algebroid isomorphisms in terms of commutative diagrams, which found a
physical interpretation as a unified tool for implementing diffeomorphisms and
gauge transformations. In a partner paper [65] we make use of this
construction to define a new geometric formalism for understanding the
extended configuration space of arbitrary gauge theories. We refer to this
construction as the configuration algebroid. We demonstrated that the
configuration algebroid provides a suitable quantification of the local
degrees of freedom in a gauge theory, leading to a fully integrable algebra of
charges associated with the local symmetries of a theory. From the point of
view of the configuration algebroid, the presence of anomalies is associated
with the question of whether the charge algebra is centrally extended. In
forthcoming work we will combine the insights of this paper with [65] to
describe anomalies as topological features of the configuration algebroid, and
demonstrate how the anomaly inflow mechanism can be incorporated into the
algebroid language.

## Acknowledgements

We thank Luca Ciambelli, Pin-Chun Pai, Manthos Karydas and Mike Stone for
conversations. This work was supported by the U.S. Department of Energy under
contract DE-SC0015655.

##  Appendix A Chain Maps

In this appendix we offer a direct computation verifying that a Lie algebroid
isomorphism $\varphi:A_{1}\rightarrow A_{2}$ satisfying the compatibility
condition (19) induces a chain map on the exterior algebras of $A_{1}$ and
$A_{2}$. It is sufficient to show that this condition holds for 0-forms and
1-forms, since $\hat{{\rm d}}$ acts as a derivation with respect to the wedge
product and the full exterior algebra is generated by the set of 1-forms along
with the wedge product. First we look at the 0-form case. Let
$\psi\in\Omega^{0}(A_{2};E_{2})$, and
$\underline{\mathfrak{X}}\in\Gamma(A_{1})$. Then,

$\displaystyle(\varphi^{*}\hat{{\rm
d}}_{2}\psi)(\underline{\mathfrak{X}})=g_{\varphi}^{-1}\big{(}\hat{{\rm
d}}_{2}\psi\circ\varphi(\underline{\mathfrak{X}})\big{)}=g_{\varphi}^{-1}\big{(}\phi_{E_{2}}\circ\varphi(\underline{\mathfrak{X}})(\psi)\big{)}$
$\displaystyle={}$ $\displaystyle
g_{\varphi}^{-1}\big{(}\phi_{E_{2}}\circ\varphi(\underline{\mathfrak{X}})\big{(}g_{\varphi}g_{\varphi}^{-1}(\psi)\big{)}\big{)}=\phi_{E_{1}}(\underline{\mathfrak{X}})\big{(}g_{\varphi}^{-1}(\psi)\big{)}=(\hat{{\rm
d}}_{1}\varphi^{*}\psi)(\underline{\mathfrak{X}})\,,$ (A.1)

where in the first equality we used (20), in the second equality we used the
definition of the Lie algebroid differential via the Koszul formula (2.2), and
in the fourth equatlity we used (19).

Now we move on to the 1-form case. Let $\eta\in\Omega^{1}(A_{2};E_{2})$, and
take $\underline{\mathfrak{X}},\underline{\mathfrak{Y}}\in\Gamma(A_{1})$. We
can write

$\displaystyle(\varphi^{*}\hat{\rm
d}_{2}\eta)(\underline{\mathfrak{X}},\underline{\mathfrak{Y}})=g_{\varphi}^{-1}[(\hat{\rm
d}_{2}\eta)(\varphi(\underline{\mathfrak{X}}),\varphi(\underline{\mathfrak{Y}}))]$
$\displaystyle={}$ $\displaystyle
g_{\varphi}^{-1}\Big{[}\phi_{E_{2}}\circ\varphi(\underline{\mathfrak{X}})(\eta\circ\varphi(\underline{\mathfrak{Y}}))-\phi_{E_{2}}\circ\varphi(\underline{\mathfrak{Y}})(\eta\circ\varphi(\underline{\mathfrak{X}}))-\eta([\varphi(\underline{\mathfrak{X}}),\varphi(\underline{\mathfrak{Y}})]_{A_{2}})\Big{]}$
$\displaystyle={}$
$\displaystyle\phi_{E_{1}}(\underline{\mathfrak{X}})\big{(}\varphi^{*}\eta(\underline{\mathfrak{Y}})\big{)}-\phi_{E_{1}}(\underline{\mathfrak{Y}})\big{(}\varphi^{*}\eta(\underline{\mathfrak{X}})\big{)}-\varphi^{*}\eta\big{(}[\underline{\mathfrak{X}},\underline{\mathfrak{Y}}]_{A_{1}}\big{)}$
$\displaystyle={}$ $\displaystyle(\hat{{\rm
d}}_{1}\varphi^{*}\underline{\eta})(\underline{\mathfrak{X}},\underline{\mathfrak{Y}})\,,$
(A.2)

where again in the first equaltiy we used (20), in the second equality we used
(2.2), and in third equality we applied (19) and (20).

##  Appendix B Free Variation and the Covariant Anomaly

In Subsection 4.3, we introduced that the covariant anomaly can be derived by
taking the free variation of the Chern-Simons form $\mathscr{C}_{Q}(\omega)$
in the covariant splitting, as shown in equation (72). We will now provide an
explicit demonstration of this derivation. Following the approach presented in
[67], we introduce a nilpotent operator
$K:\Omega^{p}(A;L)\to\Omega^{p-1}(A;L)$ that acts as follows:

$\displaystyle K\omega=0\,,\qquad K\Omega=\delta\omega\,,\qquad
K\delta\omega=0\,.$ (B.3)

Then, the variation operator on $\omega$ and $\Omega$ can be written as

$\displaystyle\delta=K\hat{\rm d}+\hat{\rm d}K\,.$ (B.4)

When performing the variation of the Chern-Simons form:

$\displaystyle\delta\mathscr{C}_{Q}=K\hat{\rm d}\mathscr{C}_{Q}+\hat{\rm
d}K\mathscr{C}_{Q}\,,$ (B.5)

the second term is a total derivative, and thus all we have to show is that
the first term in (B.5) gives rise to the first term in (72), namely
$\beta^{(2l-2,1)}(\delta\omega,\Omega)$. Using the transgression formula (64),
one finds

$\displaystyle K\hat{\rm d}\mathscr{C}_{Q}(\omega)={}$ $\displaystyle
lQ_{A_{1}\cdots A_{l}}\int_{0}^{1}{\rm
d}t\,\delta\omega^{A_{1}}\wedge_{j=2}^{l}\left(t\Omega+\frac{1}{2}(t^{2}-t)[\omega,\omega]_{L}\right)^{A_{j}}$
$\displaystyle+\frac{l-1}{2}Q_{A_{1}\cdots A_{l}}\int_{0}^{1}{\rm
d}t\,t^{2}\delta\omega^{A_{1}}[\omega,\omega]_{L}^{A_{2}}\wedge_{j=3}^{l}\left(t\Omega+\frac{1}{2}(t^{2}-t)[\omega,\omega]_{L}\right)^{A_{j}}\,.$
(B.6)

To further evaluate this, it is not difficult to first perform the integral of
the following form:

$\displaystyle\int_{0}^{1}{\rm
d}t\left[l\Big{(}tA+\frac{t^{2}-t}{2}B\Big{)}^{l-1}+\frac{l-1}{2}t^{2}B\Big{(}tA+\frac{t^{2}-t}{2}B\Big{)}^{l-2}\right]=A^{l-1}\,.$
(B.7)

Then, taking $A$ as $\Omega$ and $B$ as $[\omega,\omega]_{L}$, the integral in
(B.6) yields

$\displaystyle K\hat{\rm
d}\mathscr{C}_{Q}(\omega)=Q(\underbrace{\Omega,\ldots,\Omega}_{l-1},\delta\omega)\,.$
(B.8)

Now we can compare this with $\beta^{(2l-2,1)}(\delta\omega,\Omega)$. From
(64), one can pick up the term with a single $\omega$ and find

$\displaystyle\beta^{(2l-2,1)}(\omega,\Omega)=Q_{A_{1}\cdots
A_{l}}\int_{0}^{1}{\rm
d}t\,\omega^{A_{1}}t^{l-1}\wedge_{j=2}^{l}\Omega^{A_{j}}=\frac{1}{l}Q(\underbrace{\Omega,\ldots,\Omega}_{l-1},\omega)\,,$
(B.9)

and hence

$\beta^{(2l-2,1)}(\delta\omega,\Omega)=\frac{1}{l}Q(\underbrace{\Omega,\ldots,\Omega}_{l-1},\delta\omega)\,.$
(B.10)

Therefore, we can see that (B.5) can be written as

$\delta\mathscr{C}_{Q}(\omega)=l\beta^{(2l-2,1)}(\delta\omega,\Omega)+\hat{\rm
d}\Theta(\omega,\delta\omega)\,,$ (B.11)

where $\Theta\equiv K\mathscr{C}_{Q}$. The covariant anomaly can be read off
from the first term, while the $\Theta$ in the second term serves as the
Bardeen-Zumino polynomial which covariantizes the consistent anomaly when
added to the anomalous current [67].

## References

  * [1] C.-N. Yang and R. L. Mills, “Conservation of Isotopic Spin and Isotopic Gauge Invariance,” Phys. Rev. 96 (1954) 191–195.
  * [2] M. F. Atiyah, “Complex Analytic Connections in Fibre Bundles,” Transactions of the American Mathematical Society 85 (1957) no. 1, 181–207.
  * [3] T. T. Wu and C. N. Yang, “Concept of Nonintegrable Phase Factors and Global Formulation of Gauge Fields,” Phys. Rev. D 12 (1975) 3845–3857.
  * [4] F. Atiyah, Geometry of Yang-Mills Fields, pp. 216–221.  Springer Berlin Heidelberg, Berlin, Heidelberg, 1978.  https://doi.org/10.1007/3-540-08853-9_18.
  * [5] M. Daniel and C. M. Viallet, “The Geometrical Setting of Gauge Theories of the Yang-Mills Type,” Rev. Mod. Phys. 52 (1980) 175–197.
  * [6] J. François, L. Serge, and M. Thierry, Gauge Field Theories: Various Mathematical Approaches, pp. 177–226.  4, 2014.  arXiv:1404.4604 [math-ph].
  * [7] C. Becchi, A. Rouet, and R. Stora, “The Abelian Higgs-Kibble Model, Unitarity of the S-Operator,” Physics Letters B 52 (1974) no. 3, 344–346.
  * [8] C. Becchi, A. Rouet, and R. Stora, “Renormalization of Gauge Theories,” Ann. Phys. 98 (1976) 287–321.
  * [9] I. Tyutin, “Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism,” arXiv:0812.0580 [hep-th].
  * [10] G. Bandelloni, “Yang-Mills Cohomology in Four-dimensions,” J. Math. Phys. 27 (1986) 2551–2557.
  * [11] M. Henneaux, “Duality Theorems in BRST Cohomology,” Annals Phys. 194 (1989) 281.
  * [12] M. Henneaux, “On the Gauge Fixed BRST Cohomology,” Phys. Lett. B 367 (1996) 163–169, arXiv:hep-th/9510116.
  * [13] N. Dragon, “BRS Symmetry and Cohomology,” arXiv:hep-th/9602163.
  * [14] M. Dubois-Violette, M. Henneaux, M. Talon, and C.-M. Viallet, “Some Results on Local Cohomologies in Field Theory,” Phys. Lett. B 267 (1991) 81–87.
  * [15] F. Brandt, “Local BRST Cohomology and Covariance,” Commun. Math. Phys. 190 (1997) 459–489, arXiv:hep-th/9604025.
  * [16] G. Barnich, F. Brandt, and M. Henneaux, “Local BRST Cohomology in Gauge Theories,” Phys. Rept. 338 (2000) 439–569, arXiv:hep-th/0002245.
  * [17] S. L. Adler, “Axial Vector Vertex in Spinor Electrodynamics,” Phys. Rev. 177 (1969) 2426–2438.
  * [18] J. S. Bell and R. Jackiw, “A PCAC Puzzle: $\pi^{0}\to\gamma\gamma$ in the $\sigma$-model,” Nuovo Cim. A 60 (1969) 47–61.
  * [19] G. ’t Hooft, “Symmetry Breaking through Bell-Jackiw Anomalies,” Phys. Rev. Lett. 37 (1976) 8–11.
  * [20] K. Fujikawa, “Path Integral Measure for Gauge Invariant Fermion Theories,” Phys. Rev. Lett. 42 (1979) 1195–1198.
  * [21] L. Bonora and P. Cotta-Ramusino, “Some Remarks on BRS Transformations, Anomalies and the Cohomology of the Lie Algebra of the Group of Gauge Transformations,” Comm. Math. Phys. 87 (1983) 589.
  * [22] L. Alvarez-Gaume and P. H. Ginsparg, “The Topological Meaning of Non-Abelian Anomalies,” Nucl. Phys. B 243 (1984) 449–474.
  * [23] E. Witten, “Global Aspects of Current Algebra,” Nucl. Phys. B 223 (1983) 422–432.
  * [24] B. Zumino, “Chiral Anomalies and Differential Geometry,” in Current Algebra and Anomalies, pp. 361–392.  Princeton University Press, 1985.
  * [25] P. C. Nelson and L. Alvarez-Gaume, “Hamiltonian Interpretation of Anomalies,” Commun. Math. Phys. 99 (1985) 103.
  * [26] J. Thierry-Mieg, “Explicit Classical Construction of the Faddeev-Popov Ghost Field,” Nuovo Cim. A 56 (1980) 396.
  * [27] J. Thierry-Mieg, “Geometrical Reinterpretation of Faddeev-Popov Ghost Particles and BRS Transformations,” J. Math. Phys. 21 (1980) 2834–2838.
  * [28] M. Quiros, F. J. de Urries, J. Hoyos, M. L. Mazon, and E. Rodriguez, “Geometrical Structure of Faddeev-Popov Fields and Invariance Properties of Gauge Theories,” J. Math. Phys. 22 (1981) 1767.
  * [29] J. Thierry-Mieg, “Classical Geometrical Interpretation of Ghost Fields and Anomalies in Yang-Mills Theory and Quantum Gravity,” in Symposium on Anomalies, Geometry, Topology.  5, 1985.
  * [30] Y. Ne’eman, T. Regge, and J. Thierry-Mieg, “Ghost-Fields, BRS and Extended Supergravity as Applications of Gauge Geometry,” in 19th International Conference on High-Energy Physics, pp. 552–554.  1979\.
  * [31] L. Baulieu and J. Thierry-Mieg, “The Principle of BRS Symmetry: An Alternative Approach to Yang-Mills Theories,” Nucl. Phys. B 197 (1982) 477–508.
  * [32] J. Thierry-Mieg and L. Baulieu, “Covariant Quantization of Non-Abelian Antisymmetric Tensor Gauge Theories,” Nucl. Phys. B 228 (1983) 259–284.
  * [33] J. Thierry-Mieg, “BRS Analysis of Zamolodchikov’s Spin Two and Three Current Algebra,” Phys. Lett. B 197 (1987) 368.
  * [34] L. Bonora, P. Pasti, and M. Tonin, “Extended BRS Symmetry in Non-Abelian Gauge Theories,” Nuovo Cim. A 64 (1981) 307.
  * [35] A. C. Hirshfeld and H. Leschke, “Quantization of Yang-Mills Fields in Superspace,” Phys. Lett. B 101 (1981) 48–50.
  * [36] J. Hoyos, M. Quiros, J. Ramirez Mittelbrunn, and F. de Urries, “Superfiber Bundle Structure of Quantized Gauge Theories,” J. Math. Phys. 23 (1982) 1504.
  * [37] L. Baulieu and M. P. Bellon, “$p$ Forms and Supergravity: Gauge Symmetries in Curved Space,” Nucl. Phys. B 266 (1986) 75–124.
  * [38] C. Hull, B. J. Spence, and J. Vazquez-Bello, “The Geometry of Quantum Gauge Theories: A Superspace Formulation of BRST Symmetry,” Nucl. Phys. B 348 (1991) 108–124.
  * [39] L. Bonora and R. P. Malik, “BRST and Superfield Formalism—A Review,” Universe 7 (2021) no. 8, 280, arXiv:2107.12796 [hep-th].
  * [40] C. Chevalley and S. Eilenberg, “Cohomology Theory of Lie Groups and Lie Algebras,” Trans. Am. Math. Soc. 63 (1948) 85–124.
  * [41] B. Zumino, “Cohomology of Gauge Groups: Cocycles and Schwinger Terms,” Nucl. Phys. B 253 (1985) 477–493.
  * [42] F. Brandt, N. Dragon, and M. Kreuzer, “Lie Algebra Cohomology,” Nucl. Phys. B 332 (1990) 250–260.
  * [43] A. Weil, “Géométrie différentielle des espaces fibrés (letters to Chevalley and Koszul), 1949,” in André Weil Oeuvres Scientifiques/Collected Papers, vol. 1 (1926-1951).  Springer, Berlin, 1979.
  * [44] H. Cartan, “Cohomologie réelle d’un espace fibré principal différentiable. I : notions d’algèbre différentielle, algèbre de Weil d’un groupe de Lie,” Séminaire Henri Cartan 2 (1949-1950) .
  * [45] R. Stora, “Continuum Gauge Theories,” in New Developments in Quantum Field Theory and Statistical Mechanics Cargèse 1976, M. Lévy and P. Mitter, eds., pp. 201–224.  Springer US, Boston, MA, 1977.
  * [46] B. Zumino, Y.-S. Wu, and A. Zee, “Chiral Anomalies, Higher Dimensions, and Differential Geometry,” Nucl. Phys. B 239 (1984) 477–507.
  * [47] S. P. Sorella, “Algebraic Characterization of the Wess-Zumino Consistency Conditions in Gauge Theories,” Comm. Math. Phys. 157 (1993) 231–243, arXiv:9302136 [hep-th].
  * [48] J. Wess and B. Zumino, “Consequences of Anomalous Ward Identities,” Phys. Lett. B 37 (1971) 95–97.
  * [49] J. Pradines, “Théorie de Lie pour les groupoïdes différentiables. Relations entre propriétés locales et globales,” C. R. Acad. Sci. Paris Sér. A-B 263 (1966) A907–A910.
  * [50] J. Pradines, “Théorie de Lie pour les groupoïdes différentiables. Calcul différentiel dans la catégorie des groupoïdes infinitésimaux,” C. R. Acad. Sci. Paris Sér. A-B 264 (1967) A245–A248.
  * [51] A. Weinstein, “Poisson Geometry,” Differ. Geom. Appl. 9 (1998) no. 1, 213 – 238.
  * [52] Y. Kosmann-Schwarzbach and K. Mackenzie, “Differential Operators and Actions of Lie Algebroids,” arXiv:math/0209337 [math.DG].
  * [53] R. L. Fernandes and I. Struchiner, “Lie Algebroids and Classification Problems in Geometry,” arXiv:0712.3198 [math.DG].
  * [54] K. C. H. Mackenzie, Lie Groupoids and Lie Algebroids in Differential Geometry.  London Mathematical Society Lecture Note Series. Cambridge University Press, 1987.
  * [55] K. C. Mackenzie, General Theory of Lie Groupoids and Lie Algebroids.  Cambridge University Press, Cambridge, 2005.
  * [56] G. Ramandi and N. Boroojerdian, “Forces Unification in the Framework of Transitive Lie Algebroids,” Int. J. Theor. Phys. 54 (2015) no. 5, 1581–1593.
  * [57] C.-M. Marle, “Differential Calculus on A Lie Algebroid and Poisson Manifolds,” arXiv:0804.2451 [math.DG].
  * [58] S. Lazzarini and T. Masson, “Connections on Lie Algebroids and on Derivation-Based Noncommutative Geometry,” J. Geom. Phys. 62 (2012) no. 2, 387–402.
  * [59] C. Fournel, S. Lazzarini, and T. Masson, “Formulation of Gauge Theories on Transitive Lie Algebroids,” J. Geom. Phys. 64 (2013) 174–191, arXiv:1205.6725 [math-ph].
  * [60] U. Carow-Watamura, M. A. Heller, N. Ikeda, T. Kaneko, and S. Watamura, “Off-shell Covariantization of Algebroid Gauge Theories,” PTEP 2017 (2017) no. 8, 083B01, arXiv:1612.02612 [hep-th].
  * [61] A. Kotov and T. Strobl, “Lie Algebroids, Gauge Theories, and Compatible Geometrical Structures,” Rev. Math. Phys. 31 (2018) no. 04, 1950015, arXiv:1603.04490 [math.DG].
  * [62] J. Attard, J. François, S. Lazzarini, and T. Masson, “Cartan Connections and Atiyah Lie Algebroids,” J. Geom. Phys. 148 (2020) 103541, arXiv:1904.04915 [math-ph].
  * [63] L. Ciambelli and R. G. Leigh, “Lie Algebroids and the Geometry of Off-shell BRST,” Nucl. Phys. B 972 (2021) 115553, arXiv:2101.03974 [hep-th].
  * [64] R. L. Fernandes, “Lie Algebroids, Holonomy and Characteristic Classes,” Adv. Math. 170 (2002) no. 1, 119–179, arXiv:math/0007132 [math.DG].
  * [65] M. S. Klinger, R. G. Leigh, and P.-C. Pai, “Extended Phase Space in General Gauge Theories,” arXiv:2303.06786 [hep-th].
  * [66] H. Kanno, “Weil Algebra Structure and Geometrical Meaning of BRST Transformation in Topological Quantum Field Theory,” Z. Phys. C 43 (1989) 477.
  * [67] W. A. Bardeen and B. Zumino, “Consistent and Covariant Anomalies in Gauge and Gravitational Theories,” Nucl. Phys. B 244 (1984) 421–453.
  * [68] S.-S. Chern, “Characteristic Classes of Hermitian Manifolds,” Ann. Math. (1946) 85–121.
  * [69] S.-S. Chern, “The Geometry of G-Structures,” Bull. Amer. Math. Soc. 72 (1966) no. 2, 167–219.
  * [70] M. Nakahara, Geometry, Topology and Physics.  CRC press, Boca Raton, 2018.
  * [71] M. Göckeler and T. Schücker, Differential Geometry, Gauge Theories, and Gravity.  Cambridge Monographs on Mathematical Physics. Cambridge University Press, 10, 1989.
  * [72] A. Bilal, “Lectures on Anomalies,” arXiv:0802.0634 [hep-th].
  * [73] M. Stone, “Gravitational Anomalies and Thermal Hall Effect in Topological Insulators,” Phys. Rev. B 85 (2012) 184503, arXiv:1201.4095 [cond-mat.mes-hall].
  * [74] T. L. Hughes, R. G. Leigh, and O. Parrikar, “Torsional Anomalies, Hall Viscosity, and Bulk-Boundary Correspondence in Topological States,” Phys. Rev. D 88 (2013) no. 2, 025040, arXiv:1211.6442 [hep-th].
  * [75] N. Boulanger, “General Solutions of the Wess-Zumino Consistency Condition for the Weyl Anomalies,” JHEP 07 (2007) 069, arXiv:0704.2472 [hep-th].
  * [76] N. Boulanger, J. François, and S. Lazzarini, “A Classification of Global Conformal Invariants,” J. Phys. A 52 (2019) no. 11, 115201, arXiv:1809.05445 [math-ph].
  * [77] J. François, S. Lazzarini, and T. Masson, “Residual Weyl Symmetry out of Conformal Geometry and Its BRST Structure,” JHEP 09 (2015) 195, arXiv:1504.08297 [math-ph].
  * [78] J. François, S. Lazzarini, and T. Masson, “Becchi-Rouet-Stora-Tyutin Structure for the Mixed Weyl-Diffeomorphism Residual Symmetry,” J. Math. Phys. 57 (2016) no. 3, 033504, arXiv:1508.07666 [math-ph].
  * [79] A. Campoleoni, L. Ciambelli, A. Delfante, C. Marteau, P. M. Petropoulos, and R. Ruzziconi, “Holographic Lorentz and Carroll Frames,” JHEP 12 (2022) 007, arXiv:2208.07575 [hep-th].
  * [80] C. Fefferman and C. Graham, “Q-Curvature and Poincaré Metrics,” Math. Res. Lett. 9 (2002) no. 2-3, 139–151, arXiv:math/0110271 [math.DG].
  * [81] C. R. Graham and K. Hirachi, “The Ambient Obstruction Tensor and Q-Curvature,” IRMA Lect. Math. Theor. Phys. 8 (2005) 59–71, arXiv:math/0405068.
  * [82] C. R. Graham, “Extended Obstruction Tensors and Renormalized Volume Coefficients,” Adv. Math. 220 (2009) no. 6, 1956–1985, arXiv:0810.4203 [math.DG].
  * [83] W. Jia and M. Karydas, “Obstruction Tensors in Weyl Geometry and Holographic Weyl Anomaly,” Phys. Rev. D 104 (2021) no. 12, 126031, arXiv:2109.14014 [hep-th].
  * [84] W. Jia, M. Karydas, and R. G. Leigh, “Weyl-Ambient Geometries,” Nucl. Phys. B 991 (2023) 116224, arXiv:2301.06628 [hep-th].