RBI-ThPhys-2024-01

Athanasios Chatzistavrakidis,♣ Noriaki Ikeda,∞ Larisa
Jonke$\,{}^{{\color[rgb]{0,1,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,1,0}\pgfsys@color@cmyk@stroke{1}{0}{1}{0}\pgfsys@color@cmyk@fill{1}{0}{1}{0}{\clubsuit}},\mho}$

♣ Division of Theoretical Physics, Rudjer Bošković Institute
Bijenička 54, 10000 Zagreb, Croatia

∞Department of Mathematical Sciences, Ritsumeikan University
Kusatsu, Shiga 525-8577, Japan

℧School of Theoretical Physics, Dublin Institute for Advanced Studies
10 Burlington Road, Dublin 4, Ireland

We study twisted Courant sigma models, a class of topological field theories
arising from the coupling of 3D 0-/2-form BF theory and Chern-Simons theory
and containing a 4-form Wess-Zumino term. They are examples of theories
featuring a nonlinearly open gauge algebra, where products of field equations
appear in the commutator of gauge transformations, and they are reducible
gauge systems. We determine the solution to the master equation using a
technique, the BRST power finesse, that combines aspects of the AKSZ
construction (which applies to the untwisted model) and the general BV-BRST
formalism. This allows for a geometric interpretation of the BV coefficients
in the interaction terms of the master action in terms of an induced
generalised connection on a 4-form twisted (pre-)Courant algebroid, its
Gualtieri torsion and the basic curvature tensor. It also produces a frame
independent formulation of the model. We show, moreover, that the gauge fixed
action is the sum of the classical one and a BRST commutator, as expected from
a Schwarz type topological field theory.

###### Contents

  1. 1 Introduction
  2. 2 Dg manifolds, twists and target space covariance
    1. 2.1 Revisiting Lie algebroids and Q1 manifolds
    2. 2.2 Poisson vs. twisted Poisson structure
    3. 2.3 Pre-Courant algebroids and Q2 manifolds
  3. 3 BRST structure of twisted Courant sigma models
    1. 3.1 Action, gauge symmetry & nonlinearly open gauge algebra
    2. 3.2 On-shell closed BRST transformations
    3. 3.3 Manifestly covariant BRST transformations
  4. 4 The BRST power finesse and the master action
  5. 5 The gauge fixed master action
  6. 6 Conclusions and outlook
  7. A Classical gauge invariance
  8. B BRST transformations in local form
  9. C Technical annex to the BRST power finesse
  10. D Complete minimal solution of the master equation

##  1 Introduction

The Courant sigma model is a three-dimensional topological field theory that
was first constructed as a nontrivial coupling between 3D BF theory of
$0$-form and $2$-form and Chern-Simons theory [1]. Aspects of its quantization
as a model of topological open membranes were discussed in Refs. [2, 3] and
its construction as an AKSZ theory appears in Ref. [4]. The local gauge
structure of the model is governed by the local form of the axioms of a
Courant algebroid or equivalently by the homological vector field on a
differential graded (dg) symplectic supermanifold of degree 2 [5, 6]. The
model has appeared in several instances in physics, especially in the context
of nongeometric string backgrounds and T-duality [7, 8, 9, 10].

A direct generalization of the Courant sigma model appeared in the context of
first class constrained Hamiltonian systems together with a Wess-Zumino term
supported on a 4D world volume [11]. This is associated with membranes in
nontrivial 4-form flux and it is the direct analogon of the transition from
Poisson to Wess-Zumino-Witten or 3-form twisted Poisson sigma models in 2D
[12]. The gauge structure of this model is governed by a particular type of
_pre-Courant_ algebroids, the latter being relaxed structures originating from
Courant algebroids by dropping the Jacobi identity of the Dorfman bracket
[13]. Hence we refer to the corresponding 3D topological field theory as a
(4-form) twisted Courant sigma model.

Both Courant and twisted Courant sigma models feature all three
characteristics of a gauge system that requires treatment within the BV-BRST
formalism [14]. Their gauge algebra is closed only on the stationary surface
(on-shell), the coefficients of the gauge algebra are field-dependent and not
constant and moreover there are reducible gauge symmetries simply because
there are 2-forms in the theory. One can refer to, for instance, the textbook
[15] for the general theory of gauge systems. Therefore these constitute a
class of examples of the simplest theories that feature all these three
properties and can thus serve as prototypes to study several relations between
field theory and generalised geometric or higher structures.

On the other hand, there is an important difference between the twisted and
untwisted cases. The target space of the untwisted case is a QP manifold and
its master action within the BRST formalism need not be found by the usual
complicated procedure; instead it acquires a geometric form within the AKSZ
construction [16], see also the earlier fundamental papers [17] and [18]. This
is no longer true for the twisted model. The compatibility of the Q and P
structures is obstructed by the 4-form and the AKSZ construction does not
apply per se. One could think that it would be enough to perform the AKSZ
construction in one dimension higher and recover the twisted Courant sigma
model as a boundary theory. However, as the simpler and lower-dimensional
analogon of the twisted Poisson sigma model shows, this is not as
straightforward as one might expect. Indeed the master action of the twisted
Poisson sigma model is significantly more complicated than what one would
naively get as a boundary of the Courant sigma model, notably containing
quadratic terms in the components of the 3-form flux [19]. In fact there
exists a much larger class of 2D topological field theories based on Dirac
structures that share this feature [20].

The above discussion presents a dichotomy between the general BRST formalism
and the AKSZ construction. The former is completely general and universally
applicable, yet technically complicated and lacking a geometric intuition,
whereas the latter is remarkably simple and with a geometric interpretation by
construction, but it is not general. One would then like to develop methods
that keep the positive properties and dispense with the negative ones, or at
least come close to that. Twisted topological sigma models in various
dimensions is one possible arena to develop and test such methods. As already
mentioned, the 3D twisted Courant sigma model is perhaps the simplest class
that is general enough to allow us to draw such lessons. It is also welcome
that a subclass of this, called twisted R-Poisson sigma models in 3D, was
completely worked out with traditional methods, albeit in a very complicated
and uninspiring way [21]. It will be used here as a benchmark example to test
the more systematic method we will propose.

According to the above, the first question we would like to answer is

  * •

What is the master action of a 4-form twisted Courant sigma model?

Clearly it is not only the end result that we are interested in, but also the
method to derive it. In this respect, a crucial role is played by particular
geometrical quantities that we will encounter and use on the way. To motivate
this further, we note that although for any AKSZ sigma model the full action
is of course covariant, the coefficients of the various interaction terms in
the master action do not exhibit manifest target space covariance term by
term. Although there is nothing wrong with this, it is in fact advantageous to
determine the geometric meaning of these coefficients. One reason for this is
that it provides a clear path toward working out the twisted case.111Another
reason is that this approach can be related to the recently developed theory
of representations up to homotopy for Lie $n$-algebroids [22], whose relation
to the BRST/AKSZ formalism and twisted sigma models will be studied in [23].
We will not discuss this further in the present paper. It also provides a
globalization of the theory and its master action, with manifest target space
covariance and without need to consider a local coordinate patch. Thus the
second main question that we are interested in is:

  * •

What is the generalised geometric meaning of the coefficients in the
interaction terms of the master action for the (twisted or not) Courant sigma
model?

To answer this question, we need to consider generalised connections on
(twisted) Courant algebroids. We will then see that the main player that gives
the sought after geometric interpretation is an induced $E$-connection on $E$,
where $E$ is the vector bundle of the (twisted) Courant algebroid, together
with two associated tensors. One is its torsion and the other is the so-called
basic curvature. The former was introduced by Gualtieri in [24], after
noticing that the ordinary definition of the torsion tensor is not linear in
all its arguments for generalised connections. The basic curvature for Courant
algebroids was introduced in [25] following a similar spirit, namely
generalizing the notion of basic curvature for Lie algebroids [26, 27]222The
concept is directly related to the notion of basic connections, see Refs. [28,
29]. to make it tensorial in the realm of generalised connections on Courant
algebroids. It was also found earlier in [22] from a graded geometric
viewpoint.

Understanding the geometric meaning of these coefficients will prove important
in “finessing” the master action without solving the master equation order by
order in the number of antifields, yet in a slightly different way than the
ASKZ construction such that the twisted case is also included. This approach
is admittedly more complicated than AKSZ, yet simpler than the traditional one
and it reveals some aspects of the formalism that remain hidden in AKSZ. Since
the method we will propose involves computing powers—in the 3D case, the
square and the cube—of the only on-shell closed BRST transformation (the
longitudinal differential in [15]) in a specific way to be explained in the
main text, we call this _the BRST power finesse_.

The rest of the paper is organised in the following way. In Section 2 we
recall some basic facts about Lie and pre-Courant algebroids and their
description as dg manifolds. We pay special attention to the transformation of
their structural data under change of coordinates and introduce
$E$-connections on Lie and pre-Courant algebroids, together with their
corresponding torsion and basic curvature tensors. This provides a covariant
perspective that sets the stage for manifest target space covariance in the
field theory realization of these structures. We also discuss twists and their
particular role in each case, with emphasis in drawing a parallel between
twisted Poisson and twisted Courant structures and also discussing their
conceptual difference.

In Section 3 we turn our attention to the BRST formalism for twisted Courant
sigma models. First we discuss in detail the classical action of the theory
and the structure of its gauge algebra, showing also how it fits within a
broader graded geometric perspective to higher gauge theories. Trivial gauge
transformations are handled with care and the openness of the gauge algebra is
demonstrated. Notably we reveal and elaborate on the fact that this is a
prototype theory that exhibits the phenomenon of nonlinear openness, its gauge
algebra containing terms quadratic in field equations. In Section 3.2 we
present the on-shell closed BRST transformations from the general point of
view of dg manifolds. The main result is Proposition 3.26 which gives the on-
shell closed BRST transformation of all fields, ghosts and ghosts for ghosts
collectively in terms of the components of the homological vector field and
its derivatives. Section 3.3 contains a complete analysis of the tensorial
version of the previously found transformations. We discuss in details the
transformation of fields under changes of frame and present the fully
tensorial BRST transformation in Proposition 3.49, which yields a direct
relation between the BRST transformation and the Gualtieri torsion and Courant
algebroid basic curvature tensors. This gives an answer to the second main
question we posed above.

Section 4 contains the main results of the paper and the answer to the first
question we posed. After determining the square of the on-shell closed BRST
transformation explicitly and introducing antifields, we explain a certain
structure of higher powers of the BRST transformation that can be used to
express the master action. In the present case the highest power is 3. We
formulate Proposition 4.20 which describes a rewriting of the AKSZ-BV master
action for the untwisted Courant sigma model in terms of the elementary
operations we introduced. This is completely the same as the expanded form of
the AKSZ action in terms of number of antifields appearing in [4], but it can
be used to determine the correct master action for the twisted model as well
and moreover bring it in a manifestly covariant form term by term. The first
statement is captured in Theorem 4.22 which gives a compact form of the
minimal solution of the classical master equation for the 4-form twisted
Courant sigma model. The expanded (and complicated) form then follows simply
and it is presented in Appendix D both in non manifestly and manifestly
covariant forms. At the end of Section 4 we also comment on the quantum master
equation, solved by the same master action described earlier.

To complete the picture we present the form of the gauge fixed master action
in Section 5. Notably, we show that the result can be brought in the form of
the classical action plus a BRST commmutator thus proving that the model is a
topological field theory of Schwarz type, as expected. Section 6 contains a
discussion on our results with an outlook to future work. There are four
Appendices that contain supplementary material. Appendix A details aspects of
the gauge invariance of the classical action of the twisted Courant sigma
model and it is used to exemplify some statements in Sections 2 and 3.
Appendix B contains the local form of the on-shell closed BRST transformations
and of their squares, which appear with less details and in more abstract form
in Sections 3 and 4. Appendix C is an annex to the BRST power finesse that
explains several details of more technical nature. Finally, Appendix D
contains the expanded form of the complete minimal solution of the classical
master equation, also in a fully coordinate free form.

##  2 Dg manifolds, twists and target space covariance

In the field theory setting of the later sections we shall encounter target
spaces which are differential non-negatively graded (dg+) manifolds,—a.k.a. NQ
manifolds. These are graded (super)manifolds ${\cal M}$ endowed with a
homological vector field $Q\in\mathfrak{X}(\cal M)$ that satisfies

$Q^{2}:=\frac{1}{2}\\{Q,Q\\}=0\,.$ (2.1)

We refer to [30, 31, 32] for more precise definitions. Homological vector
fields offer an elegant and intuitive way to encode algebraic and geometric
structures and their twists. For example, a Lie algebra $\mathfrak{g}$, which
is a vector space equipped with a skew-symmetric bilinear operation satisfying
the Jacobi identity, can be viewed as a dg manifold $\mathfrak{g}[1]$ whose
homological vector field is

$Q_{\text{CE}}=-\frac{1}{2}C^{a}_{bc}a^{a}a^{b}\frac{\partial}{\partial
a^{c}}\,,$ (2.2)

where $a^{a}$ are the odd (degree 1) coordinates on $\mathfrak{g}[1]$. Then
Eq. (2.1) implies that $C^{a}_{bc}$ are the structure constants of the Lie
algebra satisfying the Jacobi identity. $Q_{\text{CE}}$ is the Chevalley-
Eilenberg differential of ordinary Lie algebra cohomology. In the rest of this
section we will take a new look to this approach for Lie, Courant and pre-
Courant algebroids, emphasizing covariance and connections and elucidating the
role of twists in case twisted geometric structures are encountered.

In the following we always work with non-negative grading and therefore we
shall simply refer to dg manifolds or Q manifolds without further reference to
non-negative grading and moreover we shall denote as Q$n$-manifolds those with
highest degree $n$.

###  2.1 Revisiting Lie algebroids and Q1 manifolds

The above correspondence between a Lie algebra with its bracket and a dg
manifold with its homological vector field is also established for Lie
algebroids [33]. This is usually presented as follows. On one side there is a
Lie algebroid, which is a triple
$(E,[\cdot,\cdot]_{\textit{\tiny{E}}},\rho:E\to TM)$ of a vector bundle $E$
over a smooth manifold $M$, of a binary skew-symmetric operation on its
sections $e$ satisfying the Jacobi identity and the Leibniz rule for
functions,

$[e,fe^{\prime}]_{\textit{\tiny{E}}}=f[e,e^{\prime}]_{\textit{\tiny{E}}}+\rho(e)(f)\,e^{\prime}$
(2.3)

and of a smooth bundle map $\rho:E\to TM$ whose role is to anchor sections to
vector fields on the manifold and it is a homomorphism of bundles,

$\rho([e,e^{\prime}]_{\textit{\tiny{E}}})=[\rho(e),\rho(e^{\prime})]\,.$ (2.4)

In a local basis of $E$, say $\mathfrak{e}_{a}$, and coordinates $x^{\mu}$ on
the base $M$, the bracket is given in terms of some structure functions
$C_{ab}^{c}(x)$ and the map $\rho$ has components $\rho_{a}{}^{\mu}(x)$ such
that

$[\mathfrak{e}_{a},\mathfrak{e}_{b}]_{\textit{\tiny{E}}}=C_{ab}^{c}(x)\mathfrak{e}_{c}\,,\quad\rho(\mathfrak{e}_{a})=\rho_{a}{}^{\mu}(x)\frac{\partial}{\partial
x^{\mu}}\,.$ (2.5)

The statement that $E$ is a Lie algebroid constrains the functions
$\rho_{a}{}^{\mu}$ and $C_{ab}^{c}$ to satisfy certain conditions. The point
now is that the very same conditions are obtained on the other side where we
have the graded Q1 manifold ${\cal M}=E[1]$ by shifting the fiber degree of
the vector bundle $E$ by 1 and also the homological vector field
$Q\in\mathfrak{X}(E[1])$

$Q=\rho_{a}{}^{\mu}(x)a^{a}\frac{\partial}{\partial
x^{\mu}}-\frac{1}{2}C_{bc}^{a}(x)a^{b}a^{c}\frac{\partial}{\partial a^{a}}\,,$
(2.6)

where $x^{\mu}$ and $a^{a}$ are the degree 0 and 1 coordinates of the graded
manifold respectively. The space of vector fields on ${\cal M}$ is spanned by
the derivations $\partial/\partial x^{\mu}$ and $\partial/\partial a^{a}$ of
degrees $0$ and $-1$ respectively. That the vector field $Q$ is homological,
equivalently that $(E[1],Q)$ is a dg manifold, is then an if and only if
statement with $E$ being a Lie algebroid.

From a geometric point of view, there is a nuisance in the above
considerations. This is that the structure functions $C_{ab}^{c}(x)$ do not
transform homogeneously under a change of basis of the vector bundle $E$, and
therefore they are not tensorial. In particular, assuming a change of basis on
the vector bundle

$\mathfrak{e}_{a}=\Lambda_{a}^{b}(\widetilde{x})\,\widetilde{\mathfrak{e}}_{b}$
(2.7)

the structure functions transform according to the rule

$C_{ab}^{c}=(\Lambda^{-1})^{c}_{c^{\prime}}\Lambda_{a}^{a^{\prime}}\Lambda_{b}^{b^{\prime}}\widetilde{C}_{a^{\prime}b^{\prime}}^{c^{\prime}}+(\Lambda^{-1})^{c}_{c^{\prime}}\widetilde{\rho}_{a^{\prime}}{}^{\mu}(\Lambda_{a}^{a^{\prime}}\widetilde{\partial}_{\mu}\Lambda_{b}^{c^{\prime}}-\Lambda_{b}^{a^{\prime}}\widetilde{\partial}_{\mu}\Lambda_{a}^{c^{\prime}})\,,$
(2.8)

where due to the coordinate transformation $x^{\mu}=x^{\mu}(\widetilde{x})$,
the partial derivatives and the coefficients of the anchor map transform as

$\displaystyle\frac{\partial}{\partial x^{\mu}}$ $\displaystyle=$
$\displaystyle\frac{\partial\widetilde{x}^{\nu}}{\partial
x^{\mu}}\frac{\partial}{\partial\widetilde{x}^{\nu}}:=\Lambda_{\mu}^{\nu}\frac{\partial}{\partial\widetilde{x}^{\nu}}\,,$
(2.9) $\displaystyle\rho_{a}{}^{\mu}$ $\displaystyle=$
$\displaystyle\Lambda_{a}^{a^{\prime}}(\Lambda^{-1})_{\mu^{\prime}}^{\mu}\widetilde{\rho}_{a^{\prime}}{}^{\mu^{\prime}}\,.$
(2.10)

Although there is nothing wrong with this per se, both for aesthetic reasons
but also to account for the invariant geometric form of the BV/BRST action for
field theories with underlying Lie algebroid structure, it is desirable to
rewrite the vector field $Q$ in a different form. With some prior knowledge,
we would like to express the vector field in terms of the quantity

$T_{bc}^{a}=2\omega_{[bc]}^{a}-C_{bc}^{a}\,,$ (2.11)

where in the non-graded differential geometric picture in terms of the Lie
algebroid $E$ the quantities $\omega^{a}_{bc}$ are coefficients of an
$E$-connection acting on $E$ itself. In invariant terms, we have a map

$\displaystyle\nabla^{\textit{\tiny{E}}}:\Gamma(E)\otimes\Gamma(E)$
$\displaystyle\to$ $\displaystyle\Gamma(E)$ $\displaystyle(e,e^{\prime})$
$\displaystyle\mapsto$
$\displaystyle\nabla^{\textit{\tiny{E}}}_{e}e^{\prime}\,,$ (2.12)

such that the following homogeneity and linearity conditions hold:

$\displaystyle\nabla^{\textit{\tiny{E}}}_{fe+e^{\prime}}\,e^{\prime\prime}$
$\displaystyle=$ $\displaystyle
f\,\nabla^{\textit{\tiny{E}}}_{e}e^{\prime\prime}+\nabla^{\textit{\tiny{E}}}_{e^{\prime}}e^{\prime\prime}\,,$
(2.13)
$\displaystyle\nabla^{\textit{\tiny{E}}}_{e}(fe^{\prime}+e^{\prime\prime})$
$\displaystyle=$ $\displaystyle
f\nabla^{\textit{\tiny{E}}}_{e}e^{\prime}+\nabla^{\textit{\tiny{E}}}_{e}e^{\prime\prime}+\rho(e)fe^{\prime}\,.$
(2.14)

In terms of the local basis, the connection coefficients are obtained as

$\nabla^{\textit{\tiny{E}}}_{{\mathfrak{e}}_{b}}{\mathfrak{e}}_{c}=\omega_{bc}^{a}{\mathfrak{e}}_{a}\,,$
(2.15)

and they transform in the usual inhomogeneous way:

$\omega^{a}_{bc}=(\Lambda^{-1})^{a}_{a^{\prime}}\Lambda_{b}^{b^{\prime}}\Lambda_{c}^{c^{\prime}}\widetilde{\omega}^{a^{\prime}}_{b^{\prime}c^{\prime}}+(\Lambda^{-1})^{a}_{a^{\prime}}\Lambda_{b}^{b^{\prime}}\widetilde{\rho}_{b^{\prime}}{}^{\mu}\widetilde{\partial}_{\mu}\Lambda_{c}^{a^{\prime}}\,.$
(2.16)

The transformation is such that it counterbalances the one of the structure
functions, allowing for a homogeneous transformation for $T^{a}_{bc}$. One may
then immediately recognize that the quantities $T_{bc}^{a}$ are the
coefficients of the Lie algebroid torsion ($E$-torsion) of this $E$-connection
on $E$ defined as

$T^{\textit{\tiny{E}}}(e,e^{\prime})=\nabla^{\textit{\tiny{E}}}_{e}e^{\prime}-\nabla^{\textit{\tiny{E}}}_{e^{\prime}}e-[e,e^{\prime}]_{\textit{\tiny{E}}}\,,$
(2.17)

in other words $T^{a}_{bc}=\langle
T^{\textit{\tiny{E}}}(\mathfrak{e}_{b},\mathfrak{e}_{c}),\mathfrak{e}^{a}\rangle\,,$
where $\mathfrak{e}^{a}$ is the dual basis of the vector bundle $E^{\ast}$.
Clearly $T^{\textit{\tiny{E}}}$ is a tensor, as desired. Then we can rewrite
the vector field $Q$ in a fully covariant form as

$Q=a^{a}\,\mathrm{D}^{(0)}_{a}+\frac{1}{2}\,T^{a}_{bc}\,a^{b}a^{c}\,\mathrm{D}^{(-1)}_{a}\,,$
(2.18)

where we have defined the natural degree $0$ and degree $-1$ derivations

$\displaystyle\mathrm{D}^{(0)}_{a}=\rho_{a}{}^{\mu}\frac{\partial}{\partial
x^{\mu}}-\omega^{c}_{ab}a^{b}\frac{\partial}{\partial
a^{c}}\,,\qquad\mathrm{D}^{(-1)}_{a}=\frac{\partial}{\partial a^{a}}\,.$
(2.19)

These are natural in the sense that they satisfy

$\displaystyle\mathrm{D}^{(0)}_{a}x^{\mu}=\rho_{a}{}^{\mu}\,,\quad\mathrm{D}^{(0)}_{a}a^{b}=-\,\omega^{b}_{ac}a^{c}\,,$
(2.20)
$\displaystyle\mathrm{D}^{(-1)}_{a}x^{\mu}=0\,,\quad\,\,\mathrm{D}^{(-1)}_{a}a^{b}=\delta_{a}^{b}\,.$
(2.21)

In particular, keeping in mind that the degree $1$ coordinate $a^{b}$ on the
graded side corresponds to the local basis of the dual bundle $E^{\ast}$ on
the non-graded side, it should transform as a section of $E^{\ast}$, which is
indeed the case since we find $-\omega^{b}_{ac}$ on the right-hand side. It
should be emphasized that this is a rewriting of the previous expression in
covariant terms, no additional terms have been added for covariantization; in
other words, the vector field of (2.6) is already covariant albeit not in a
manifest way.333When an $E$-connection on some vector bundle $V$ is
considered, the graded geometric formulation goes through the dg manifold
$E[1]\oplus V$ equipped with a suitable homological vector field. This is not
what we do here and in the rest of this section.

There exists, moreover, a special and very simple case where the above
considerations simplify further. This corresponds to the choice of
$E$-connection being the induced one by an ordinary vector bundle connection
$\nabla$ on $E$, possibly with torsion, through the anchor. Denoting this
induced connection with a solid dot over it, it is

$\accentset{\hbox{\textbullet}}{\nabla}^{\textit{\tiny{E}}}_{e}e^{\prime}=\nabla_{\rho(e)}e^{\prime}\,.$
(2.22)

Although special, this is the connection of interest in topological sigma
models as will become transparent in the ensuing. It is convenient to define
the difference of an arbitrary $E$-connection on $E$ from this induced one,

$\phi(e,e^{\prime}):=(\nabla^{\textit{\tiny{E}}}_{e}-\accentset{\hbox{\textbullet}}{\nabla}^{\textit{\tiny{E}}}_{e})e^{\prime}\,.$
(2.23)

This $\phi$ is evidently an endomorphism and its components are
$\phi^{c}_{ab}=\omega^{c}_{ab}-\rho_{a}{}^{\mu}\,\omega^{c}_{\mu b}\,.$
Equivalently, we consider general $E$-connections of the form

$\nabla^{\textit{\tiny{E}}}=\accentset{\hbox{\textbullet}}{\nabla}^{\textit{\tiny{E}}}+\phi\,.$
(2.24)

The torsion tensor for the induced $E$-connection becomes

$\accentset{\hbox{\textbullet}}{T}^{a}_{bc}=2\rho_{[b}{}^{\mu}\omega_{\mu
c]}^{a}-C_{bc}^{a}\,,$ (2.25)

and the homological vector field is written as

$Q=a^{a}\rho_{a}{}^{\mu}\,\mathrm{D}^{(0)}_{\mu}+\frac{1}{2}\,\accentset{\hbox{\textbullet}}{T}^{a}_{bc}\,a^{b}a^{c}\,\mathrm{D}^{(-1)}_{a}\,,$
(2.26)

with $\mathrm{D}^{(0)}_{\mu}=\frac{\partial}{\partial x^{\mu}}-\omega_{\mu
b}^{c}a^{b}\frac{\partial}{\partial a^{c}}\,.$ More details on this are found
in [25].

###  2.2 Poisson vs. twisted Poisson structure

It is useful to briefly revisit a simple nontrivial example that can moreover
be twisted. Consider the Lie algebroid on the cotangent bundle $E=T^{\ast}M$
of a Poisson manifold $(M,\Pi)$ with Poisson bivector
$\Pi\in\Gamma(\wedge^{2}TM)$. The Lie bracket is the Koszul-Schouten bracket
of 1-forms, given as

$[\eta,\eta^{\prime}]_{\text{KS}}={\cal
L}_{\Pi^{\sharp}(\eta)}\eta^{\prime}-{\cal
L}_{\Pi^{\sharp}(\eta^{\prime})}\eta-\mathrm{d}\left(\Pi(\eta,\eta^{\prime})\right)\,,\quad\eta,\eta^{\prime}\in\Omega^{1}(M)\,,$
(2.27)

and where the anchor map $\Pi^{\sharp}:T^{\ast}M\to TM$ is induced by the
Poisson structure $\Pi$. In addition, choose the induced $E$-connection on
$E$. Then the $E$-torsion is given as

$\accentset{\hbox{\textbullet}}{T}^{\mu\nu}_{\rho}=-\mathring{\nabla}_{\rho}\Pi^{\mu\nu}\,,$
(2.28)

where $\mathring{\nabla}$ is an ordinary vector bundle connection on
$T^{\ast}M$ induced by an ordinary connection on $M$ without torsion, which we
denote by the same symbol.444One could denote the dual connection on
$T^{\ast}M$ as $\mathring{\nabla}^{\ast}$ instead, but we hope no confusion
arises. The homological vector field in covariant form turns out to be

$Q=a_{\mu}\Pi^{\mu\nu}\left(\frac{\partial}{\partial
x^{\nu}}+\mathring{\Gamma}^{\sigma}_{\rho\nu}a_{\sigma}\frac{\partial}{\partial
a_{\rho}}\right)-\frac{1}{2}\mathring{\nabla}_{\rho}\Pi^{\mu\nu}a_{\mu}a_{\nu}\frac{\partial}{\partial
a_{\rho}}\,,$ (2.29)

where $a_{\mu}$ is the degree $1$ coordinate and $\partial/\partial a_{\mu}$
is the degree $-1$ derivation. This rewriting appeared without much
explanation in Ref. [34].

In presence of a closed 3-form background $H\in\Omega_{\text{cl}}^{3}(M)$,
there exists a geometric structure that departs from vanilla Poisson to what
is called twisted Poisson structure [35]. This is once more given by a
bivector $\Pi$ but this time it satisfies the defining property

$\frac{1}{2}[\Pi,\Pi]_{\text{SN}}=\langle\Pi\otimes\Pi\otimes\Pi,H\rangle\,,$
(2.30)

in terms of the Schouten-Nijenhuis bracket, the extension of the Lie bracket
of vector fields to multivector fields. In other words, the Jacobi identity
for the would-be Poisson bracket of functions is violated and the violation is
precisely controlled by the 3-form $H$ three times contracted with the
bivector $\Pi$ in its first slot. Nevertheless, the cotangent bundle
$T^{\ast}M$ continues to bear a Lie algebroid structure, this time with the
Koszul-Schouten Lie bracket being suitably twisted to

$[\eta,\eta^{\prime}]_{\text{HKS}}=[\eta,\eta^{\prime}]_{\text{KS}}+H(\Pi(\eta),\Pi(\eta^{\prime}))\,,$
(2.31)

which satisfies the Jacobi identity by virtue of the twisted Poisson condition
(2.30).

The fact that a Lie algebroid underlies both Poisson and twisted Poisson
manifolds also means that there exists a Q1 manifold description for both.
Indeed, the homological vector field that gives rise to the twisted Poisson
case is simply of the same form as the untwisted case,

$Q^{H}=a_{\mu}\Pi^{\mu\nu}\left(\frac{\partial}{\partial
x^{\nu}}+\Gamma^{\sigma}_{\rho\nu}a_{\sigma}\frac{\partial}{\partial
a_{\rho}}\right)-\frac{1}{2}\mathring{\nabla}_{\rho}\Pi^{\mu\nu}a_{\mu}a_{\nu}\frac{\partial}{\partial
a_{\rho}}\,,$ (2.32)

with the important difference that the connection on $M$ that induces the
connection on the Lie algebroid now has torsion given by the closed 3-form $H$
once contracted with the bivector $\Pi$ [19]. In particular, the connection
coefficients are now given as

$\Gamma_{\mu\nu}^{\rho}=\mathring{\Gamma}_{\mu\nu}^{\rho}+\frac{1}{2}\Pi^{\rho\sigma}H_{\mu\nu\sigma}\,.$
(2.33)

Note that regardless of whether the ordinary connection has torsion or not,
the $E$-connection always has $E$-torsion which is independent of the 3-form
(2.28). In practice, the torsion of $\nabla$ counterbalances the twist of the
Lie algebroid bracket in the definition of the $E$-torsion.

The above example teaches us two lessons. First, when the Poisson structure is
twisted by a 3-form the covariant expression for the homological vector field
that describes it retains its form at the expense of introducing a connection
with torsion on the manifold. Second, the 3-form twists various algebraic and
geometric operations in different way. Note specifically that it can be
contracted once, twice or three times with the bivector $\Pi$ and each of
these combinations plays a role in modifying some operation or quantity. Let
us use the notation

$H^{\mu}_{\nu\rho}=\Pi^{\mu\kappa}H_{\kappa\nu\rho}\,,\quad
H^{\mu\nu}_{\rho}=\Pi^{\mu\kappa}\Pi^{\nu\lambda}H_{\kappa\lambda\rho}\,,\quad
H^{\mu\nu\rho}=\Pi^{\mu\kappa}\Pi^{\nu\lambda}\Pi^{\rho\sigma}H_{\kappa\lambda\sigma}\,.$
(2.34)

Then we observe that from the classical differential geometric point of view
the first modifies the symmetric connection $\nabla$ on $M$ to one with
torsion, the second modifies the binary bracket on $E=T^{\ast}M$ and the third
modifies the Schouten-Nijenhuis bracket of the bivector with itself, i.e. the
Jacobi identity for the Poisson bracket. This is of course a special case in
the category of Lie algebroids, since there exists a natural symplectic form
and in absence of $H$ the Q1 manifold is in fact a QP1 manifold [6].

###  2.3 Pre-Courant algebroids and Q2 manifolds

The picture of Lie algebroids as dg manifolds extends to Courant algebroids
too, and even further to pre-Courant algebroids. In the latter case it may
happen that a closed 4-form is responsible for the violation of the Jacobi
identity of the Courant-Dorfman bracket, a structure called (4-form) twisted
Courant algebroid in Ref. [11]. Let us give a definition. First we recall that
a Courant vector bundle is a pseudo-Euclidean anchored vector bundle
$(E,\langle\cdot,\cdot\rangle,\rho:E\to TM)$ such that the transpose map
$\rho^{\ast}:T^{\ast}M\to E^{\ast}\simeq E$ to the anchor satisfies
$\rho\circ\rho^{\ast}=0$ [13].

###### Definition 2.35

[13] A pre-Courant algebroid is a Courant vector bundle together with a binary
operation $\circ$ on the space of sections $\Gamma(E)$ that satisfies the
following three axioms:

  * •

$\rho(e\circ e^{\prime})=[\rho(e),\rho(e^{\prime})]\,,$

  * •

$\langle e\circ e,e^{\prime}\rangle=\frac{1}{2}\rho(e^{\prime})\langle
e,e\rangle\,,$

  * •

$\rho(e)\langle e^{\prime},e^{\prime\prime}\rangle=\langle e\circ
e^{\prime},e^{\prime\prime}\rangle+\langle e^{\prime},e\circ
e^{\prime\prime}\rangle\,,$

for all $e,e^{\prime},e^{\prime\prime}\in\Gamma(E)$.

In [36] it was shown that under mild assumptions (transitivity, i.e. that the
kernel of the map $\rho$ is trivial) the definition of a pre-Courant algebroid
is equivalent to the later definition of a 4-form twisted Courant algebroid of
[11] that appeared in the context of first class constrained systems. In
particular for a 4-form twisted Courant algebroid the Jacobiator of the binary
operation is given as

$\text{Jac}(e,e^{\prime},e^{\prime\prime}):=e\circ(e^{\prime}\circ
e^{\prime\prime})-(e\circ e^{\prime})\circ
e^{\prime\prime}-e^{\prime}\circ(e\circ
e^{\prime\prime})=\rho^{\ast}H(\rho(e),\rho(e^{\prime}),\rho(e^{\prime\prime}))\,.$
(2.36)

For vanishing 4-form the definition reduces to the one for an ordinary Courant
algebroid. As usual one may write a definition in terms of the skew-symmetric
operation

$[e,e^{\prime}]_{\textit{\tiny{E}}}:=\frac{1}{2}(e\circ
e^{\prime}-e^{\prime}\circ e)\,,$ (2.37)

also for pre-Courant algebroids. In this paper we assume transitivity and
therefore we do not distinguish pre- and twisted Courant algebroids any
longer.

Below we mostly work with the alternative definition in terms of dg manifolds.
We begin with Courant algebroids. The properties of the Courant algebroid
data, namely the binary bracket $[\cdot,\cdot]_{\textit{\tiny{E}}}$ on
sections of a vector bundle $E$ over $M$ with anchor $\rho$ and additional
fiber metric $\langle\cdot,\cdot\rangle$ can be neatly encoded in a
homological vector field $Q\in\mathfrak{X}{(\cal M)}$, this time with the dg
manifold being the Q2 (in fact QP2) manifold ${\cal
M}=T^{\ast}[2]E_{0}[1]$.555This is up to isomorphism, since one can of course
model this on the second-order cotangent bundle $T^{\ast}[2]T^{\ast}[1]M$
instead [6, 37]. Although for Courant algebroids this is the same, this ceases
to be true in higher geometric cases. For example, the second-order cotangent
bundle can be used to accommodate R-Poisson geometries in arbitrary dimensions
[34, 38], whereas the cotangent bundle of the tangent is not sufficient for
this unless restricted to Courant algebroids. We work with $E_{0}=TM$ in the
following, but the more general case follows suit. The coordinates on this
graded manifold are $(x^{\mu},a^{a},b_{\mu})$ of respective degrees $(0,1,2)$
and as usual we have packaged the degree $1$ coordinates into a single one
$a^{a}=(a^{\mu},a_{\mu})$. Note that the Latin index $a$ is now one on $E$,
not $E_{0}$. The space of vector fields on ${\cal M}$ is spanned by the
corresponding derivations of inverted degrees $(0,-1,-2)$. Then the most
general homological vector field has the form666If one works with general Q2
manifolds without necessarily having a symplectic structure, then the most
general case proliferates to semi-strict Lie 2-algebroids. This was discussed
in detail in Ref. [39]. [32]

$\displaystyle Q=\rho_{a}{}^{\mu}a^{a}\frac{\partial}{\partial
x^{\mu}}-\big{(}\eta^{ab}\rho_{b}{}^{\mu}b_{\mu}+\frac{1}{2}C^{a}_{bc}a^{b}a^{c}\big{)}\frac{\partial}{\partial
a^{a}}-\big{(}\partial_{\mu}\rho_{a}{}^{\nu}b_{\nu}a^{a}+\frac{1}{3!}\partial_{\mu}C_{abc}a^{a}a^{b}a^{c}\big{)}\frac{\partial}{\partial
b_{\mu}}\,.$ (2.38)

Thinking in terms of ordinary differential geometric terms, we have introduced
again a basis $\mathfrak{e}_{a}$ for the vector bundle $E$ of a Courant
algebroid. In this basis, $C^{a}_{bc}(x)$ are the structure functions of the
Courant bracket and $\eta_{ab}$ are the components of the fiber metric, in our
conventions

$\eta_{ab}=2\langle\mathfrak{e}_{a},\mathfrak{e}_{b}\rangle_{\textit{\tiny{E}}}\,.$
(2.39)

The $\eta^{ab}$ are the inverse metric components and Latin indices are raised
and lowered with this metric, for instance $C_{abc}=\eta_{cd}C^{d}_{ab}$.
Finally, $\rho_{a}{}^{\mu}(x)$ are once more the components of the anchor in
this basis. The statement now is that the condition $Q^{2}=0$ is equivalent to
the local coordinate form of the axioms of a Courant algebroid laid down in
[41], as given in [1] in suitable conventions.

For our purposes here, the homological vector field given in (2.38) requires
covariantization in the same spirit as the Lie algebroid homological vector
field of the previous section. Indeed, there are two obvious sources of
noncovariance, the structure functions $C_{bc}^{a}(x)$ and the partial
derivative on the components of the anchor. This evidently calls for the
introduction of a suitable connection. Moreover, we should also recall that
this is once more a case of non-manifest covariance. Our goal is to make
covariance manifest and show then the relation to appropriate Courant
algebroid tensors.

An important difference to the case of Lie algebroids regards the
transformation of the structure functions under a change of basis on the
bundle. In the present case this transformation reads

$\displaystyle
C_{ab}^{c}=(\Lambda^{-1})^{c}_{c^{\prime}}\Lambda_{a}^{a^{\prime}}\Lambda_{b}^{b^{\prime}}\widetilde{C}_{a^{\prime}b^{\prime}}^{c^{\prime}}+(\Lambda^{-1})^{c}_{c^{\prime}}\widetilde{\rho}_{a^{\prime}}{}^{\mu}(\Lambda_{a}^{a^{\prime}}\widetilde{\partial}_{\mu}\Lambda_{b}^{c^{\prime}}-\Lambda_{b}^{a^{\prime}}\widetilde{\partial}_{\mu}\Lambda_{a}^{c^{\prime}})-$
$\displaystyle\qquad\qquad-\,{\textstyle\frac{1}{2}}\widetilde{\eta}_{a^{\prime}b^{\prime}}\widetilde{\eta}^{c^{\prime}d^{\prime}}\widetilde{\rho}_{d^{\prime}}{}^{\mu}(\Lambda^{-1})^{c}_{c^{\prime}}(\Lambda_{a}^{a^{\prime}}\widetilde{\partial}_{\mu}\Lambda_{b}^{b^{\prime}}-\Lambda_{b}^{a^{\prime}}\widetilde{\partial}_{\mu}\Lambda_{a}^{b^{\prime}})\,,$
(2.40)

with the new term appearing due to the unconventional, anomalous Leibniz rule
that the Courant bracket satisfies:

$[e,fe^{\prime}]_{\textit{\tiny{E}}}=f[e,e^{\prime}]_{\textit{\tiny{E}}}+\rho(e)f\,e^{\prime}-\mathrm{d}_{\textit{\tiny{E}}}f\langle
e,e^{\prime}\rangle\,,$ (2.41)

where $\mathrm{d}_{\textit{\tiny{E}}}:C^{\infty}(M)\to\Gamma(E)$ is the
Courant algebroid differential defined via
$\langle\mathrm{d}_{\textit{\tiny{E}}}f,e\rangle=\frac{1}{2}\rho(e)f$. We
remark that $\eta_{ab}$ transforms homogeneously, namely

$\displaystyle\eta_{ab}=\Lambda_{a}^{a^{\prime}}\Lambda_{b}^{b^{\prime}}\widetilde{\eta}_{a^{\prime}b^{\prime}}\,.$
(2.42)

Note that the unconventional transformation of the structure functions does
not improve for the Dorfman bracket, since it regards both entries in the
bracket and even though the Leibniz rule is standard in the second entry it is
not so in the first one.

Then the first thing to note is that we need an $E$-connection on $E$, where
$E$ is the Courant algebroid at hand. We use the same notation as for Lie
algebroids and in the local basis the connection coefficients are as in Eq.
(2.15). Famously, for Courant algebroid connections the naive torsion tensor
is not suitable since it fails to exhibit linearity in all its arguments. A
suitable Courant algebroid $E$-torsion tensor was defined in [24] and it reads
as

$\displaystyle\mathcal{T}^{\textit{\tiny{E}}}(e_{1},e_{2},e_{3})=\langle\nabla^{\textit{\tiny{E}}}_{e_{1}}e_{2}-\nabla^{\textit{\tiny{E}}}_{e_{2}}e_{1}-[e_{1},e_{2}]_{\textit{\tiny{E}}},e_{3}\rangle_{\textit{\tiny{E}}}+\frac{1}{2}\left(\langle\nabla^{\textit{\tiny{E}}}_{e_{3}}e_{1},e_{2}\rangle_{\textit{\tiny{E}}}-\langle\nabla^{\textit{\tiny{E}}}_{e_{3}}e_{2},e_{1}\rangle_{\textit{\tiny{E}}}\right)$
(2.43)

in terms of the skew-symmetric Courant bracket and analogously for the Dorfman
bracket. Other possible options for a good definition of a torsion tensor for
Courant algebroid connections were suggested in [42]. In a chosen local basis,
the components of the Gualtieri $E$-torsion are

$\mathcal{T}_{abc}=\omega^{d}_{[ab]}\eta_{cd}-\frac{1}{2}C_{ab}^{d}\eta_{dc}+\frac{1}{2}\omega^{d}_{c[a}\eta_{b]d}\,.$
(2.44)

We could read this formula in an upside down way by solving in terms of the
structure functions of the bracket,

$C^{d}_{ab}=\big{(}-2\mathcal{T}_{abc}+2\omega^{e}_{[ab]}\eta_{ec}+\omega^{e}_{c[a}\eta_{b]e}\big{)}\eta^{cd}\,.$
(2.45)

This can be used to substitute the structure functions with the $E$-torsion in
the vector field $Q$, plus additional terms that must and will be accounted
for. Secondly, since there is a differentiation on functions on $M$ in the
components of $Q$, we need a vector bundle connection $\nabla$ on $TM\oplus E$
as well. This connection can be extended to act on differential forms in the
usual way and therefore we have an analog of the tetrad connection for
Vielbeins, now on the components of the anchor.777To avoid misunderstanding of
this analogy, we mention that no “tetrad postulate” is imposed here.
Specifically,

$\nabla_{\mu}\rho_{a}{}^{\nu}=\partial_{\mu}\rho_{a}{}^{\nu}+\Gamma^{\nu}_{\mu\rho}\rho_{a}{}^{\rho}-\omega_{\mu
a}^{b}\rho_{b}{}^{\nu}\,.$ (2.46)

We shall denote the torsionless part of this connection by $\mathring{\nabla}$
as before.

We are now ready to express $Q$ in a different form using (2.45) and (2.46).
To reach a manifestly covariant result, it is necessary to perform a
coordinate transformation (corresponding to a nonlinear field redefinition in
the gauge theory to be discussed below) as follows

$b_{\mu}^{\scriptscriptstyle\nabla}:=b_{\mu}+\frac{1}{2}\omega_{\mu
ab}a^{a}a^{b}\,,$ (2.47)

mixing the degree $2$ and degree $1$ coordinates. Let us pause to further
justify this transformation. First note that the coordinate and bundle
transformations of the Courant algebroid in the classical differential
geometric side correspond to graded coordinate transformations in the Q2
manifold side. Specifically, these coordinate transformations are

$\displaystyle x^{\mu}$ $\displaystyle=$ $\displaystyle
x^{\mu}(\widetilde{x})\,$ (2.48) $\displaystyle a^{a}$ $\displaystyle=$
$\displaystyle(\Lambda^{-1})_{a^{\prime}}^{a}\widetilde{a}^{a^{\prime}}\,$
(2.49) $\displaystyle b_{\mu}$ $\displaystyle=$
$\displaystyle\Lambda_{\mu}^{\mu^{\prime}}\,\widetilde{b}_{\mu^{\prime}}+\frac{1}{2}\partial_{\mu}(\Lambda^{-1})_{a}^{a^{\prime}}\Lambda^{c}_{a^{\prime}}\widetilde{\eta}_{cb}\widetilde{a}^{a}\widetilde{a}^{b}$
(2.50)

They are obtained by integrating the infinitesimal counterparts which are
generated by a quadratic Hamiltonian and the canonical Poisson bracket on
$\mathcal{M}$ which exists because it is a cotangent bundle [6]. Observing the
complicated transformation of the degree 2 coordinate we can ask whether it
can be undone by a suitable redefinition. Indeed this is the redefinition of
(2.47). The fact that

$b_{\mu}^{\scriptscriptstyle\nabla}=\Lambda_{\mu}^{\mu^{\prime}}\,\widetilde{b}_{\mu^{\prime}}^{\scriptscriptstyle\nabla}$
(2.51)

justifies it. In the following we will work with this “covariant” degree 2
coordinate and the corresponding degree $-2$ derivation $\partial/\partial
b^{\scriptscriptstyle\nabla}_{\mu}$.

Taking all these into account, a straightforward calculation leads to the
following alternative form of the homological vector field:

$\displaystyle
Q=a^{a}\rho_{a}{}^{\mu}\mathrm{D}^{(0)}_{\mu}-\bigg{(}\eta^{ac}\rho_{a}{}^{\mu}b_{\mu}^{\scriptscriptstyle\nabla}+\big{(}-\mathcal{T}_{ab}{}^{c}+\phi_{ab}{}^{c}+\frac{1}{2}\phi^{c}{}_{ab}\big{)}a^{a}a^{b}\bigg{)}\mathrm{D}^{(-1)}_{c}+$
$\displaystyle\qquad+\,\bigg{(}-\nabla_{\nu}\rho_{a}{}^{\mu}b^{\scriptscriptstyle\nabla}_{\mu}a^{a}+\big{(}\frac{1}{3}\nabla_{\nu}\mathcal{T}_{abc}-\frac{1}{2}\nabla_{\nu}\phi_{abc}+\frac{1}{2}\rho_{a}{}^{\mu}\eta_{cd}R^{d}{}_{b\nu\mu}\big{)}a^{a}a^{b}a^{c}\bigg{)}\mathrm{D}^{\nu}_{(-2)}\,,\qquad$
(2.52)

where we have defined the natural degree $0$, degree $-1$ and degree $-2$
derivations in the present case, which have the form

$\displaystyle\mathrm{D}^{(0)}_{\mu}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial
x^{\mu}}-\frac{1}{2}\frac{\partial\omega_{\nu bc}}{\partial
x^{\mu}}a^{b}a^{c}\frac{\partial}{\partial
b_{\nu}}+\Gamma_{\nu\mu}^{\sigma}b_{\sigma}^{\scriptscriptstyle\nabla}\frac{\partial}{\partial
b_{\nu}}-\omega_{\mu b}{}^{c}a^{b}\mathrm{D}_{c}^{(-1)}\,,$ (2.53)
$\displaystyle\mathrm{D}_{a}^{(-1)}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial a^{a}}+\omega_{\mu
ba}a^{b}\frac{\partial}{\partial b_{\mu}}\,,$ (2.54)
$\displaystyle\mathrm{D}^{\mu}_{(-2)}$ $\displaystyle=$
$\displaystyle\frac{\partial}{\partial b_{\mu}^{\scriptscriptstyle\nabla}}\,.$
(2.55)

Once more these are natural in the sense of the following relations when they
act on the coordinates,

$\displaystyle\mathrm{D}_{\mu}^{(0)}x^{\nu}$ $\displaystyle=$
$\displaystyle\delta_{\mu}{}^{\nu}\,,\qquad\,\,\quad\qquad\,\,\mathrm{D}^{(-1)}_{a}x^{\mu}=0\,,\quad\,\quad\,\,\,\,\mathrm{D}^{\nu}_{(-2)}x^{\mu}=0\,,$
(2.56) $\displaystyle\mathrm{D}_{\mu}^{(0)}a^{c}$ $\displaystyle=$
$\displaystyle-\,\omega_{\mu
b}{}^{c}a^{b}\,,\quad\quad\quad\,\,\mathrm{D}_{a}^{(-1)}a^{c}=\delta_{a}^{c}\,,\quad\quad\,\,\,\,\mathrm{D}^{\nu}_{(-2)}a^{c}=0\,,$
(2.57) $\displaystyle\mathrm{D}_{\nu}^{(0)}b_{\mu}^{\scriptscriptstyle\nabla}$
$\displaystyle=$
$\displaystyle\Gamma^{\sigma}_{\mu\nu}b_{\sigma}^{\scriptscriptstyle\nabla}\,,\,\quad\qquad\,\quad\mathrm{D}_{a}^{(-1)}b^{\scriptscriptstyle\nabla}_{\mu}=0\,,\,\,\,\,\quad\,\quad\mathrm{D}^{\mu}_{(-2)}b^{\scriptscriptstyle\nabla}_{\nu}=\delta^{\mu}_{\nu}\,.$
(2.58)

In case the induced connection
$\accentset{\hbox{\textbullet}}{\nabla}^{\textit{\tiny{E}}}$ is chosen, the
endomorphism $\phi$ vanishes and the vector field takes the simpler form

$\displaystyle
Q=a^{a}\rho_{a}{}^{\mu}\mathrm{D}^{(0)}_{\mu}-\bigg{(}\eta^{ac}\rho_{a}{}^{\mu}b_{\mu}^{\scriptscriptstyle\nabla}-\accentset{\hbox{\textbullet}}{\mathcal{T}}_{ab}{}^{c}a^{a}a^{b}\bigg{)}\mathrm{D}^{(-1)}_{c}+$
$\displaystyle\qquad+\,\bigg{(}-\nabla_{\nu}\rho_{a}{}^{\mu}b^{\scriptscriptstyle\nabla}_{\mu}a^{a}+\big{(}\frac{1}{3}\nabla_{\nu}\accentset{\hbox{\textbullet}}{\mathcal{T}}_{abc}+\frac{1}{2}\rho_{a}{}^{\mu}\eta_{cd}R^{d}{}_{b\nu\mu}\big{)}a^{a}a^{b}a^{c}\bigg{)}\mathrm{D}^{\nu}_{(-2)}\,.\qquad$
(2.59)

There is a further geometrical association to make. Recall that from the non-
manifestly covariant expression (2.38) we observe that a derivative on the
structure functions appears. This turned to a derivative on the components of
the $E$-torsion $\mathcal{T}^{\textit{\tiny{E}}}$ through (2.45) and in turn
this derivative got covariantized in (2.52). However, the covariant derivative
on the Courant algebroid torsion has an independent geometrical essence. As
proven in [25], it is related to the tensor that measures the compatibility
between the connection on $M$ and the bracket on the Courant algebroid—see
[26] for introducing this notion for Lie algebroids and [27, 43] for more
detailed explanations. This tensor is called the basic curvature tensor and
for Courant algebroids it is defined as [22, 25]

$\displaystyle\mathcal{S}^{\textit{\tiny{E}}}(e_{1},e_{2},e_{3})X$
$\displaystyle=$
$\displaystyle\langle\nabla_{X}[e_{1},e_{2}]_{\textit{\tiny{E}}}-[\nabla_{X}e_{1},e_{2}]_{\textit{\tiny{E}}}-[e_{1},\nabla_{X}e_{2}]_{\textit{\tiny{E}}}-\nabla_{\overline{\nabla}^{\textit{\tiny{E}}}_{e_{2}}X}e_{1}+\nabla_{\overline{\nabla}^{\textit{\tiny{E}}}_{e_{1}}X}e_{2},e_{3}\rangle_{\textit{\tiny{E}}}\,\,$
(2.60) $\displaystyle+$
$\displaystyle\frac{1}{2}\left(\langle\nabla_{\overline{\nabla}^{\textit{\tiny{E}}}_{e_{3}}X}e_{1},e_{2}\rangle_{\textit{\tiny{E}}}-\langle\nabla_{\overline{\nabla}^{\textit{\tiny{E}}}_{e_{3}}X}e_{2},e_{1}\rangle_{\textit{\tiny{E}}}\right)\,,$

in terms of an $E$-connection on $TM$ (the opposite to an ordinary vector
bundle connection on $E$) given as

$\overline{\nabla}^{\textit{\tiny{E}}}_{e}X=\rho(\nabla_{X}e)+[\rho(e),X]\,.$
(2.61)

It is also proven in [25] that it may be expressed as

$\displaystyle\mathcal{S}^{\textit{\tiny{E}}}(e_{1},e_{2},e_{3})X$
$\displaystyle=$
$\displaystyle-\nabla_{X}\mathcal{T}^{\textit{\tiny{E}}}(e_{1},e_{2},e_{3})+\langle\mathcal{C}(e_{2},X)e_{1}-\mathcal{C}(e_{1},X)e_{2},e_{3}\rangle_{\textit{\tiny{E}}}$
(2.62)
$\displaystyle-\,\frac{1}{2}\bigg{(}\langle\mathcal{C}(e_{3},X)e_{1},e_{2}\rangle_{\textit{\tiny{E}}}-\langle\mathcal{C}(e_{3},X)e_{2},e_{1}\rangle_{\textit{\tiny{E}}}\bigg{)}\,,$

in terms of the Gualtieri torsion and a tensor $\mathcal{C}$ defined through

${\cal
C}(e_{1},X)e_{2}:=[\nabla^{\textit{\tiny{E}}}_{e_{1}},\nabla_{X}]e_{2}-\nabla_{[\rho(e_{1}),X]}e_{2}+\phi(\nabla_{X}e_{1},e_{2})\,,$
(2.63)

where we recall that $\phi$ is the difference of the $E$-connection we work
with from the induced $E$-connection. Note that $\phi$ obviously vanishes when
the induced connection is chosen. However, in the general case the last term
in the definition of $\mathcal{C}$ is absolutely necessary to guarantee that
it transforms tensorially in all arguments.

In a local basis the components of the Courant algebroid basic curvature
tensor read

$\displaystyle\mathcal{S}_{\mu
abc}=-\nabla_{\mu}\mathcal{T}_{abc}-\mathcal{C}_{\mu[ab]c}-\frac{1}{2}\mathcal{C}_{\mu
c[ab]}\,,$ (2.64)

with

$\displaystyle\mathcal{C}_{\mu
ab}{}^{c}=\rho_{a}{}^{\nu}\partial_{\nu}\omega^{c}_{\mu
b}-\partial_{\mu}\omega^{c}_{ab}+2\omega^{d}_{\mu
b}\omega^{c}_{(ad)}-\omega^{d}_{ab}\omega^{c}_{\mu
d}+\partial_{\mu}\rho_{a}{}^{\nu}\omega^{c}_{\nu
b}-\rho_{d}{}^{\nu}\omega^{d}_{\mu b}\omega^{c}_{\nu a}\,.$ (2.65)

Caution is drawn to the fact that the coefficients of two different
connections appear in this formula, both denoted as $\omega$ and distinguished
by their indices. One is the ordinary vector bundle connection and one is the
$E$-connection, which can be completely general. In the case of induced
$E$-connection, the tensor simplifies greatly to

$\accentset{\hbox{\textbullet}}{\mathcal{C}}_{\mu
ab}{}^{c}=\rho_{a}{}^{\nu}R^{c}{}_{b\nu\mu}\,,$ (2.66)

thus being the anchored curvature tensor. One should recall that the usual
expression for the curvature for Courant algebroid connections is a tensor if
and only if the induced connection is chosen, which explains this relation.

We are now ready to express $Q$ in an alternative manifestly covariant form
using the definitions related to the Courant algebroid basic curvature tensor
$\mathcal{S}$. For the induced $E$-connection the result is remarkably simple,

$\displaystyle
Q=a^{a}\rho_{a}{}^{\mu}\mathrm{D}^{(0)}_{\mu}-\big{(}\eta^{ac}\rho_{a}{}^{\mu}b_{\mu}^{\scriptscriptstyle\nabla}-\accentset{\hbox{\textbullet}}{\mathcal{T}}_{ab}{}^{c}a^{a}a^{b}\big{)}\mathrm{D}^{(-1)}_{c}-\big{(}\nabla_{\nu}\rho_{a}{}^{\mu}b^{\scriptscriptstyle\nabla}_{\mu}+\frac{1}{3}\accentset{\hbox{\textbullet}}{\mathcal{S}}_{\nu
abc}a^{b}a^{c}\big{)}a^{a}\mathrm{D}^{\nu}_{(-2)}\,.\,\,\,\,\qquad$ (2.67)

We have thus expressed the homological vector field corresponding to a Courant
algebroid in terms of the Gualtieri torsion and the basic curvature of the
induced Courant algebroid connection on itself. In the following we will
always refer to the connection $\accentset{\hbox{\textbullet}}{\nabla}$ and
hence we denote $\accentset{\hbox{\textbullet}}{\mathcal{T}}$ and
$\accentset{\hbox{\textbullet}}{\mathcal{S}}$ simply as $\mathcal{T}$ and
$\mathcal{S}$.

Let us now turn to twisted Courant algebroids. As discussed earlier, the main
difference to Courant algebroids is that the Jacobi identity for the Dorfman
bracket is modified by a section obtained by contracting a closed 4-form $H$
with three vector fields in the image of the anchor map and then producing a
section from this 1-form with the transpose map $\rho^{\ast}:T^{\ast}M\to E$.
The description of this structure in terms of a Q2 manifold is as follows. We
consider the same graded manifold $\mathcal{M}=T^{\ast}[2]E_{0}[1]$ and the
homological vector field

$\displaystyle
Q^{H}=Q-\frac{1}{3!}\rho_{a}{}^{\nu}\rho_{b}{}^{\sigma}\rho_{c}{}^{\lambda}H_{\mu\nu\sigma\lambda}a^{a}a^{b}a^{c}\frac{\partial}{\partial
b_{\mu}}\,,$ (2.68)

where $Q$ is the one of the Courant algebroid. In other words, the vector
field (2.68) is homological if and only if the following conditions hold:

$\displaystyle\eta^{ab}\rho_{a}{}^{\mu}\rho_{b}{}^{\nu}=0\,,$ (2.69)
$\displaystyle\rho_{b}{}^{\mu}\partial_{\mu}\rho_{a}{}^{\nu}-\rho_{a}{}^{\mu}\partial_{\mu}\rho_{b}{}^{\nu}+\eta^{ef}\rho_{e}{}^{\nu}C_{fab}=0\,,$
(2.70)
$\displaystyle\rho_{d}{}^{\mu}\partial_{\mu}C_{abc}-\rho_{a}{}^{\mu}\partial_{\mu}C_{bcd}+\rho_{b}{}^{\mu}\partial_{\mu}C_{cda}-\rho_{c}{}^{\mu}\partial_{\mu}C_{dab}\,+$
$\displaystyle\qquad+\eta^{ef}C_{eab}C_{cdf}+\eta^{ef}C_{eac}C_{dbf}+\eta^{ef}C_{ead}C_{bcf}=-\,\rho_{d}{}^{\mu}\rho_{a}{}^{\mu}\rho_{b}{}^{\kappa}\rho_{c}{}^{\lambda}H_{\mu\nu\kappa\lambda}\,.$
(2.71)

These are nothing else but the local coordinate form of the axioms of a
twisted Courant algebroid. The third, $H$-dependent equation is the local
coordinate form of the Jacobiator (2.36).

It is useful to write down the covariant form of these conditions in terms of
the ordinary connection $\nabla$ on $E$ and of the basic curvature and
Gualtieri torsion tensors. For (2.70), the covariant expression in terms of
the components of the Gualtieri torsion is

$\rho_{a}{}^{\mu}\nabla_{\mu}\rho_{b}{}^{\nu}-\rho_{b}{}^{\mu}\nabla_{\mu}\rho_{a}{}^{\nu}+2\eta^{ef}\rho_{e}{}^{\nu}{\mathcal{T}}_{abf}=0\,.$
(2.72)

Similarly, the covariant expression for (2.71) can be written in terms of the
Gualtieri torsion and the basic curvature tensor, yielding the following
elegant formula:

$\displaystyle\rho_{[d}{}^{\mu}{\mathcal{S}}_{\mu
abc]}+{\textstyle\frac{3}{2}}\eta^{ef}{\mathcal{T}}_{ea[b}{\mathcal{T}}_{cd]f}=-{\textstyle\frac{1}{8}}\rho_{d}{}^{\mu}\rho_{a}{}^{\nu}\rho_{b}{}^{\rho}\rho_{c}{}^{\sigma}H_{\mu\nu\rho\sigma}\,,$
(2.73)

which is essentially an algebraic Bianchi identity for the Courant algebroid
basic curvature. Furthermore, prompted by Eq. (2.68) we define

$\widetilde{\mathcal{S}}_{\mu abc}:=\mathcal{S}_{\mu abc}+\frac{1}{2}H_{\mu
abc}\,,$ (2.74)

having introduced the notation

$\displaystyle H_{\mu\nu\rho a}=\rho_{a}{}^{\kappa}H_{\mu\nu\rho\kappa}\,,$
$\displaystyle H_{\mu\nu
ab}=\rho_{a}{}^{\kappa}\rho_{b}{}^{\lambda}H_{\mu\nu\kappa\lambda}\,,$
$\displaystyle H_{\mu
abc}=\rho_{a}{}^{\kappa}\rho_{b}{}^{\lambda}\rho_{c}{}^{\nu}H_{\mu\kappa\lambda\nu}\,,$
$\displaystyle
H_{abcd}=\rho_{a}{}^{\mu}\rho_{b}{}^{\nu}\rho_{c}{}^{\kappa}\rho_{d}{}^{\lambda}H_{\mu\nu\kappa\lambda}\,.$
(2.75)

As usual, the bundle indices in these quantities are raised with the (inverse)
metric $\eta^{ab}$, and $H$ remains totally antisymmetric in all indices, e.g
$H_{\mu abc}=-H_{a\mu bc}=H_{ab\mu c}=-H_{abc\mu}$. Similar to the discussion
of the twisted Poisson case following Eq. (2.34), these quantities have a
definite geometric interpretation. The last quantity, $H_{abcd}$, modifies the
Jacobi identity for the Courant bracket (2.36). The next to last, $H_{\mu
bcd}$, modifies the basic curvature (2.74). In different terms, akin to the
$L_{\infty}$ description of a Courant sigma model [44], this term twists the
ternary bracket.888We recall that the $L_{\infty}$ description of a Courant
algebroid was given in Ref. [45]. More generally, the relation of $L_{\infty}$
to perturbative (gauge) field theories was explored and reviewed in [46, 47].
We shall see later that $H_{\mu\nu cd}$ modifies the ordinary curvature of the
connection $\nabla$ on $E$, while the first term $H_{\mu\nu\rho a}$ controls
the non-exact part of $H$, as discussed in App. A. Then we observe that the
homological vector field for a twisted Courant algebroid may be written as

$\displaystyle
Q^{H}=a^{a}\rho_{a}{}^{\mu}\mathrm{D}^{(0)}_{\mu}-\big{(}\eta^{ac}\rho_{a}{}^{\mu}b_{\mu}^{\scriptscriptstyle\nabla}-{\mathcal{T}}_{ab}{}^{c}a^{a}a^{b}\big{)}\mathrm{D}^{(-1)}_{c}-\big{(}\nabla_{\nu}\rho_{a}{}^{\mu}b^{\scriptscriptstyle\nabla}_{\mu}+\frac{1}{3}\widetilde{\mathcal{S}}_{\nu
abc}a^{b}a^{c}\big{)}a^{a}\mathrm{D}^{\nu}_{(-2)}\,.$ (2.76)

Notice that the form is the same as for the usual Courant algebroid with the
difference that $\mathcal{S}$ is replaced by the $H$-dependent
$\widetilde{\mathcal{S}}$.

##  3 BRST structure of twisted Courant sigma models

###  3.1 Action, gauge symmetry & nonlinearly open gauge algebra

The Courant sigma model can be described as a membrane sigma model with target
space $\mathcal{M}=T^{\ast}[2]T[1]M$, or more generally $T^{\ast}[2]E_{0}[1]$
for some Lie algebroid $E_{0}$ over $M$ [1, 2, 4]. As such, its gauge
structure is encoded in the axioms of a Courant algebroid and vice versa; or,
in other words, the solution to the classical master equation in the BV/BRST
formulation of the theory is encoded in the graded symplectic structure and
the homological vector field on $\mathcal{M}$, which is the original spirit of
the AKSZ construction [16]. Here we take the above logic one step further and
slightly depart from the usual AKSZ construction by considering topological
membrane sigma models with a Wess-Zumino term. This was originally suggested
in [11] in the context of Hamiltonian systems with first class constraints and
also implicitly in [48] in studies related to fluxes in double field theory.
We refer to them as twisted Courant sigma models here because it turns out
that their gauge structure is governed by a twisted Courant algebroid.

In this section we revisit the classical action of the model using dg manifold
language and filling in some gaps regarding the structure of its gauge
algebra. The source space (world volume) of the model is the dg manifold
$T[1]\Sigma$, where $\Sigma$ is a three-dimensional manifold (the membrane).
For convenience, we introduce coordinates on $T[1]\Sigma$ denoted as
$\sigma^{m}$ of degree 0 and $\theta^{m}$ of degree 1 for the fibre
coordinates. In accord with the notation introduced in the previous section,
the classical fields are real functions $X^{\mu}(\sigma)$, $X^{\ast}E$-valued
1-forms $A=A^{a}_{m}(\sigma)\theta^{m}e_{a}$, where $e_{a}$ is a local basis
of the pullback bundle $X^{\ast}E$ and $T^{\ast}M$-valued 2-forms
$B=\frac{1}{2}B_{\mu mn}(\sigma)\theta^{m}\theta^{n}\mathrm{d}X^{\mu}$. The
latter two fields can also be viewed as pullback coordinates once a suitable
map $\phi:T[1]\Sigma\to T^{\ast}[2]T[1]M$ is introduced and extended to a map

$\phi_{\times}:T[1]\Sigma\to T[1]\Sigma\times T^{\ast}[2]T[1]M\,,$ (3.1)

which is the identity on the first factor [39]. Then
$A^{a}=\phi^{\ast}_{\times}(a^{a})$ and
$B_{\mu}=\phi^{\ast}_{\times}(b_{\mu})$. The classical action functional of
maps in our conventions reads

$S_{\text{cl}}=\int_{T[1]\Sigma}\left(-B_{\mu}\mathrm{d}X^{\mu}+\frac{1}{2}\eta_{ab}A^{a}\mathrm{d}A^{b}+\rho^{\mu}_{a}(X)B_{\mu}A^{a}+\frac{1}{3!}C_{abc}(X)A^{a}A^{b}A^{c}\right)+\int_{\widehat{\Sigma}}X^{\ast}H\,$
(3.2)

where $H\in\Omega_{\text{cl}}^{4}(M)$ and we make the usual assumptions the
guarantee the existence of the extension of the world volume to a 4D one
$\widehat{\Sigma}$ and the independence of the action functional of this
extension, see e.g. [49] for a clear exposition in any dimension.

The topological membrane action (3.2) is invariant under a prescribed set of
infinitesimal gauge symmetries provided that
$\rho_{a}^{\mu},\eta_{ab},C_{abc}$ and $H_{\mu\nu\rho\sigma}$ are the
components of the anchor map, fibre metric, binary bracket and 4-form of a
twisted Courant algebroid $E$. This means that they satisfy the set of
algebraic and differential equations (2.69), (2.70) and (2.71). These gauge
symmetries may be neatly encoded in compact form using the homological vector
field $Q^{H}$, Eq.(2.76). Denote the fields of the model collectively as
$\phi^{\alpha}=(X^{\mu},A^{a},B_{\mu})$, understood as pull-backs via the map
$\phi_{\times}$ defined above of the coordinates
$x^{\alpha}=(x^{\mu},a^{a},b_{\mu})$, namely

$\phi^{\alpha}=\phi^{\ast}_{\times}(x^{\alpha})\,.$ (3.3)

Note that $\phi^{\alpha}=\phi^{\alpha}(\sigma,\theta)$ are dependent on the
coordinates of the source Q manifold $T[1]\Sigma$. The homological vector
field on $T[1]\Sigma$ is simply the de Rham differential
$\mathrm{d}=\theta^{m}\partial/\partial\sigma^{m}$. Define a collective gauge
parameter as $\epsilon^{\alpha}=(0,\epsilon^{a},-\psi_{\mu})$ with components
of degrees $0$ and $1$ respectively—we denoted explicitly the obvious absence
of gauge parameter for the lowest field which is a scalar. Then we can rewrite

$\mathrm{d}\epsilon^{\alpha}=[\mathrm{d},\epsilon]^{\alpha}=(\mathcal{L}_{\mathrm{d}}\epsilon)^{\alpha}\,,$
(3.4)

in terms of the (graded) Lie bracket of vector fields, provided we introduce
the vector field $\epsilon=\epsilon^{\alpha}\partial_{\alpha}$ on
$T[1]\Sigma\times\mathcal{M}$. Then it becomes advantageous to introduce the
following distinguished vector field on the product manifold $T[1]\Sigma\times
T^{\ast}[2]T[1]M$,

$\widehat{Q}=\mathrm{d}+Q^{H}\,.$ (3.5)

The gauge transformations of the fields are then given in terms of this vector
field as

$\delta\phi^{\alpha}=(\phi_{\times}^{\ast}\circ\mathcal{L}_{\widehat{Q}}\epsilon)(x^{\alpha})+\delta_{\text{triv}}\phi^{\alpha}\,,$
(3.6)

in the spirit of [39] as revisited in [25]. Here $\mathcal{L}_{\widehat{Q}}$
is the Lie derivative along the homological vector field $\widehat{Q}$. The
second term denotes the contributions that are proportional to the field
equations $F^{\mu}$ and $G^{a}$ obtained by varying with respect to the 2-form
and the 1-form respectively—the remaining field equation has degree 3, one
higher than the top form field in the model and does not contribute. It is
useful to mention that the field strengths of the various fields take the form

${\mathcal{F}}^{\alpha}=\mathrm{d}\phi^{\alpha}-\phi^{\ast}(Q^{\alpha})\,,$
(3.7)

in terms of the components of the homological vector field $Q^{H}$ [39].

Note that as long as $\epsilon$ is expressed in the canonical basis
$(\partial_{\alpha})$ these gauge transformations are not in manifestly
covariant form. We are not going to present the expanded form of this set of
gauge transformations at this stage. For Courant algebroids it is found in
various works in the literature, for example [1]. For twisted Courant
algebroids it is instructive to isolate the $H$-dependence in this gauge
transformations in order to understand its geometric significance. Note that
only the top form field acquires an $H$-dependence in its gauge
transformation, see App. A, therefore we only write this one:

$\delta B_{\mu}=\delta B_{\mu}|_{H=0}\,-\frac{1}{2}H_{\mu
abc}\,\epsilon^{a}A^{b}A^{c}-\frac{1}{2}H_{\mu\nu
ab}\,\epsilon^{a}A^{b}F^{\nu}-\frac{1}{3!}H_{\mu\nu\rho
a}\,\epsilon^{a}F^{\nu}F^{\rho}$ (3.8)

where $\delta B_{\mu}|_{H=0}$ is the transformation of the 2-form in the
Courant sigma model and $F$ is given as

$F^{\mu}=\mathrm{d}X^{\mu}-\rho_{a}{}^{\mu}A^{a}\,.$ (3.9)

Then $F^{\mu}=0$ is the field equation of the 2-form field. Thus we observe
that in presence of the 4-form $H$, there are field-equation dependent
(trivial) gauge transformations in $\delta B_{\mu}$. What is more, these
include a somewhat unorthodox nonlinear term in the field equations.

Two important remarks are in order. First, the algebra of gauge
transformations is both open and soft, in other words it only closes on the
stationary surface and its structure “constants” are field dependent.
Moreover, since the theory features a 2-form field, it is a first stage
reducible Hamiltonian system. Let us take a closer look to the gauge algebra.
We focus on the most complicated commutator of two gauge transformations on
the highest degree field $B_{\mu}$. This is

$\displaystyle[\delta_{1},\delta_{2}]B_{\mu}=\delta_{12}B_{\mu}+U_{\mu\nu}\wedge
F^{\nu}+V_{\mu\nu\rho}F^{\nu}\wedge F^{\rho}+W_{\mu a}G^{a}\,,$ (3.10)

where $G^{a}$ is the field equation for the field $A^{a}$ given in non-
covariant form as

$G^{a}:=\mathrm{d}A^{a}+\eta^{ab}\rho_{b}{}^{\mu}B_{\mu}+\frac{1}{2}{C}^{a}_{bc}A^{b}\wedge
A^{c}=0\,,$ (3.11)

$\delta_{12}$ is the gauge transformation with gauge parameters
$\epsilon^{a}_{12}$ and $\psi_{\mu 12}$, which are found to be

$\displaystyle\epsilon_{12}^{a}$ $\displaystyle=$ $\displaystyle
C_{bc}^{a}\epsilon_{1}^{b}\epsilon_{2}^{c}\,,$ (3.12) $\displaystyle\psi_{\mu
12}$ $\displaystyle=$ $\displaystyle
2\partial_{\mu}\rho_{a}{}^{\nu}\epsilon_{[1}^{a}\psi_{\nu
2]}-(\partial_{\mu}C_{abc}+H_{\mu
abc})\epsilon_{1}^{a}\epsilon_{2}^{b}A^{c}-{\textstyle\frac{1}{2}}H_{\mu\nu
ab}\epsilon_{1}^{a}\epsilon_{2}^{b}F^{\nu}\,,$ (3.13)

and the three coefficients $U,V$ and $W$ are given by999We note in advance
that tensors such as the basic curvature appear in the covariant version of
these coefficients. That the basic curvature appears as a coefficient in the
gauge algebra of the simpler case of Lie algebroid gauge theories was first
noticed in [50], even though the geometric meaning of the tensor was not yet
identified there.

$\displaystyle U_{\mu\nu}$ $\displaystyle=$ $\displaystyle
2\partial_{\mu}\partial_{\nu}\rho_{a}{}^{\sigma}\psi_{\sigma[1}\epsilon^{a}_{2]}+\epsilon_{1}^{a}\epsilon_{2}^{b}A^{c}(\partial_{\mu}\partial_{\nu}C_{abc}+\partial_{(\mu}H_{\nu)abc})\,,$
(3.14) $\displaystyle V_{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle-{\textstyle\frac{1}{3}}\epsilon^{a}_{1}\epsilon^{b}_{2}\,\partial_{(\mu}H_{\nu)\rho
ab}\,,$ (3.15) $\displaystyle W_{\mu a}$ $\displaystyle=$
$\displaystyle-\epsilon_{1}^{b}\epsilon_{2}^{c}(\partial_{\mu}C_{abc}+{\textstyle\frac{1}{2}}H_{\mu
abc})\,.$ (3.16)

We observe that a term quadratic in the field equations is generated in the
gauge algebra (3.10). This term is absent for vanishing 4-form. Thus we see
that twisted Courant algebroids feature a new property compared with standard
gauge theories, that of nonlinear openness of their gauge algebra. This was
also identified before in the context of twisted R-Poisson models [34],
further studied in [38, 21]. Related to this is the fact that the 1-form gauge
parameter $\psi_{\mu 12}$ receives a field equation contribution, one more
unconventional feature of the model. In principle one should further calculate
the Jacobi identity for the gauge algebra and higher identities thereof and
identify all the structural quantities that appear on the way. We refrain from
embarking in this complicated task for the moment, since an equivalent way of
addressing this goes through the BRST transformation and its various powers,
as we discuss below. Another feature of the gauge algebra which is worth
highlighting is that the coefficient of the term which is nonlinear in the
field equations is explicitly symmetric in its first two indices, see Eq.
(3.15). This will also play a crucial role in understanding the systematics of
the BRST formalism for the model.

The second remark is that the gauge transformation $\delta\phi^{\alpha}$ is
not tensorial. To express it in a basis-independent form one must take into
account changes of frame in the target space. This can be parametrized by a
connection on the twisted Courant algebroid, which is one of the reasons we
introduced such objects in the previous section. We will also account for this
directly within the BRST formalism.

###  3.2 On-shell closed BRST transformations

Following the standard steps of the BRST procedure, we now promote the gauge
transformations discussed previously to BRST transformations which will be
nilpotent on-shell, namely on the constrained surface. The two gauge
parameters $\epsilon$ and $\psi$ are assigned a ghost degree 1 (we do not
introduce new notation for them) and moreover due to the first stage
reducibility of the system we must introduce an additional ghost of degree 2
which we denote as $\widetilde{\psi}$. This is the ghost for the ghost $\psi$.
We summarize the fields and ghosts, which are the same as for the usual
Courant sigma model, in Table 1.

Field/Ghost |  |  |  |  |  |
---|---|---|---|---|---|---
$X^{\mu}$ | $A^{a}$ | $B_{\mu}$ | $\epsilon^{a}$ | $\psi_{\mu}$ | $\widetilde{\psi}_{\mu}$
Ghost degree |  |  |  |  |  |
$0$ | $0$ | $0$ | $1$ | $1$ | $2$
Form degree |  |  |  |  |  |
$0$ | $1$ | $2$ | $0$ | $1$ | $0$
Table 1: The eight fields and ghosts of the twisted Courant sigma model (due
to the index $a$ the fields $A$ and $\epsilon$ are doubled). The ghosts
$\epsilon^{a},\psi_{\mu}$ correspond to the scalar and 1-form gauge parameters
and the degree 2 ghost $\widetilde{\psi}_{\mu}$ is the ghost for the ghost
$\psi_{\mu}$.

We denote the on-shell closed BRST transformations as $s_{0}$ to avoid
confusion with previous and later notation. We have not introduced antifields
yet but we can already mention that $s_{0}$ is the “longitudinal differential”
in the sense of [15] and therefore it annihilates antifields. According to our
earlier discussion, the BRST transformation of the fields in non manifestly
covariant form is

$s_{0}\phi^{\alpha}=\mathrm{d}\epsilon^{\alpha}+\epsilon^{\beta}\partial_{\beta}Q^{\alpha}+s_{\text{triv}}\phi^{\alpha}\,,$
(3.17)

where $Q^{\alpha}$ are the components of $Q^{H}$. The coefficients are
controlled by the first derivatives of the homological vector field, as
expected by the general form that contains the Lie derivative along it.
Anticipating a generalization of the present situation to any higher gauge
theory in the sense of Ref. [39], we move on to determine the analogous
general collective expressions for the ghosts and ghost for ghost, a task that
was not performed in [39]. To determine the general form of the BRST
transformation on the ghosts we make the following degree 2 Ansatz, neglecting
the pull-back map for the time being:

$s_{0}\epsilon^{\alpha}=\mathrm{d}\widetilde{\epsilon}^{\,\alpha}+\widetilde{\epsilon}^{\,\beta}\kappa^{\alpha}_{\beta}(\phi)+\frac{1}{2}\epsilon^{\beta}\epsilon^{\gamma}\lambda^{\alpha}_{\beta\gamma}(\phi)+s_{\text{triv}}\epsilon^{\alpha}\,,$
(3.18)

where $\widetilde{\epsilon}^{\,\alpha}=(0,0,-\widetilde{\psi}_{\mu})$ is an
alternative notation for the ghost for ghost that aligns with the previously
introduced notation. Consistency of the BRST formalism imposes constraints on
the undetermined coefficients $\kappa$ and $\lambda$. To determine $\kappa$
and $\lambda$ in Eq. (3.18) it is enough to require that the square of the
BRST transformation on the fields vanishes weakly,

$s_{0}^{2}\phi^{\alpha}\approx 0\,,$ (3.19)

with $\approx$ denoting weak equality. Note that
$s_{\text{triv}}\phi^{\alpha}\approx 0$ anyway and it does not influence the
rest of the calculation. The result of imposing this condition is that the
coefficients $\kappa$ and $\lambda$ are given as

$\displaystyle\kappa^{\alpha}_{\beta}$ $\displaystyle=$
$\displaystyle-\partial_{\beta}Q^{\alpha}\,,$ (3.20)
$\displaystyle\lambda^{\alpha}_{\beta\gamma}$ $\displaystyle=$
$\displaystyle-\partial_{\beta}\partial_{\gamma}Q^{\alpha}\,.$ (3.21)

We observe that they are controlled by the first and second derivatives of the
homological vector field $Q^{H}$. We note in passing that the second
derivatives are related to the Atiyah cocycle on the dg manifold
$\mathcal{M}$, here for the trivial connection. This is in accord with the
relation of the Atiyah cocycle to gauge theory found in Ref. [25].

To complete this local analysis, we need to determine the BRST transformation
on the ghosts for ghosts $\widetilde{\epsilon}^{\,\alpha}$. Since in the
present case the single component of this quantity is a scalar field, we do
not need to include any exterior derivatives. We make the degree 3 Ansatz

$s_{0}\widetilde{\epsilon}^{\,\alpha}=\widetilde{\epsilon}^{\,\beta}\epsilon^{\gamma}\widetilde{\kappa}^{\alpha}_{\beta\gamma}(\phi)+{\textstyle\frac{1}{3!}}\epsilon^{\beta}\epsilon^{\gamma}\epsilon^{\delta}\widetilde{\lambda}^{\alpha}_{\beta\gamma\delta}(\phi)\,.$
(3.22)

Consistency requires that

$s_{0}^{2}\epsilon^{\alpha}\approx 0\,,$ (3.23)

which fixes the new undetermined coefficients to be

$\displaystyle\widetilde{\kappa}^{\alpha}_{\beta\gamma}$ $\displaystyle=$
$\displaystyle\partial_{\beta}\partial_{\gamma}Q^{\alpha}\,,$ (3.24)
$\displaystyle\widetilde{\lambda}^{\alpha}_{\beta\gamma\delta}$
$\displaystyle=$
$\displaystyle\partial_{\beta}\partial_{\gamma}\partial_{\delta}Q^{\alpha}\,.$
(3.25)

It is thus found that also third derivatives of the homological vector field
appear in the BRST transformation of the ghost for ghost. This is also
expected in view of the relation of these transformations to the Kapranov
$L_{\infty}[1]$ algebra that governs the BRST structure of any higher gauge
theory as proposed in Ref. [25] to which we refer for more details.
Summarizing the above discussion and ignoring pullbacks for the time being, we
have proven that consistency of the BRST formalism leads to the following:

###### Proposition 3.26

The BRST transformations of the fields
$(\phi^{\alpha})=(X^{\mu},A^{a},B_{\mu})$, ghosts
$(\epsilon^{\alpha})=(0,\epsilon^{a},-\psi_{\mu})$ and ghost for ghost
$(\widetilde{\epsilon}^{\,\alpha})=(0,0,-\widetilde{\psi}_{\mu})$ for a 4-form
twisted Courant sigma model take the following form in terms of the components
$(Q^{\alpha})$ of the homological vector field $Q^{H}$ of a twisted Courant
algebroid (Eq. (2.76)) and their derivatives:

$\displaystyle s_{0}\phi^{\alpha}$
$\displaystyle=\mathrm{d}\epsilon^{\alpha}+\epsilon^{\beta}\partial_{\beta}Q^{\alpha}+s_{\text{triv}}\phi^{\alpha}\,,$
(3.27) $\displaystyle s_{0}\epsilon^{\alpha}$
$\displaystyle=\mathrm{d}\widetilde{\epsilon}^{\,\alpha}-\widetilde{\epsilon}^{\,\beta}\partial_{\beta}Q^{\alpha}-\frac{1}{2}\epsilon^{\beta}\epsilon^{\gamma}\partial_{\beta}\partial_{\gamma}Q^{\alpha}+s_{\text{triv}}\epsilon^{\alpha}\,,$
(3.28) $\displaystyle s_{0}\widetilde{\epsilon}^{\,\alpha}$
$\displaystyle=\widetilde{\epsilon}^{\,\beta}\epsilon^{\gamma}\partial_{\beta}\partial_{\gamma}Q^{\alpha}+\frac{1}{3!}\epsilon^{\beta}\epsilon^{\gamma}\epsilon^{\delta}\partial_{\beta}\partial_{\gamma}\partial_{\delta}Q^{\alpha}\,,$
(3.29)

with $s_{\text{triv}}$ vanishing on the stationary surface.

###  3.3 Manifestly covariant BRST transformations

We would now like to understand the global meaning of the various coefficients
in Proposition 3.26 and reach a manifestly target space covariant formulation.
Let us first explain what it would mean to implement manifest target space
covariance and obtain inherently tensorial expressions for the gauge
symmetries. One may think of this as the following three-step algorithm: (i)
Consider the field $\phi^{\alpha}$ and its non-tensorial gauge transformation
$\delta\phi^{\alpha}$ and rewrite it in terms of covariant quantities using
the connection $\nabla$; (ii) consider the index-free field
$\phi=\phi^{\alpha}\otimes{e}_{\alpha}$ and isolate the transformation of the
pull-back basis ${e}_{\alpha}$ in $\delta\phi^{\alpha}$; (iii) define a
_different_ , tensorial transformation $\delta^{\scriptscriptstyle\nabla}\phi$
by absorbing suitable field equation dependent terms such that the final
expression is manifestly covariant and tensorial.

To illustrate this procedure let us consider an example. In the twisted
Courant sigma model take the gauge transformation of the 1-form as given from
Eq. (3.6)101010A gauge transformation always has ambiguity of trivial gauge
transformations, $s_{\text{triv}}\phi^{\alpha}$. We fix this ambiguity of the
gauge transformation of $A^{a}$ to the standard form. Such ambiguities of
gauge transformations in the BV and BFV formalisms have been discussed for the
twisted PSM in [40].:

$\delta
A^{a}=\mathrm{d}\epsilon^{a}+C_{bc}^{a}A^{b}\epsilon^{c}+\eta^{ab}\rho_{b}{}^{\mu}\psi_{\mu}\,.$
(3.30)

Step (i) of the above algorithm instructs us to rewrite this transformation in
the alternative form

$\delta
A^{a}=\mathrm{D}\epsilon^{a}-2\eta^{ad}\mathcal{T}_{bcd}A^{b}\epsilon^{c}+\eta^{ab}\rho_{b}{}^{\mu}\psi^{\scriptscriptstyle\nabla}_{\mu}-\omega^{a}_{\mu
b}F^{\mu}\epsilon^{b}-\rho_{b}{}^{\mu}\omega_{\mu c}^{a}A^{c}\epsilon^{b}\,,$
(3.31)

where $\mathrm{D}$ is the exterior covariant derivative induced by $\nabla$,
$\mathcal{T}$ is the Gualtieri torsion tensor and
$\psi^{\scriptscriptstyle\nabla}$ is a redefined 1-form gauge parameter. We
take this opportunity to introduce the following redefined field and
parameters in accord with the analysis of Section 2:111111Of course it is no
accident that these three quantities appear together in the same superfield of
degree 2 in the AKSZ construction of Courant sigma models.

$\displaystyle B_{\mu}^{\scriptscriptstyle\nabla}$ $\displaystyle=$
$\displaystyle B_{\mu}+{\textstyle\frac{1}{2}}\omega_{\mu ab}A^{a}A^{b}\,,$
(3.32) $\displaystyle{\psi}_{\mu}^{\scriptscriptstyle\nabla}$ $\displaystyle=$
$\displaystyle{\psi}_{\mu}+\omega_{\mu ab}A^{a}\epsilon^{b}\,,$ (3.33)
$\displaystyle\widetilde{\psi}_{\mu}^{\scriptscriptstyle\nabla}$
$\displaystyle=$
$\displaystyle\widetilde{\psi}_{\mu}+{\textstyle\frac{1}{2}}\omega_{\mu
ab}\epsilon^{a}\epsilon^{b}\,.$ (3.34)

Note now that this completely equivalent form of the gauge transformation for
the 1-form $A^{a}$ contains two additional terms generated during the
covariantization process. If we now consider the pull-back basis $e_{a}$ of
the vector bundle $X^{\ast}E$, this transforms under a change of frame
according to

$\delta e_{a}=\omega_{\mu a}^{b}\delta X^{\mu}e_{b}\,.$ (3.35)

We now observe that for the field $A=A^{a}\otimes e_{a}$ we can apply the
Leibniz rule to obtain

$\delta A=\delta A^{a}\otimes e_{a}+A^{a}\otimes\delta
e_{a}:=\bar{\delta}A^{a}\otimes e_{a}\,,$ (3.36)

where we defined the tensorial transformation (see e.g. [51])

$\bar{\delta}A^{a}=\delta A^{a}+\omega_{\mu b}^{a}A^{b}\delta X^{\mu}\,.$
(3.37)

It is a matter of mere inspection to see that the second term in this Leibniz
rule corresponds precisely to the last term in the transformation $\delta
A^{a}$ in the form of Eq. (3.31). This isolates the transformation of the
pull-back basis as instructed in step (ii) of the algorithm. Finally we define
the new transformation

$\delta^{\scriptscriptstyle\nabla}A^{a}:=\delta A^{a}+\omega^{a}_{\mu
b}F^{\mu}\epsilon^{b}\,,$ (3.38)

and the associated tensorial one

$\bar{\delta}^{\scriptscriptstyle\nabla}A^{a}:=\delta^{\scriptscriptstyle\nabla}A^{a}+\omega_{\mu
b}^{a}A^{b}\delta X^{\mu}\,.$ (3.39)

This is step (iii) of the algorithm and together with the previous steps leads
to the final tensorial and manifestly covariant result

$\delta^{\scriptscriptstyle\nabla}A=(\delta^{\scriptscriptstyle\nabla}A)^{a}\otimes
e_{a}=\bar{\delta}^{\scriptscriptstyle\nabla}A^{a}\otimes
e_{a}=\mathrm{D}\epsilon-2\mathcal{T}(A,\epsilon)+\rho^{\ast}(\psi^{\scriptscriptstyle\nabla})\,,$
(3.40)

where $\rho^{\ast}$ is the transpose map to the anchor $\rho$. In the BRST
setting we should replace $\delta$ and $\delta^{\scriptscriptstyle\nabla}$
with operators $s_{0}$ and $s_{0}^{\scriptscriptstyle\nabla}$ and the gauge
parameters with the corresponding ghosts.

Even though the analysis above illustrates the general procedure, it hides the
advantages of the covariantized homological vector field for twisted Courant
algebroids introduced in Section 2. Indeed this rewriting obviates the need to
apply the above algorithm. Instead one obtains _automatically_ the correct
tensorial transformations on the fields of the model as follows. We start by
revisiting the example of the field $A^{a}$ that was detailed above. Note
first that there are two expansions of the homological vector field $Q^{H}$,
one in the basis $(\partial_{\alpha})$ and one in the basis
$(\mathrm{D}_{\alpha})$ introduced by the differential operators
$\mathrm{D}^{(0)}$, $\mathrm{D}^{(-1)}$ and $\mathrm{D}^{(-2)}$ in Eqs.
(2.53), (2.54) and (2.55). The coefficients in each expansions are of course
different:

$Q^{H}=Q^{\alpha}\partial_{\alpha}=Q^{\alpha}_{\scriptscriptstyle\nabla}\mathrm{D}_{\alpha}\,.$
(3.41)

According to Eq. (2.76) the “covariant” components are given as

$\displaystyle Q_{\scriptscriptstyle\nabla}^{\mu}$ $\displaystyle=$
$\displaystyle a^{a}\rho_{a}{}^{\mu}\,,$ (3.42) $\displaystyle
Q_{\scriptscriptstyle\nabla}^{a}$ $\displaystyle=$
$\displaystyle-\eta^{ab}\rho_{b}{}^{\mu}b^{\scriptscriptstyle\nabla}_{\mu}+{\mathcal{T}}_{bc}{}^{a}a^{b}a^{c}\,,$
(3.43) $\displaystyle Q^{\scriptscriptstyle\nabla}_{\mu}$ $\displaystyle=$
$\displaystyle-\nabla_{\mu}\rho_{a}{}^{\nu}b^{\scriptscriptstyle\nabla}_{\nu}a^{a}-\frac{1}{3}\widetilde{\mathcal{S}}_{\mu
abc}a^{a}a^{b}a^{c}\,.$ (3.44)

Notice now that

$\displaystyle\mathrm{D}^{\mu}_{(-2)}Q_{\scriptscriptstyle\nabla}^{a}$
$\displaystyle=$ $\displaystyle-\eta^{ab}\rho_{b}{}^{\mu}\,,$ (3.45)
$\displaystyle\mathrm{D}_{b}^{(-1)}Q_{\scriptscriptstyle\nabla}^{a}$
$\displaystyle=$ $\displaystyle 2{\mathcal{T}}_{bc}{}^{a}a^{c}\,.$ (3.46)

Putting these together we directly observe that

$\bar{\delta}^{\scriptscriptstyle\nabla}A^{a}=\mathrm{D}\epsilon^{a}+\epsilon_{\scriptscriptstyle\nabla}^{\alpha}\mathrm{D}_{\alpha}Q_{\scriptscriptstyle\nabla}^{a}\,,$
(3.47)

where we used the coefficients in the alternative expansion of the collective
gauge parameter:

$\epsilon=\epsilon^{\alpha}\partial_{\alpha}=\epsilon^{\alpha}_{\scriptscriptstyle\nabla}\mathrm{D}_{\alpha}\,,$
(3.48)

which are nothing but
$(\epsilon^{\alpha}_{\scriptscriptstyle\nabla})=(0,\epsilon^{a},-\psi^{\scriptscriptstyle\nabla}_{\mu})$.
Doing the same analysis for all fields and ghosts in the classical basis of
the model, it is straightforward to show that the tensorial version of
Proposition 3.26 (with implicit pull-backs) is:

###### Proposition 3.49

The BRST transformations on the fields
$\phi=(\phi_{\scriptscriptstyle\nabla}^{\alpha})=(X^{\mu},A^{a},B^{\scriptscriptstyle\nabla}_{\mu})$,
the ghosts
$\epsilon=(\epsilon_{\scriptscriptstyle\nabla}^{\alpha})=(0,\epsilon^{a},-\psi^{\scriptscriptstyle\nabla}_{\mu})$
and the ghost for ghost
$\widetilde{\epsilon}=(\widetilde{\epsilon}_{\scriptscriptstyle\nabla}^{\,\alpha})=(0,0,-\widetilde{\psi}^{\scriptscriptstyle\nabla}_{\mu})$
of a 4-form twisted Courant sigma model with source $T[1]\Sigma$ and target
$\mathcal{M}=T^{\ast}[2]T[1]M$ with coordinates
$(x^{\alpha})=(x^{\mu},a^{a},b^{\scriptscriptstyle\nabla}_{\mu})$ of degrees
$(0,1,2)$ are given in terms of the homological vector field $Q^{H}$ of a
4-form twisted Courant algebroid given in (2.76) and in tensorial form as
$s_{0}^{\scriptscriptstyle\nabla}\phi=s_{0}^{\scriptscriptstyle\nabla}\phi^{\alpha}_{\scriptscriptstyle\nabla}\otimes
e_{\alpha}$ (and similarly for $\epsilon$ and $\widetilde{\epsilon}$) with

$\displaystyle
s_{0}^{\scriptscriptstyle\nabla}\phi_{\scriptscriptstyle\nabla}^{\alpha}$
$\displaystyle=\mathrm{D}\epsilon^{\alpha}_{\scriptscriptstyle\nabla}+\epsilon^{\beta}_{\scriptscriptstyle\nabla}\mathrm{D}_{\beta}Q^{\alpha}_{\scriptscriptstyle\nabla}+s^{\scriptscriptstyle\nabla}_{\text{triv}}\phi^{\alpha}_{\scriptscriptstyle\nabla}\,,$
(3.50) $\displaystyle
s_{0}^{\scriptscriptstyle\nabla}\epsilon_{\scriptscriptstyle\nabla}^{\alpha}$
$\displaystyle=\mathrm{D}\widetilde{\epsilon}^{\,\alpha}_{\scriptscriptstyle\nabla}-\widetilde{\epsilon}^{\,\beta}_{\scriptscriptstyle\nabla}\mathrm{D}_{\beta}Q^{\alpha}_{\scriptscriptstyle\nabla}-\frac{1}{2}\epsilon_{\scriptscriptstyle\nabla}^{\beta}\epsilon_{\scriptscriptstyle\nabla}^{\gamma}\mathrm{D}_{\beta}\mathrm{D}_{\gamma}Q^{\alpha}_{\scriptscriptstyle\nabla}+s^{\scriptscriptstyle\nabla}_{\text{triv}}\epsilon^{\alpha}_{\scriptscriptstyle\nabla}\,,$
(3.51) $\displaystyle
s_{0}\widetilde{\epsilon}_{\scriptscriptstyle\nabla}^{\,\alpha}$
$\displaystyle=\widetilde{\epsilon}^{\,\beta}_{\scriptscriptstyle\nabla}\epsilon^{\gamma}_{\scriptscriptstyle\nabla}\mathrm{D}_{\beta}\mathrm{D}_{\gamma}Q_{\scriptscriptstyle\nabla}^{\alpha}+\frac{1}{3!}\epsilon^{\beta}_{\scriptscriptstyle\nabla}\epsilon^{\gamma}_{\scriptscriptstyle\nabla}\epsilon^{\delta}_{\scriptscriptstyle\nabla}\mathrm{D}_{\beta}\mathrm{D}_{\gamma}\mathrm{D}_{\delta}Q^{\alpha}_{\scriptscriptstyle\nabla}\,,$
(3.52)

where $\mathrm{D}$ is the exterior covariant derivative on $\Sigma$, the
differential operators $\mathrm{D}_{\alpha}$ are as in Eqs. (2.53), (2.54) and
(2.55) and $s_{\text{triv}}^{\scriptscriptstyle\nabla}$ vanish on the
stationary surface.

The proof of the proposition is completed using the following set of equations
for first derivatives

$\displaystyle\mathrm{D}^{\nu}Q_{\scriptscriptstyle{\nabla}}^{\mu}=0\leavevmode\nobreak\
,\qquad\qquad\qquad\mathrm{D}_{a}Q_{\scriptscriptstyle{\nabla}}^{\mu}=\rho_{a}{}^{\mu}\leavevmode\nobreak\
,$
$\displaystyle\mathrm{D}^{\nu}Q_{\scriptscriptstyle{\nabla}}^{a}=-\eta^{ab}\rho_{b}{}^{\nu}\leavevmode\nobreak\
,\quad\qquad\,\mathrm{D}_{b}Q_{\scriptscriptstyle{\nabla}}^{a}=2{\cal
T}^{a}_{bc}a^{c}\,,$
$\displaystyle\mathrm{D}^{\nu}Q^{\scriptscriptstyle{\nabla}}_{\mu}=-\nabla_{\mu}\rho_{a}{}^{\nu}a^{a}\leavevmode\nobreak\
,\qquad\mathrm{D}_{a}Q^{\scriptscriptstyle{\nabla}}_{\mu}=-\nabla_{\mu}\rho_{a}{}^{\nu}b^{\scriptscriptstyle{\nabla}}_{\nu}-\widetilde{\cal
S}_{\mu abc}a^{b}a^{c}\leavevmode\nobreak\ ,$

and the following set for second and third derivatives

$\displaystyle\mathrm{D}_{a}\mathrm{D}^{\nu}Q^{\scriptscriptstyle{\nabla}}_{\mu}=-\nabla_{\mu}\rho_{a}{}^{\nu}\leavevmode\nobreak\
,\qquad\mathrm{D}_{c}\mathrm{D}_{b}Q_{\scriptscriptstyle{\nabla}}^{a}=2{\cal
T}^{a}_{bc}\leavevmode\nobreak\ ,$
$\displaystyle\mathrm{D}_{b}\mathrm{D}_{a}Q^{\scriptscriptstyle{\nabla}}_{\mu}=-2\widetilde{\cal
S}_{\mu abc}a^{c}\leavevmode\nobreak\
,\quad\,\mathrm{D}^{\nu}\mathrm{D}_{a}Q^{\scriptscriptstyle{\nabla}}_{\mu}=-\nabla_{\mu}\rho_{a}{}^{\nu}\,,$
$\displaystyle\mathrm{D}_{c}\mathrm{D}_{b}\mathrm{D}_{a}Q_{\scriptscriptstyle{\nabla}\mu}=-2\widetilde{\cal
S}_{\mu abc}\leavevmode\nobreak\ .$

These allow to express the tensorial form of the BRST transformations on the
fields and ghosts in terms of the classical differential geometric data on a
twisted Courant algebroid, namely the anchor map $\rho$ and its transpose
$\rho^{\ast}$, the Gualtieri torsion $\mathcal{T}$ and the 4-form twisted
basic curvature $\widetilde{\mathcal{S}}$. We present some additional details
and expressions in a coordinate basis as supplementary material in Appendix B;
here we report the final manifestly tensorial and basis-independent
result—with suitable grouping of terms:

$\displaystyle s_{0}^{\scriptscriptstyle\nabla}X$ $\displaystyle=$
$\displaystyle\rho(\epsilon)\,,$ (3.53) $\displaystyle
s_{0}^{\scriptscriptstyle\nabla}\epsilon$ $\displaystyle=$
$\displaystyle\,\,\quad-\,\rho^{\ast}(\widetilde{\psi}^{\scriptscriptstyle\nabla})+\mathcal{T}(\epsilon,\epsilon)\,,$
(3.54) $\displaystyle s_{0}^{\scriptscriptstyle\nabla}A$ $\displaystyle=$
$\displaystyle\mathrm{D}\epsilon+\rho^{\ast}(\psi^{\scriptscriptstyle\nabla})-2{\mathcal{T}}(A,\epsilon)\,,$
(3.55) $\displaystyle
s_{0}^{\scriptscriptstyle\nabla}\widetilde{\psi}^{\,\scriptscriptstyle\nabla}$
$\displaystyle=$ $\displaystyle\hskip
106.0pt-\,\nabla\rho\,(\widetilde{\psi}^{\,\scriptscriptstyle\nabla},\epsilon)-{\textstyle\frac{1}{3}}\widetilde{\mathcal{S}}(\epsilon,\epsilon,\epsilon)\,,$
(3.56) $\displaystyle
s_{0}^{\scriptscriptstyle\nabla}\psi^{\scriptscriptstyle\nabla}$
$\displaystyle=$ $\displaystyle\hskip 10.0pt{\rm
D}\widetilde{\psi}^{\,\scriptscriptstyle\nabla}+\nabla\rho(\widetilde{\psi}^{\,\scriptscriptstyle\nabla},A)+\nabla\rho({\psi}^{\scriptscriptstyle\nabla},\epsilon)+\widetilde{\mathcal{S}}(A,\epsilon,\epsilon)+s_{\text{triv}}^{\scriptscriptstyle\nabla}\psi^{\scriptscriptstyle\nabla}\,,$
(3.57) $\displaystyle
s_{0}^{\scriptscriptstyle\nabla}B^{\scriptscriptstyle\nabla}$ $\displaystyle=$
$\displaystyle-\mathrm{D}\psi^{\scriptscriptstyle\nabla}+\nabla\rho(\psi^{\scriptscriptstyle\nabla},A)-\nabla\rho(B^{\scriptscriptstyle\nabla},\epsilon)-\widetilde{\mathcal{S}}(A,A,\epsilon)+s_{\text{triv}}^{\scriptscriptstyle\nabla}B^{\scriptscriptstyle\nabla}\,,$
(3.58)

where the final terms in the BRST transformation of the 1-form ghost and the
2-form field are given by

$\displaystyle
s_{\text{triv}}^{\scriptscriptstyle\nabla}\psi^{\scriptscriptstyle\nabla}$
$\displaystyle=$
$\displaystyle\frac{1}{4}(H(\rho(\epsilon),\rho(\epsilon))-R(\epsilon,\epsilon))(-,F)\,,$
(3.59) $\displaystyle
s_{\text{triv}}^{\scriptscriptstyle\nabla}B^{\scriptscriptstyle\nabla}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}(H(\rho(\epsilon),\rho(A))-R(\epsilon,A))(-,F)-\frac{1}{3!}H(-,\rho(\epsilon),F,F)$
(3.60)

To completely clarify and explain our notation, first note that all geometric
objects mentioned below are understood in composition with the map $X$ and
their components in a coordinate basis are therefore functions of the pull-
back coordinates $X^{\mu}$. One can understand this better by noting that for
example the scalar parameter $\epsilon\in\Gamma(X^{\ast}E)$ (which should not
be confused with the collective gauge parameter introduced earlier) is also a
section of the pull-back bundle, and thus $\rho$ cannot just act on it since
its domain is the bundle $E$; a composition with $X$ is due. The same remark
holds in all other cases.

The covariant BRST transformation $s_{0}^{\scriptscriptstyle\nabla}$ (obtained
directly from Proposition 3.49 and corresponding to
$\bar{\delta}^{\scriptscriptstyle\nabla}$ on the classical fields) is
_different_ off-shell than the BRST transformation $s_{0}$ (corresponding to
$\bar{\delta}$ on fields) for all non-scalar fields, but remains unchanged for
the scalar fields $X,\epsilon$ and
$\widetilde{\psi}^{\,\scriptscriptstyle\nabla}$. To explain this, first note
that the field equations of the fields $B_{\mu}$ and $A^{a}$ written in
covariant form are

$\displaystyle F^{\mu}$ $\displaystyle:=$
$\displaystyle\mathrm{d}X^{\mu}-\rho_{a}{}^{\mu}A^{a}=0\,,$ (3.61)
$\displaystyle G_{{\scriptscriptstyle\nabla}}^{a}$ $\displaystyle:=$
$\displaystyle G^{a}+\omega_{\mu b}^{a}F^{\mu}A^{b}={\rm
D}A^{a}+\eta^{ab}\rho_{b}{}^{\mu}B_{\mu}^{\scriptscriptstyle\nabla}-\eta^{ab}{\mathcal{T}}_{cdb}A^{c}A^{d}=0\,.$
(3.62)

Our definition of covariant BRST transformation is then given for the three
non-scalar fields as:

$\displaystyle s_{0}^{\scriptscriptstyle\nabla}A^{a}$ $\displaystyle=$
$\displaystyle s_{0}A^{a}+\omega^{a}_{\mu b}F^{\mu}\epsilon^{b}\,,$ (3.63)
$\displaystyle
s_{0}^{\scriptscriptstyle\nabla}\psi^{\scriptscriptstyle\nabla}_{\mu}$
$\displaystyle=$ $\displaystyle
s_{0}{\psi}_{\mu}^{\scriptscriptstyle\nabla}-\mathring{\Gamma}_{\mu\sigma}^{\nu}{\widetilde{\psi}}_{\nu}^{\,\scriptscriptstyle\nabla}F^{\sigma}+{\textstyle\frac{1}{2}}(\partial_{(\mu}\omega_{\nu)ab}+\omega_{[\mu
a}^{d}\omega_{\nu]bd})F^{\nu}\epsilon^{a}\epsilon^{b}\,,$ (3.64)
$\displaystyle
s_{0}^{\scriptscriptstyle\nabla}{B}_{\mu}^{\scriptscriptstyle\nabla}$
$\displaystyle=$ $\displaystyle
s_{0}{B}_{\mu}^{\scriptscriptstyle\nabla}-\mathring{\Gamma}_{\mu\sigma}^{\nu}{\psi}_{\nu}^{\scriptscriptstyle\nabla}F^{\sigma}-(\partial_{(\mu}\omega_{\nu)ab}+\omega_{[\mu
a}^{d}\omega_{\nu]bd})F^{\nu}A^{a}\epsilon^{b}-\omega_{\mu
ab}G^{a}\epsilon^{b}\,,$ (3.65)

where $s_{0}\phi^{\alpha}$ is the tensorial transformation corresponding to
the gauge transformation $\bar{\delta}\phi^{\alpha}$. Obviously on scalar
fields there is no difference since there are no scalar equations of motion
and also the field equation of $X^{\mu}$ plays no role in this since it is of
too high degree (a 3-form).

##  4 The BRST power finesse and the master action

The BRST transformation is nilpotent on-shell, a fact that does not depend on
whether a manifest target space covariant formulation is employed or not—we
shall mostly be working on a local patch in this section but the globalisation
can be performed in a fairly straightforward way as in previous sections. Off-
shell its square contains contributions proportional to the field equations of
the model, a fact we already used in order to fix it. It is instructive and in
fact very useful to determine these expressions, see Appendix B for more
technical details. In terms of the components $Q^{\alpha}$ of the homological
vector field $Q^{H}$, using definition (3.7) for the various field strengths
that correspond to the field equations of the model, first we find
that121212The consistent choice of grading is dictated by the form degree of
the field in question:
$|\phi|=0,\,|\epsilon|=-1,\,|\widetilde{\epsilon}|=-2,\,|\mathcal{F}|=1$. Note
also that we use Deligne’s sign convention for fields with multiple grading,
e.g. for differential form degree $f$ and ghost degree $g$:
$\alpha_{1}\alpha_{2}=(-1)^{f_{1}f_{2}+g_{1}g_{2}}\alpha_{2}\alpha_{1}$.

$s_{0}{\cal
F}^{\alpha}=(-1)^{|\epsilon^{\beta}|}\epsilon^{\beta}{\mathcal{F}}^{\gamma}\partial_{\gamma}\partial_{\beta}Q^{\alpha}\leavevmode\nobreak\
,$ (4.1)

where ${\cal F}^{\alpha}=\\{F^{\mu},G^{a}\\}$ are the field equations defined
in (3.61) and (3.62). This is simply the statement that the field strengths
transform covariantly. Furthermore, the Bianchi identities have the simple
form

$\mathrm{d}{\mathcal{F}}^{\alpha}+{\mathcal{F}}^{\beta}\partial_{\beta}Q^{\alpha}=0\leavevmode\nobreak\
.$ (4.2)

We need these transformations in order to proceed with determining the
explicit form of the square of the BRST transformation on the classical basis.
Indeed it is straightforward to show that

$\displaystyle s_{0}^{2}\phi^{\alpha}$ $\displaystyle=$
$\displaystyle-\widetilde{\epsilon}^{\beta}{\mathcal{F}}^{\gamma}\partial_{\gamma}\partial_{\beta}Q^{\alpha}-{\textstyle\frac{1}{2}}(-1)^{|\epsilon^{\beta}|+|\epsilon^{\gamma}|}\epsilon^{\beta}\epsilon^{\gamma}{\mathcal{F}}^{\delta}\partial_{\delta}\partial_{\beta}\partial_{\gamma}Q^{\alpha}$
(4.3) $\displaystyle+\,\mathrm{d}(s_{\rm triv}\epsilon^{\alpha})+s_{0}(s_{\rm
triv}\phi^{\alpha})+s_{\rm
triv}\epsilon^{\beta}\partial_{\beta}Q^{\alpha}-\epsilon^{\beta}s_{\rm
triv}\phi^{\gamma}\partial_{\gamma}\partial_{\beta}Q^{\alpha}\,,$
$\displaystyle s_{0}^{2}\epsilon^{\alpha}$ $\displaystyle=$
$\displaystyle(-1)^{|\epsilon^{\gamma}|}\widetilde{\epsilon}^{\beta}\epsilon^{\gamma}{\mathcal{F}}^{\delta}\partial_{\delta}\partial_{\beta}\partial_{\gamma}Q^{\alpha}+{\textstyle\frac{1}{3!}}(-1)^{|\epsilon^{\beta}|+|\epsilon^{\gamma}|+|\epsilon^{\delta}|}\epsilon^{\beta}\epsilon^{\gamma}\epsilon^{\delta}{\mathcal{F}}^{\kappa}\partial_{\kappa}\partial_{\beta}\partial_{\gamma}\partial_{\delta}Q^{\alpha}$
(4.4) $\displaystyle+\,s_{0}(s_{\rm triv}\epsilon^{\alpha})-s_{\rm
triv}\epsilon^{\beta}\epsilon^{\gamma}\partial_{\beta}\partial_{\gamma}Q^{\alpha}-{\textstyle\frac{1}{2}}\epsilon^{\beta}\epsilon^{\gamma}s_{\rm
triv}\phi^{\delta}\partial_{\delta}\partial_{\beta}\partial_{\gamma}Q^{\alpha}\,.$

The square of the BRST transformation on all scalar fields vanishes
identically, which also explains the absence of
$s_{0}^{2}\widetilde{\epsilon}^{\,\alpha}$. Using (4.1) and (4.2) we can
easily see that both $s_{0}^{2}\phi^{\alpha}$ and $s_{0}^{2}\epsilon^{\alpha}$
vanish weakly. Additionally, expression (4.3) is used to find the trivial
transformation of the ghosts from the trivial transformations of the physical
fields by demanding the mutual cancellation of the
$\mathrm{d}\epsilon^{\alpha}$-dependent terms arising from the first two terms
in the second line of (4.3). This results in the general expressions

$\displaystyle s_{\text{triv}}\phi^{\alpha}$ $\displaystyle=$
$\displaystyle-(-1)^{(|Q^{\beta}|+1){\mathcal{F}}^{\alpha}}\frac{1}{3!}H^{\alpha}{}_{\beta\gamma\delta}\epsilon^{\varepsilon}\partial_{\varepsilon}Q^{\beta}\left(3{\mathcal{F}}^{\gamma}Q^{\delta}+{\mathcal{F}}^{\gamma}{\mathcal{F}}^{\delta}\right)\leavevmode\nobreak\
,$ (4.5) $\displaystyle s_{\text{triv}}\epsilon^{\alpha}$ $\displaystyle=$
$\displaystyle(-1)^{(|Q^{\beta}|+1){\mathcal{F}}^{\alpha}}\frac{1}{4}H^{\alpha}{}_{\beta\gamma\delta}\epsilon^{\varepsilon}\partial_{\varepsilon}Q^{\beta}\epsilon^{\varepsilon^{\prime}}\partial_{\varepsilon^{\prime}}Q^{\gamma}{\mathcal{F}}^{\delta}\leavevmode\nobreak\
,$ (4.6)

with only nonvanishing component of $H^{\alpha}{}_{\beta\gamma\delta}$ being
the 4-form $H_{\mu\nu\kappa\lambda}$. Although this is a somewhat redundant
presentation of the already found ones in (3.59) and (3.60), it shows the
appearance of the derivatives of the homological vector field in the trivial
transformations. Furthermore, the explicit form of the non-vanishing squares
on the non-scalar fields is:

$\displaystyle s_{0}^{2}A^{a}$ $\displaystyle=$
$\displaystyle-\eta^{ab}F^{\mu}\bigg{(}\partial_{\mu}\rho_{b}{}^{\nu}\widetilde{\psi}_{\nu}+{\textstyle\frac{1}{2}}(\partial_{\mu}C_{bcd}+{\textstyle\frac{1}{2}}H_{\mu
bcd})\epsilon^{c}\epsilon^{d}\bigg{)}\leavevmode\nobreak\ ,$ $\displaystyle
s_{0}^{2}\psi_{\mu}$ $\displaystyle=$
$\displaystyle-F^{\lambda}\bigg{(}\partial_{\mu}\partial_{\lambda}\rho_{a}{}^{\nu}\widetilde{\psi}_{\nu}\epsilon^{a}+{\textstyle\frac{1}{6}}(\partial_{\mu}\partial_{\lambda}C_{abc}+\partial_{(\mu}H_{\lambda)abc})\epsilon^{a}\epsilon^{b}\epsilon^{c}\bigg{)}\leavevmode\nobreak\
,$ $\displaystyle s_{0}^{2}B_{\mu}$ $\displaystyle=$
$\displaystyle-F^{\lambda}\bigg{(}\partial_{\mu}\partial_{\lambda}\rho_{a}{}^{\nu}(\psi_{\nu}\epsilon^{a}+\widetilde{\psi}_{\nu}A^{a})+{\textstyle\frac{1}{2}}(\partial_{\mu}\partial_{\lambda}C_{abc}+\partial_{(\mu}H_{\lambda)abc})A^{a}\epsilon^{b}\epsilon^{c}\bigg{)}$
(4.7)
$\displaystyle-\,G^{a}\bigg{(}\partial_{\mu}\rho_{a}{}^{\nu}\widetilde{\psi}_{\nu}+{\textstyle\frac{1}{2}}(\partial_{\mu}C_{abc}+{\textstyle\frac{1}{2}}H_{\mu
abc})\epsilon^{b}\epsilon^{c}\bigg{)}-{\textstyle\frac{1}{6}}\partial_{(\mu}H_{\lambda)\sigma
ab}\epsilon^{a}\epsilon^{b}F^{\lambda}F^{\sigma}\,.$

These reflect the openness of the gauge algebra of the model, even more so the
nonlinear openness as obvious from the last term in Eq. (4), where we note
again the symmetrization in the first two indices in the derivative of the
4-form components.

At the end of the day we should have a BV-BRST transformation which is
nilpotent off-shell. The usual procedure invokes antifields such that they
transform to equations of motion. In that way the contributions of the square
$s_{0}^{2}$ will appear in the solution of the classical master equation
together with the prescribed combination of antifields. We shall return to
this in more detail below. What is important to mention here already is that
besides the square of the BRST transformation, higher powers of it which are
also weakly vanishing can (and in the present case will) in principle play the
same role. In other words, one should also compute all necessary $s_{0}^{n}$
for $n>2$, in the precise way we will explain below. In practice it is not
necessary (or possible) to compute all these powers. However, there is a
maximum power $n$ relevant in each case. To determine this power we need to
introduce the antifields, even though we will continue working with the
longitudinal differential (which annihilates them) and not with the Koszul-
Tate one. There is one antifield for each field, ghost and ghost for ghost of
the model. In the present case of twisted Courant sigma models the set of
antifields together with their ghost number and form degree appear in Table 2.

Antifield |  |  |  |  |  |
---|---|---|---|---|---|---
$X^{\dagger}_{\mu}$ | $A^{\dagger}_{a}$ | $B^{\dagger\,\mu}$ | $\epsilon^{\dagger}_{a}$ | $\psi^{\dagger\,\mu}$ | $\widetilde{\psi}^{\dagger\,\mu}$
Ghost degree |  |  |  |  |  |
$-1$ | $-1$ | $-1$ | $-2$ | $-2$ | $-3$
Form degree |  |  |  |  |  |
$3$ | $2$ | $1$ | $3$ | $2$ | $3$
Table 2: The set of antifields of the twisted Courant sigma model.

We recall the main properties of these antifields. For a given field or ghost
$\varphi$ the corresponding antifield $\varphi^{\dagger}$ has ghost number
$\text{gh}(\cdot)$ and form degree $\text{deg}(\cdot)$ such that

$\displaystyle\text{gh}(\varphi)+\text{gh}(\varphi^{\dagger})=-1\,,$ (4.8)
$\displaystyle\text{deg}(\varphi)+\text{deg}(\varphi^{\dagger})=3\,.$ (4.9)

In words, the sum of the form degrees of any field and its antifield must be
equal to the spacetime dimension where the model is defined, here 3, because
they appear as a pair in the sector of the master action with one antifield,
whereas the ghost degrees sum to $-1$ since the BRST formalism introduces the
extended cotangent bundle $T^{\ast}[-1]\mathcal{M}$ with negative degree shift
that accomodates the fields and antifields together. The BV-BRST
transformation131313Our terminology is such that the off-shell nilpotent
transformation is called BV-BRST and the only on-shell nilpotent one is called
just BRST. of each antifield is an equation of motion for its associated
field, in our conventions

$s\varphi^{\dagger}=(-1)^{\rm deg(\varphi^{\dagger})}{\cal
F}_{\varphi}+\dots\,,$ (4.10)

where the ellipses denote further terms that are antifield dependent and
guarantee that the BV-BRST transformation is nilpotent. Recall that the
$s_{0}$ (or $s_{0}^{\scriptscriptstyle\nabla}$) part of $s$ annihilates the
antifields,

$s_{0}\varphi^{\dagger}=0\,,$ (4.11)

and the part of it that is responsible for Eq. (4.10) is the Koszul-Tate
differential [15].

In the present work we will not have to deal with the full BV-BRST
transformation. The above properties of the antifields already dictate which
powers of the BRST transformation before the introduction of antifields are
sufficient to determine the master action. To find this all we need to know is
that the master action must have vanishing ghost degree. Obviously the
classical action $S_{\text{cl}}$ already satisfies this requirement.
Furthermore the generic term $\varphi^{\dagger}s_{0}\varphi$ also satisfies it
for any field and ghost due to (4.8) and the fact that the BRST transformation
has ghost degree $1$. This is the well known term in the master action with
one antifield. Taking this logic one step further, let us consider the square
of the BRST transformation on some field or ghost $\varphi$. This is either
zero or it contains terms proportional to one or more equations of motion,
including products thereof in the case of nonlinear openness. To include these
terms in the action we need to replace equations of motion by corresponding
antifields that transform to them under the Koszul-Tate differential. Then one
must consider higher powers of the BRST transformation and do the same as long
as the result is of vanishing ghost degree. To remain general, assume that we
include in the action a term of the type

$\varphi^{\dagger}\mathcal{O}_{(n)}\varphi\,,$ (4.12)

of ghost degree zero, and with $\mathcal{O}_{(1)}\varphi=s_{0}\varphi$.
Assuming absence of scalar antifields and of terms in the action that would
contain a Hodge star, the lowest form degree of an antifield is 1 and we
conclude that there is a bound on the integer $n$, equivalently on the power
$n-1$:

$n-1\leq\text{deg}(\varphi)\,.$ (4.13)

Thus the total power of the BRST transformation we must consider is field
dependent and the absolute maximum power in a model is equal to the highest
form degree. For instance in the case of the twisted Courant sigma model this
power amounts to $n=3$, whereas for the twisted Poisson sigma model it is not
necessary to go beyond $n=2$.

Based on the above discussion and disregarding scalar fields for which this is
obviously irrelevant, we see that for the 1-forms $A^{a}$ and $\psi_{\mu}$ we
only need to know the square of their BRST transformation, which we have
already found, see (4). These will contribute to the action as

$\displaystyle\varphi^{\dagger}{\mathcal{O}}_{(2)}\varphi=\varphi^{\dagger}\mathfrak{f}\left(s_{0}(\mathcal{O}_{(1)}\varphi)\right)\leavevmode\nobreak\
,$ (4.14)

where we defined the operation of taking any expression $\varphi$ (here of the
form $(s_{0})^{2}\varphi$) that contains an equation of motion and replacing
the equation of motion with the corresponding antifield:

$\mathfrak{f}(\varphi)=\varphi|_{{\cal F}\to(-1)^{{\rm deg}{\cal
F}+1}\varphi^{\dagger}_{\cal F}}\,.$ (4.15)

Specifically,

$\displaystyle A^{\dagger}_{a}{\mathcal{O}}_{(2)}A^{a}$ $\displaystyle=$
$\displaystyle\eta^{ab}A^{\dagger}_{a}B^{\dagger\mu}(-\partial_{\mu}\rho_{b}{}^{\nu}\widetilde{\psi}_{\nu}-{\textstyle\frac{1}{2}}(\partial_{\mu}C_{bcd}+{\textstyle\frac{1}{2}}H_{\mu
bcd})\epsilon^{c}\epsilon^{d})\leavevmode\nobreak\ ,$
$\displaystyle\psi^{\dagger\,\mu}{\mathcal{O}}_{(2)}\psi_{\mu}$
$\displaystyle=$
$\displaystyle\psi^{\dagger\,\mu}B^{\dagger\,\lambda}(-\partial_{\mu}\partial_{\lambda}\rho_{b}{}^{\nu}\widetilde{\psi}_{\nu}\epsilon^{a}-{\textstyle\frac{1}{6}}(\partial_{\mu}\partial_{\lambda}C_{abc}+\partial_{(\mu}H_{\lambda)abc})\epsilon^{a}\epsilon^{b}\epsilon^{c})\leavevmode\nobreak\
.$ (4.16)

Moving on, for the 2-form $B_{\mu}$ there is a subtlety due to the nonlinear
openness of the gauge algebra. Recall that in the commutator of gauge
transformations on the 2-form, Eq. (3.10), we encountered a nonlinear term
with symmetrization in the first two indices of its coefficient. The same
structure appears in the square of the BRST transformation, the last term in
Eq. (4). Since this will have to appear together with a product of two
commuting antifields, we are prompted to replace only the equation of motion
whose index is within the symmetrization. In other words the operator
$\mathfrak{f}$ acts only on one of the two field equations in the product as
specified. This gives the term

$\displaystyle B^{\dagger\,\mu}{\mathcal{O}}_{(2)}B_{\mu}$ $\displaystyle=$
$\displaystyle
B^{\dagger\,\mu}B^{\dagger\lambda}(-\partial_{\mu}\partial_{\lambda}\rho_{a}{}^{\nu}(\psi_{\nu}\epsilon^{a}+\widetilde{\psi}_{\nu}A^{a})-{\textstyle\frac{1}{2}}(\partial_{\mu}\partial_{\lambda}C_{abc}+\partial_{\mu}H_{\lambda
abc})A^{a}\epsilon^{b}\epsilon^{c})$ (4.17)
$\displaystyle-\,\eta^{ad}B^{\dagger\,\mu}A^{\dagger}_{d}(-\partial_{\mu}\rho_{a}{}^{\nu}\widetilde{\psi}_{\nu}-{\textstyle\frac{1}{2}}(\partial_{\mu}C_{abc}+{\textstyle\frac{1}{2}}H_{\mu
abc})\epsilon^{b}\epsilon^{c})$
$\displaystyle-\,{\textstyle\frac{1}{6}}B^{\dagger\,\mu}B^{\dagger\lambda}\partial_{\mu}H_{\lambda\sigma
ab}\epsilon^{a}\epsilon^{b}F^{\sigma}\leavevmode\nobreak\ .$

Furthermore, for the 2-form field $B_{\mu}$ we expect a cubic antifield
contribution of ghost degree zero141414The cubic contributions of higher ghost
degree contribute to potential anomalies controlled by Bianchi identities. We
shall not analyse these issues here.. The cubic contribution is obtained by
replacing the remaining equation of motion in $O_{(2)}\varphi$ with the
corresponding antifield, performing a (third) BRST transformation and again
replacing the newly generated field equation with the corresponding antifield.
The result contains terms of various ghost degrees and to isolate the correct
term we should project to degree 0. What we explained in words takes the form:

$\varphi^{\dagger}{\mathcal{O}}_{(3)}\varphi=\varphi^{\dagger}\mathfrak{f}\left(s_{0}\left(\mathfrak{f}\left({\mathcal{O}}_{(2)}\varphi\right)\right)\right)|_{{}_{\text{gh}=0}}\,.$
(4.18)

This is the right way in which the cube of the BRST transformation
contributes. Specializing to the only field for which this general procedure
is relevant in the present case, the contribution to the master action is

$\displaystyle B^{\dagger\mu}{\mathcal{O}}_{(3)}B_{\mu}$ $\displaystyle=$
$\displaystyle
B^{\dagger\mu}B^{\dagger\lambda}B^{\dagger\sigma}\bigg{(}\partial_{\mu}\partial_{\lambda}\partial_{\sigma}\rho_{a}{}^{\nu}\widetilde{\psi}_{\nu}\epsilon^{a}+{\textstyle\frac{1}{6}}\partial_{\mu}\partial_{\lambda}\partial_{\sigma}C_{abc}\epsilon^{a}\epsilon^{b}\epsilon^{c}$
(4.19)
$\displaystyle\qquad\qquad\quad+\,{\textstyle\frac{1}{6}}\big{(}\partial_{\sigma}\partial_{\mu}H_{\lambda
abc}-{\textstyle\frac{3}{2}}\partial_{\mu}\partial_{\lambda}\rho_{a}{}^{\nu}H_{\nu\sigma
bc}\big{)}\epsilon^{a}\epsilon^{b}\epsilon^{c}\bigg{)}\,.$

The procedure is described in more technical detail in App.C. Then as a
preliminary, calibrating result which is simple to prove we state the
following for the case of vanishing 4-form:

###### Proposition 4.20

The AKSZ-BV master action for the (untwisted) Courant sigma model can be
expressed as

$\displaystyle S_{\text{\tiny{\rm BV,0}}}=S_{\rm
cl}|_{{}_{H=0}}+\sum_{n=1}^{3}\frac{(-1)^{gh\varphi+n}}{\beta_{\dagger}!\beta_{n}}\,\varphi^{\dagger}\cdot\mathcal{O}_{(n)}\varphi|_{{}_{H=0}}\,,$
(4.21)

where $S_{\rm cl}$ is the classical action,
$\varphi=(\phi,\epsilon,\widetilde{\epsilon})$ is the collection of fields,
ghosts and ghost for ghost, and the $\mathcal{O}_{(n)}\varphi$ is defined as
above with $H=0$. The factor $\beta_{\dagger}$ counts the number of antifields
of the same kind in any of the terms in the sum and the symmetry factor
$\beta_{n}$ counts the terms that appear more than once for a fixed $n$.

The proof is a direct comparison151515Up to irrelevant differences in
conventions and notation. of each term in the sum of Eq. (4.21) with the
corresponding terms with 1, 2 and 3 antifields in Ref. [4]. This is then
nothing but an alternative rewriting of the AKSZ theory in 3D, in which the
expansion in the antifields is obtained at face value. This slightly
alternative perspective is useful for two reasons. First, it can be upgraded
relatively easily to a manifestly target space covariant form where the
various coefficients in the master action acquire a geometric interpretation
in terms of the Courant algebroid structures and tensors, namely the Gualtieri
torsion and the basic curvature. The second advantage, in which we focus here,
is that the above form of the master action can be used to account for the
_4-form twisted_ Courant sigma model for which there is no three-dimensional
AKSZ construction.161616Even though one can consider that it can be obtained
as a boundary theory of the AKSZ construction in four dimensions, see e.g. the
approach developed in Ref. [52], it would be difficult to find the right
boundary conditions for this to happen, especially to do this in a systematic
way. This is accounted for by the following:

###### Theorem 4.22

The minimal solution of the classical master equation for the 4-form twisted
Courant sigma model is

$\displaystyle S_{\text{\tiny{\rm BV}}}=S_{\rm
cl}+\sum_{n=1}^{3}\frac{(-1)^{gh\varphi+n}}{\beta_{\dagger}!\beta_{n}}\,\varphi^{\dagger}\cdot\mathcal{O}_{(n)}\varphi\,,$
(4.23)

where $S_{\rm cl}$ is the classical action,
$\varphi=(\phi,\epsilon,\widetilde{\epsilon})$ is the collection of fields,
ghosts and ghost for ghost, and the $\mathcal{O}_{(n)}\varphi$ is defined as
above including the 4-form contributions. The factor $\beta_{\dagger}$ counts
the number of antifields of the same kind in any of the terms in the sum and
the symmetry factor $\beta_{n}$ counts the terms that appear more than once
for a fixed $n$.

We call this procedure for obtaining the master action the “BRST power
finesse” because instead of the standard approach of homological perturbation
theory where the master equation is solved order by order by adding terms to
the master action with increasing number of antifields, here all essential
information is obtained directly from the on-shell closed BRST transformation.
This is of course completely equivalent, it just avoids some technically
challenging steps at the price of computing the higher powers of the BRST
transformation as explained above. In cases where the AKSZ construction is not
directly applicable or when a global geometric interpretation of the
coefficients in the master action is desired, this procedure is technically
advantageous.

The detailed expression for the BV master action is given in App.D. Obviously
it reduces to the AKSZ-BV action when the 4-form vanishes. Moreover, it is
directly checked that for the special case of $M$ being a twisted R-Poisson
manifold of order 3 in the sense of Ref. [34] it reduces to the master action
of the three-dimensional twisted R-Poisson sigma model, which was tediously
determined in Ref. [21], see in particular Section 4 in that paper. The
agreement of the presently more general case to the special case there is
established with the following identifications,

$\rho=\Pi^{\sharp}\,,\quad
C_{\rho}^{\mu\nu}=\partial_{\rho}\Pi^{\mu\nu}\,,\quad
C^{\mu\nu\rho}=R^{\mu\nu\rho}\,,$ (4.24)

where $\Pi^{\sharp}:T^{\ast}M\to TM$ is the map induced by a Poisson structure
$\Pi$ and $R^{\mu\nu\rho}$ are the components of a 3-vector that satisfies

$[\Pi,R]_{\text{SN}}+\langle\Pi^{\otimes 4},H\rangle\,,$ (4.25)

which is the defining condition of a 4-form twisted R-Poisson structure of
order 3, giving rise to a special case of a 4-form twisted Courant algebroid.

The general result may also be written as

$S_{\text{\tiny{\rm BV}}}=S_{\text{\tiny{\rm BV,0}}}+S_{\text{\tiny{\rm
BV,H}}}\leavevmode\nobreak\ ,$ (4.26)

where the AKSZ action is what we denoted above as $S_{\text{\tiny{BV,0}}}$ and
$S_{\text{\tiny{\rm BV,H}}}$ contains only $H$-dependent terms. This is useful
in isolating the new contributions and eventually proving that

$\displaystyle\left(S_{0},S_{3}\right)_{\text{\tiny{\rm
BV}}}+\left(S_{1},S_{2}\right)_{\text{\tiny{\rm BV}}}=0\,,$ (4.27)
$\displaystyle 2(S_{1},S_{3})_{\text{\tiny{\rm
BV}}}+(S_{2},S_{2})_{\text{\tiny{\rm BV}}}=0\,,$ (4.28)

where $S_{i}$ is the sector of $S_{\text{\tiny{BV}}}$ with $i$ antifields and
$(\cdot,\cdot)_{\text{\tiny{\rm BV}}}$ is the odd BV bracket. Using the axioms
of the pre-Courant algebroid in the covariant form (2.72) and (2.73), their
derivatives and the following identity,

$\displaystyle\rho_{[a}{}^{\mu}\nabla_{\mu}\widetilde{\mathcal{S}}_{\nu
bcd]}=-\nabla_{\nu}\rho_{[a}{}^{\mu}\widetilde{\mathcal{S}}_{\mu
bcd]}-3\,\widetilde{\mathcal{S}}_{\nu
e[ab}{\mathcal{T}}^{e}_{cd]}\leavevmode\nobreak\ ,$ (4.29)

we have validated by direct computations that these nontrivial conditions
hold. Together with the remaining, simpler identities, this establishes that
this is a solution to the classical master equation and that the above theorem
is valid.

#### Comment on the quantum master equation.

Before closing this section, let us turn to the question of the solution to
the quantum master equation. Recall that the BV Laplacian is defined as

$\Delta=\sum_{\alpha}(-1)^{\text{gh}(\varphi^{\alpha})}\frac{\delta^{2}}{\delta\varphi^{\alpha}\varphi^{\dagger}_{\alpha}}\,,$
(4.30)

where the sum is taken over all fields and ghosts and the functional
derivative acts from left to right. Then the quantum master equation is

$i\hbar\Delta W=\frac{1}{2}(W,W)_{\text{BV}}\,,$ (4.31)

where $W$ is the quantum action whose loop expansion starts with the classical
master action $S_{\text{\tiny{\rm{BV}}}}$. It was shown in Ref. [53] that the
minimal solution to the classical master equation of the Poisson sigma model
satisfies the quantum master equation too. Note, moreover, that this remains
true for the twisted Poisson sigma model, as can be directly seen via its
master action in Eq. (4.36) of Ref. [19].

Let us check what happens in the Courant or twisted Courant sigma model. There
are six terms in the minimal solution to the classical master action on which
the BV Laplacian does not vanish, namely those with simultaneous appearance of
a field and its antifield:

$S_{h}=\int\big{(}-X^{\dagger}_{\mu}\rho_{a}{}^{\mu}(X)\epsilon^{a}-(A^{\dagger}_{a}A^{b}+\frac{1}{2}\epsilon^{\dagger}_{a}\epsilon^{b})C^{a}_{bc}\epsilon^{c}+(B^{\dagger\mu}B_{\nu}+\psi^{\dagger\mu}\psi_{\nu}+\widetilde{\psi}^{\dagger\mu}\widetilde{\psi}_{\nu})\partial_{\mu}\rho_{a}{}^{\nu}\epsilon^{a}\big{)}\,.$
(4.32)

This is easier to see in the expanded form of Appendix B. The rest of the
terms are in the kernel of the Laplacian. All terms in (4.32) are independent
of the 4-form and therefore they are the same for both the twisted and the
untwisted models. A simple inspection shows that the $X^{\dagger}X$ and
$\widetilde{\psi}^{\dagger}\widetilde{\psi}$ terms cancel out after the action
of the BV Laplacian. Noting moreover that a 2-form and an 1-form have the same
number of components in three dimensions, it turns out that the $B^{\dagger}B$
term cancels out with the $\psi^{\dagger}\psi$ term. What remains are the
$A^{\dagger}A$ and $\epsilon^{\dagger}\epsilon$ terms, which vanish separately
under the action of $\Delta$ because the result is proportional to
$C^{a}_{ab}$. Thus the classical master action we determined for the (twisted)
Courant sigma model is also a solution of the quantum master equation
171717Note that the calculation of the $\Delta S$ includes divergences since
the BV Laplacian includes the trace of the functional derivative. In fact,
$\Delta S_{h}=C\cdot 0$ with an infinite constant $C$ similar to the case of
the Poisson sigma model [53]. We can prove that the proper gauge invariant
regularization exists and the (twisted or not) Courant sigma model satisfies
the quantum master equation under this regularization.. We clarify that by
this we mean the naive quantum master equation in the sense of Ref. [54],
since in the infinite dimensional setting of field theory the odd Laplacian is
not well defined.

##  5 The gauge fixed master action

To complete the picture painted in the previous sections we embark in gauge
fixing the large gauge symmetry of the twisted Courant sigma model. This is a
necessary task if one wants to compute correlation functions, even though we
shall not do this in the present paper—see [2] for computations of
bulk/boundary correlators in the open membrane case. It also serves as an
illustrating example for the general theory of gauge fixing within the BRST
formalism [15, 55] and the AKSZ construction in particular [56].

The first step toward gauge fixing is to further enlarge the minimal set of
fields and ghosts to a non-minimal set. A non-minimal set contains more fields
than it is sufficient to solve the classical master equation. However the
classical master equation does not have a unique solution and indeed the
larger set still generates a solution. These additional fields are usually
called trivial pairs and they contain antighosts181818We remind here that
antighosts are not the same fields as the antifields of the ghosts. and
Lagrange multipliers. We will use the general notation $\xi$ for antighosts
and $\zeta$ for Lagrange multipliers. They also come together with their
corresponding antifields so that $\xi^{\dagger}$ will be antifields for the
antighosts and $\zeta^{\dagger}$ antifields for the Lagrange multipliers.

Let us briefly recall the general properties of these additional fields. For
an ordinary 1-form gauge field such as $A^{a}$, we have a scalar gauge
parameter such as $\epsilon^{a}$ which we replaced by a ghost of ghost number
1 with the same name in the BRST treatment. To each such ghost we assign a
trivial pair $(\xi_{a},\zeta_{a})$. The ghost number of the antighost
$\xi_{a}$ is the opposite of one for its ghost, in the present case
$\text{gh}(\xi_{a})=-1$. We also require that the ghost number of the Lagrange
multiplier is in general

$\text{gh}(\zeta)=\text{gh}(\xi)+1\,.$ (5.1)

The reason for this is that the Lagrange multiplier will eventually pair with
the antifield of the antighost in the non-minimal solution of the classical
master equation and therefore the two must have opposite ghost number. Since
$\text{gh}(\xi^{\dagger})=-1-\text{gh}(\xi)$, this means that we require
$\text{gh}(\zeta)=-\text{gh}(\xi^{\dagger})$ which is nothing but Eq. (5.1).
This means in particular that for a usual gauge field the ghost number of the
Lagrange multipler vanishes. Counting degrees of freedom, an 1-form in $d$
dimensions has $d$ components and we reduce $1$ for the ghost and $1$ for the
antighost to reach the correct number of $d-2$ polarizations for a massless
gauge field.

For 2-form gauge fields the situation is a little more involved due to the
reducibility of the gauge theory. An example is the 2-form191919In this
section we consider the original fields before the redefinition. $B_{\mu}$ in
the twisted Courant sigma model but the discussion is general. In this case we
have a 1-form gauge parameter $\psi_{\mu}$ which we turn into a ghost of the
same name and ghost number 1. To this we assign an 1-form antighost
$\xi^{\mu}$ of ghost number $-1$ and the Lagrange multiplier $\zeta^{\mu}$ of
ghost number $0$. We also have the scalar ghost for ghost
$\widetilde{\psi}_{\mu}$ of ghost number $2$ to which we assign a scalar
antighost $\widetilde{\xi}^{\,\mu}$ of ghost number $-2$ and a Lagrange
multiplier $\widetilde{\zeta}^{\,\mu}$ of ghost number $-1$. This is however
not enough to yield the correct degree of freedom count. There is one
additional field to introduce, what is usually called an extraghost that we
denote as $\bar{\xi}_{\mu}$. This is an auxiliary scalar field of vanishing
ghost number together with the Lagrange multiplier $\bar{\zeta}_{\mu}$ of
ghost number $+1$. To justify this proliferation of fields, recall that a
2-form in $d$ dimensions has $d(d-1)/2$ components from which we subtract $2d$
from the two 1-form (anti)ghosts and we add $3$ at the next level from the
ghost for ghost, antighost for ghost and extraghost. The result is
$(d-2)(d-3)/2$ which is the correct number of polarizations for a massless
2-form. For the field content of the (twisted or not) Courant sigma model we
summarize the trivial pair content in Table 3 and their antifields in Table 4.
Although in the case at hand the situation is not exceedingly complicated, we
also present the rather simple BV triangles for clarity:

${A^{a}}$${B_{\mu}}$${\xi_{a}}$${\epsilon^{a}}$${\xi^{\mu}}$${\psi_{\mu}}$${\bar{\xi}_{\mu}}$${\widetilde{\xi}^{\,\mu}}$${\widetilde{\psi}_{\mu}}$
(5.2)

Antighost/Multiplier |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
$\xi_{a}$ | $\xi^{\mu}$ | $\widetilde{\xi}^{\,\mu}$ | $\bar{\xi}_{\mu}$ | $\zeta_{a}$ | $\zeta^{\mu}$ | $\widetilde{\zeta}^{\,\mu}$ | $\bar{\zeta}_{\mu}$
Ghost degree |  |  |  |  |  |  |  |
$-1$ | $-1$ | $-2$ | $0$ | $0$ | $0$ | $-1$ | $+1$
Form degree |  |  |  |  |  |  |  |
$0$ | $1$ | $0$ | $0$ | $0$ | $1$ | $0$ | $0$
Table 3: Antighosts/extraghost $\xi$ and Lagrange multipliers $\zeta$ for the twisted Courant sigma model. Antifield |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---
$\xi^{\dagger\,a}$ | $\xi^{\dagger}_{\mu}$ | $\widetilde{\xi}^{\,\dagger}_{\mu}$ | $\bar{\xi}_{\dagger}^{\mu}$ | $\zeta^{\dagger\,a}$ | $\zeta^{\dagger}_{\mu}$ | $\widetilde{\zeta}^{\,\dagger}_{\mu}$ | $\bar{\zeta}_{\dagger}^{\mu}$
Ghost degree |  |  |  |  |  |  |  |
$0$ | $0$ | $+1$ | $-1$ | $-1$ | $-1$ | $0$ | $-2$
Form degree |  |  |  |  |  |  |  |
$3$ | $2$ | $3$ | $3$ | $3$ | $2$ | $3$ | $3$
Table 4: Antifields of the antighosts/extraghost $\xi$ and of the Lagrange
multipliers $\zeta$ for the twisted Courant sigma model.

Now we are ready to write down a non-minimal solution to the classical master
equation which is

$S^{+}_{\text{\tiny{\rm BV}}}=S_{\text{\tiny{\rm
BV}}}+\int\xi^{\dagger\,\alpha}\zeta_{\alpha}\,,$ (5.3)

where the collective notation
$\xi^{\dagger\alpha}=(\xi^{\dagger\,a},\xi^{\dagger}_{\mu},\widetilde{\xi}^{\,\dagger}_{\mu},\bar{\xi}_{\dagger}^{\mu})$
and
$\zeta_{\alpha}=(\zeta_{a},\zeta^{\mu},\widetilde{\zeta}^{\,\mu},\bar{\zeta}_{\mu})$
was introduced. This is the action that we would like to gauge fix. To do so
we must introduce a Lagrangian of degree $-1$ usually called the gauge fixing
fermion and denoted by $\Psi$. This is a functional independent of all
antifields and it is used to fix all antifields as

$\varphi^{\dagger}=\frac{\partial\Psi}{\partial\varphi}$ (5.4)

and thus eliminate them from the action as

$S_{\text{\tiny{\rm GF}}}=S^{+}_{\text{\tiny{\rm
BV}}}\bigg{[}\varphi,\varphi^{\dagger}=\frac{\partial\Psi}{\partial\varphi}\bigg{]}\,.$
(5.5)

The gauge fixing fermion is of course not unique and its choice implements
different gauges. Nevertheless there are some basic requirements that have to
be satisfied, stemming from the necessity to have properly defined propagators
in the theory. We refer to [57] and to the review [55] for more details on
this general point. For a $\delta$-function gauge fixing the gauge fixing
fermion is independent of the Lagrange multipliers and an admissible choice
for a first stage reducible system has the following form in our notation:

$\Psi_{\delta}=\int\bigg{(}\xi_{a}f_{1}^{a}(\phi)+\xi^{\mu}f_{2\,\mu}(\phi)+\widetilde{\xi}^{\,\mu}f_{3\,\mu}^{\nu}(\phi)\psi_{\nu}+\xi^{\mu}f_{4\,\mu}^{\nu}(\phi)\bar{\xi}_{\nu}\bigg{)}\,,$
(5.6)

where all four $f_{i}$ are functionals only of the fields and they are
required to have proper rank [57] so that propagators exist. As remarked e.g.
in [58] a standard gauge fixing in the AKSZ construction is to consider a
Riemannian metric that defines a Hodge star operator and a gauge fixing
fermion with fields in the image of the codifferential
$\mathrm{d}^{\dagger}=-\ast\mathrm{d}\,\ast$. This is in line with Eq. (5.6)
and it was already used in [2] to consider the following gauge fixing fermion
for the Courant sigma model

$\Psi_{\delta,\text{\tiny{\rm{CSM}}}}=\int\bigg{(}\mathrm{d}\xi_{a}\ast
A^{a}+\mathrm{d}\xi^{\mu}\ast
B_{\mu}+\mathrm{d}\widetilde{\xi}^{\,\mu}\ast\psi_{\mu}+\mathrm{d}\bar{\xi}_{\mu}\ast\xi^{\mu}\bigg{)}\,,$
(5.7)

which up to total derivatives is of the same form as (5.6) for suitable choice
of $f_{i}$, that is $f_{1}^{a}=\ast\,\mathrm{d}^{\dagger}A^{a}$,
$f_{2\,\mu}=\ast\,\mathrm{d}^{\dagger}B_{\mu}$,
$f_{3\,\mu}^{\nu}=\ast\,\mathrm{d}^{\dagger}\delta_{\mu}^{\nu}$ and
$f_{4\,\mu}^{\nu}=\mathrm{d}^{\dagger}\ast\delta_{\mu}^{\nu}$. We can use the
same one for the twisted Courant sigma model too. This choice fixes the
antifields as

$\displaystyle
X^{\dagger\,\mu}=0\,,\qquad\quad\,\,A^{\dagger}_{a}=\ast\,\mathrm{d}\xi_{a}\,,\qquad\qquad\,\,\,\,\,B^{\dagger\,\mu}=\ast\,\mathrm{d}\xi^{\mu}\,,$
(5.8)
$\displaystyle\epsilon^{\dagger}_{a}=0\,,\qquad\,\,\,\quad\,\,\,\,\,\,\psi^{\dagger\,\mu}=-\ast\,\mathrm{d}\widetilde{\xi}^{\,\mu}\,,\,\qquad\,\,\,\,\widetilde{\psi}^{\,\dagger\,\mu}=0\,,$
(5.9) $\displaystyle\xi^{\dagger\,a}=-\mathrm{d}\ast
A^{a}\,,\quad\xi^{\dagger}_{\mu}=\mathrm{d}\ast
B_{\mu}+\ast\,\mathrm{d}\bar{\xi}_{\mu}\,,\,\,\,\,\,\,\widetilde{\xi}^{\,\dagger}_{\mu}=-\mathrm{d}\ast\psi_{\mu}\,,\qquad\bar{\xi}^{\mu}_{\dagger}=-\mathrm{d}\ast\xi^{\mu}\,,\qquad\quad$
(5.10)

and all antifields of the Lagrange multipliers vanish:
$\zeta^{\dagger\,\alpha}=0$. Then we can write the complete form of the gauge
fixed master action as

$\displaystyle
S_{\text{\tiny{GF}}}=S_{\text{\tiny{kin}}}+S_{\text{\tiny{int,0}}}+S_{\text{\tiny{int,H}}}\,,$
(5.11)

where the kinetic, $H$-independent interaction and $H$-dependent interaction
terms are given in non manifestly covariant form as:

$\displaystyle S_{\text{\tiny{kin}}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\int\bigg{(}-B_{\mu}(\mathrm{d}X^{\mu}-\ast\mathrm{d}\zeta^{\mu})+A^{a}({\textstyle\frac{1}{2}}\eta_{ab}\mathrm{d}A^{a}+\ast\mathrm{d}\zeta_{a})+\psi_{\mu}(\ast\mathrm{d}\widetilde{\zeta}^{\,\mu}-\mathrm{d}\ast\mathrm{d}\xi^{\mu})$
(5.12)
$\displaystyle\qquad\,\,\,\,-\epsilon^{a}\mathrm{d}\ast\mathrm{d}\xi_{a}+\widetilde{\psi}_{\mu}\mathrm{d}\ast\mathrm{d}\widetilde{\xi}^{\,\mu}+\zeta^{\mu}\ast\mathrm{d}\bar{\xi}_{\mu}+\bar{\zeta}_{\mu}\mathrm{d}\ast\xi^{\mu}\bigg{)}\leavevmode\nobreak\
,$ $\displaystyle S_{\text{\tiny{int,0}}}\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\int\bigg{(}\rho_{a}{}^{\mu}B_{\mu}A^{a}+{\textstyle\frac{1}{3!}}C_{abc}\epsilon^{a}\epsilon^{b}\epsilon^{c}+(\eta^{ab}\rho_{b}{}^{\mu}\psi_{\mu}+C^{a}_{bc}A^{b}\epsilon^{c})\ast\mathrm{d}\xi_{a}$
(5.13)
$\displaystyle\qquad\,\,\,\,+(\partial_{\mu}\rho_{a}{}^{\nu}(B_{\nu}\epsilon^{a}-\psi_{\nu}A^{a})+{\textstyle\frac{1}{2}}\partial_{\mu}C_{abc}A^{a}A^{b}\epsilon^{c})\ast\mathrm{d}\xi^{\mu}$
$\displaystyle\qquad\,\,\,\,-(\partial_{\mu}\rho_{a}{}^{\nu}(\widetilde{\psi}_{\nu}A^{a}+\psi_{\nu}\epsilon^{a})+{\textstyle\frac{1}{2}}\partial_{\mu}C_{abc}A^{a}\epsilon^{b}\epsilon^{c})\ast\mathrm{d}\widetilde{\xi}^{\,\mu}$