BRST and Superfield Formalism—A Review

Abstract

This article, which is a review with substantial original material, is meant to offer a comprehensive description of the superfield representations of BRST and anti-BRST algebras and their applications to some field-theoretic topics. After a review of the superfield formalism for gauge theories, we present the same formalism for gerbes and diffeomorphism invariant theories. The application to diffeomorphisms leads, in particular, to a horizontal Riemannian geometry in the superspace. We then illustrate the application to the description of consistent gauge anomalies and Wess–Zumino terms for which the formalism seems to be particularly tailor-made. The next subject covered is the higher spin YM-like theories and their anomalies. Finally, we show that the BRST superfield formalism applies as well to the $N=1$ super-YM theories formulated in the supersymmetric superspace, for the two formalisms go along with each other very well.

1. Introduction

The discovery of the BRST symmetry in gauge field theories, refs. [1,2,3,4], is a fundamental achievement in quantum field theory. This symmetry is not only the building block of the renormalization programs, but it has opened the way to an incredible number of applications. Besides gauge field theories, all theories with a local symmetry are characterized by a BRST symmetry: theories of gerbes, sigma models, topological field theories, string and superstring theories, to name the most important ones. Whenever a classical theory is invariant under local gauge transformations, its quantum counterpart has a BRST-type symmetry that governs its quantum behavior. Two main properties characterize the BRST symmetry. The first is its group theoretical nature: performing two (different) gauge transformations one after the other and then reversing the order of them does not lead to the same result (unless the original symmetry is Abelian), but the two different results are related by a group theoretical rule. This is contained in the nilpotency of the BRST transformations.

The second important property of the BRST transformations is nilpotency itself. It is inherited, via the Faddeev–Popov quantization procedure, from the anticommuting nature of the ghost and anti-ghost fields. This implies that, applying twice the same transformation, we obtain 0. The two properties together give rise to the Wess–Zumino consistency conditions, a fundamental tool in the study of anomalies. It must be noted that while the first property is classical, the second is entirely quantum. In other words, the BRST symmetry is a quantum property.

It was evident from the very beginning that the roots of BRST symmetry are geometrical. The relevant geometry was linked at the beginning to the geometry of the principal fiber bundles [5,6,7,8] (see also [9]). Although it is deeply rooted in it, the BRST symmetry is rather connected to the geometry of infinite dimensional bundles and groups, in particular, the Lie group of gauge transformations [10,11]. One may wonder where this infinite dimensional geometry is stored in a perturbative quantum gauge theory. The answer is, in the anticommutativity of the ghosts and in the nilpotency of the BRST transformations themselves, together with their group theoretical nature. There is a unique way to synthesize these quantum properties, and this is the superfield formalism. The BRST symmetry calls for the introduction of the superfield formulation of quantum field theories. One might even dare say that the superfield formalism is the genuine language of a quantum gauge theory. This is the subject of the present article, which is both a review of old results and a collection of new ones, with the aim of highlighting the flexibility of the superfield approach to BRST symmetry (it is natural to extend it to include also the anti-BRST symmetry [12,13,14]). Here, the main focus is on the algebraic aspects and on the ample realm of applications, leaving the more physical aspects (functional integral and renormalization) for another occasion. We will meet general features—we can call them universal—which appear in any application and for any symmetry. One is the so-called horizontality conditions, i.e., the vanishing of the components along the anticommuting directions, which certain quantities must satisfy. Another is the so-called Curci–Ferrari conditions [12], which always appear when both (non-Abelian) BRST and anti-BRST symmetries are present.

Before passing to a description of how the present review is organized, let us comment on the status of anti-BRST. It is an algebraic structure that comes up naturally as a companion to the BRST one, but it is not necessarily a symmetry of any gauge-fixed action. It holds, for instance, for linear gauge fixing, and some implications have been studied to some extent in [14] and also in [15,16,17]. However, it is fair to say that no fundamental role for this symmetry has been uncovered so far, although it is also fair to say that the research in this field has never overcome a preliminary stage1. In this review, we consider BRST and anti-BRST together in the superfield formalism but whenever it is more convenient and expedient to use only the BRST symmetry, we focus only on it.

We start in Section 2 with a review of the well-known superfield formulation of BRST and anti-BRST of non-Abelian gauge theories, which is obtained by enlarging the spacetime with two anticommuting coordinates, $\vartheta$ and $\overline{\vartheta }$. Section 3 is devoted to gerbe theories, which are close to ordinary gauge theories. After a short introduction, we show that it is simple and natural to reproduce the BRST and anti-BRST symmetries with the superfield formalism. As always, when both BRST and anti-BRST are involved, we come across specific CF conditions. The next two sections are devoted to diffeomorphisms. Diffeomorphisms are a different kind of local transformation; therefore, it is interesting to see, first of all, if the superfield formalism works. In fact, in Section 4, we find horizontality and CF conditions for which BRST and anti-BRST transformations are reproduced by the superfield formalism. We show, however, that the super-metric, i.e., the metric with components in the anticommuting directions, is not invertible. So a super-Riemannian geometry is not possible in the superspace but, in exchange, we can define a horizontal super-geometry, with Riemann and Ricci tensors defined on the full superspace. In Section 5 we deal with frame superfields and define fermions in superspace. In summary, there are no obstructions to formulate quantum gravitational theories in the superspace.

The second part of the paper concerns applications of the superfield method to some practical problems, notably to anomalies. Consistent anomalies are a perfect playground for the superfield method, as we show in Section 6. We show that not only are all the formulas concerning anomalies in any even dimension easily reproduced, but in fact, the superfield formalism seems to be tailor-made for them. A particularly sleek result is the way one can extract Wess–Zumino terms from it. In Section 7, we apply the superfield formalism to HS-YM-like theories. After a rather detailed introduction to such novel models, we show that the superfield method fits perfectly well and is instrumental in deriving the form of anomalies, which would otherwise be of limited access. Section 8 is devoted to the extension of the superfield method in still another direction—that of supersymmetry. We show, as an example, that the supersymmetric superspace formulation of $N=1$ SYM theory in 4D can be easily enlarged by extending the superspace with the addition of $\vartheta ,\overline{\vartheta }$, while respecting the supersymmetric geometry (constraints). In Section 9, we make some concluding remarks and comments on some salient features of our present work.

The appendices contain auxiliary materials, except the first (Appendix A), which might seem a bit off topic with respect to the rest of the paper. We deem it useful to report in order to clarify the issue of the classical geometric description of the BRST symmetry. As mentioned above, this description is possible. However, one must formulate this problem in the framework of the geometry of the infinite dimensional groups of gauge transformations (which are, in turn, rooted in the geometry of principal fiber bundles). The appropriate mathematical tool is the evaluation map. One can easily see how the superfield method formulation parallels the geometrical description.

Finally, let us add that this review covers only a part of the applications of the superfield approach that have appeared in the literature. We must mention [21,22,23,24,25,26,27,28,29,30,31,32,33,34,35] for further extensions of the method and additional topics not presented here. A missing subject in this paper, as well as, to the best of our knowledge, in the present literature, is the exploration of the possibility to extend the superfield method to the Batalin–Vilkovisky approach to field theories with local symmetries.

Notations and Conventions. The superspace is represented by super-coordinates $X^M=\{x^{\mu },\vartheta ,\overline{\vartheta }\}$, where $x^{\mu }$$(\mu =0,1,\dots ,d-1)$ are ordinary commuting coordinates, while $\vartheta$ and $\overline{\vartheta }$ are anticommuting: ${\vartheta }^2={\overline{\vartheta }}^2=\vartheta \overline{\vartheta }+\overline{\vartheta }\vartheta =0$, but commute with $x^{\mu }$. We make use of a generalized differential geometric notation: the exterior differential $d=\frac{\partial }{\partial x^{\mu }}d{x}^{\mu }$ is generalized to $\tilde{d}=d+\frac{\partial }{\partial \vartheta }d\vartheta +\frac{\partial }{\partial \overline{\vartheta }}d\overline{\vartheta }$. Correspondingly, mimicking the ordinary differential geometry, we introduce super-forms; for instance, $\tilde{\omega }={\omega }_{\mu }\left(x\right)d{x}^{\mu }+{\omega }_{\vartheta }\left(x\right)d\vartheta +{\omega }_{\overline{\vartheta }}\left(x\right)d\overline{\vartheta }$, where ${\omega }_{\mu }$ are ordinary commuting intrinsic components, while ${\omega }_{\vartheta },{\omega }_{\overline{\vartheta }}$ anticommute with each other and commute with ${\omega }_{\mu }$. In the same tune, we introduce also super-tensors, such as the super-metric; see Section 4.4. As far as commutativity properties (gradings) are concerned, the intrinsic components of forms and tensors on one side and the symbols $d^{\mu },d\vartheta ,d\overline{\vartheta }$, on the other constitute separate, mutually commuting sets. When $\tilde{d}$ acts on a super-function $\tilde{F}\left(X\right)$, it is understood that the derivatives act on it from the left to form the components of a 1-super-form:

$$\begin{array}{c}\hfill \tilde{d}\tilde{F}\left(X\right)=\frac{\partial }{\partial x^{\mu }}\tilde{F}\left(X\right)d{x}^{\mu }+\frac{\partial }{\partial \vartheta }\tilde{F}\left(X\right)d\vartheta +\frac{\partial }{\partial \overline{\vartheta }}\tilde{F}\left(X\right)d\overline{\vartheta }\end{array}$$

(1)

When it acts on a 1-super-form, it is understood that the derivatives act on the intrinsic components from the left, and the accompanying symbol $d{x}^{\mu },d\vartheta ,d\overline{\vartheta }$ becomes juxtaposed to the analogous symbols of the super-form from the left to form the combinations $d{x}^{\mu }\wedge d{x}^{\nu },d{x}^{\mu }\wedge d\vartheta ,d\vartheta \wedge d\vartheta ,d\vartheta \wedge d\overline{\vartheta },\dots$, with the usual rule for the spacetime symbols, and $d{x}^{\mu }\wedge d\vartheta =-d\vartheta \wedge d{x}^{\mu },d{x}^{\mu }\wedge d\overline{\vartheta }=-d\overline{\vartheta }\wedge d{x}^{\mu }$, but $d\vartheta \wedge d\overline{\vartheta }=d\overline{\vartheta }\wedge d\vartheta$, and $d\vartheta \wedge d\vartheta$ and $d\overline{\vartheta }\wedge d\overline{\vartheta }$ are non-vanishing symbols. In a similar way, one proceeds with higher degree super-forms. More specific notations will be introduced later when necessary.

2. The Superfield Formalism in Gauge Field Theories

The superfield formulation of the BRST symmetry in gauge field theories was proposed in [15]; for an earlier version, see [36,37]. Here, we limit ourselves to a summary. Let us consider a generic gauge theory in $\mathsf{d}$ dimensional Minkowski spacetime $\mathsf{M}$, with connection $A_{\mu }^a{T}^a$ ($\mu =0,1,\dots ,\mathsf{d}-1$), valued in a Lie algebra $\mathfrak{g}$ with anti-hermitean generators $T^a$, such that $[T^a,T^b]=f^{abc}{T}^c$. In the following, it is convenient to use the more compact form notation and represent the connection as a one-form $A=A_{\mu }^a{T}^{a}d{x}^{\mu }$. The curvature and gauge transformation are as follows:

$$\begin{array}{c}\hfill F=dA+\frac{1}{2}[A,A]\mathrm{and}{\delta }_{\lambda }A=d\lambda +[A,\lambda ],\end{array}$$

(2)

with $\lambda \left(x\right)={\lambda }^a\left(x\right)T^a$ and $d=d{x}^{\mu }\frac{\partial }{\partial x^{\mu }}$. The infinite dimensional Lie algebra of gauge transformations and its cohomology can be formulated in a simpler and more effective way if we promote the gauge parameter $\lambda$ to an anticommuting ghost field $c=c^a{T}^a$ and define the BRST transform as follows2:

$$\begin{array}{c}\hfill \mathfrak{s}A\equiv dc+[A,c],\mathfrak{s}c=-\frac{1}{2}[c,c].\end{array}$$

(3)

As a consequence of this, we have the following:

$$\begin{array}{c}\hfill \mathcal{F}\equiv (d-\mathfrak{s})(A+c)+\frac{1}{2}[A+c,A+c]=F,\end{array}$$

(4)

which is sometime referred to as the Russian formula [7,15,38,39]. Equation (A10) is true, provided we assume the following:

$$\begin{array}{c}\hfill [A,c]=[c,A].\end{array}$$

(5)

i.e., if we assume that c behaves like a one-form in the commutator with ordinary forms and with itself. It can be, in fact, related to the Maurer–Cartan form in $\mathsf{G}$. This explains its anticommutativity.

A very simple way to reproduce the above formulas and properties is by enlarging the space to a superspace with coordinates $(x^{\mu },\vartheta )$, where $\vartheta$ is anticommuting, and promoting the connection A to a one-form superconnection $\tilde{A}=\varphi (x,\vartheta )+{\varphi }_{\vartheta }(x,\vartheta )d\vartheta$ with the following expansions:

$$\begin{array}{c}\hfill \varphi (x,\vartheta )=\mathsf{A}\left(x\right)+\vartheta \Gamma \left(x\right),{\varphi }_{\vartheta }(x,\vartheta )=\mathsf{c}\left(x\right)+\vartheta G\left(x\right),\end{array}$$

(6)

and two-form supercurvature

$$\begin{array}{c}\hfill \tilde{F}=\tilde{d}\tilde{A}+\frac{1}{2}[\tilde{A},\tilde{A}],\tilde{F}=\Phi (x,\vartheta )+{\Phi }_{\vartheta }(x,\vartheta )d\vartheta +{\Phi }_{\vartheta \vartheta }(x,\vartheta )d\vartheta \wedge d\vartheta ,\end{array}$$

(7)

with $\Phi (x,\vartheta )=\mathsf{F}\left(x\right)+\vartheta \Lambda \left(x\right)$ and $\tilde{d}=d+\frac{\partial }{\partial \vartheta }d\vartheta$. Notice that since ${\vartheta }^2=0$, $d\vartheta \wedge d\vartheta \ne 0$, while $d{x}^{\mu }\wedge d\vartheta =-d\vartheta \wedge d{x}^{\mu }$. Then, we impose the ‘horizontality’ condition:

$$\begin{array}{c}\hfill \tilde{F}=\Phi (x,\vartheta ),\mathrm{i}.\mathrm{e}.,{\Phi }_{\vartheta }(x,\vartheta )=0={\Phi }_{\vartheta \vartheta }(x,\vartheta ).\end{array}$$

(8)

The last two conditions imply the following:

$$\begin{array}{c}\hfill \Gamma \left(x\right)=dc\left(x\right)+[\mathsf{A}\left(x\right),c\left(x\right)],G\left(x\right)=-\frac{1}{2}[\mathsf{c}\left(x\right),\mathsf{c}\left(x\right)].\end{array}$$

Moreover, $\Lambda \left(x\right)=\left[\mathsf{F}\right(x),\mathsf{c}(x\left)\right]$.

This means that we can identify $\mathsf{c}\left(x\right)\equiv c\left(x\right)$, $\mathsf{A}\equiv A$, $\mathsf{F}\equiv F$, and the $\vartheta$ translation with the BRST transformation $\mathfrak{s}$, i.e., $\mathfrak{s}\equiv \frac{\partial }{\partial \vartheta }$. In this way all the previous transformations, including Equation (5)—which, at first sight, is strange looking—are naturally explained. It is also possible to push further the use of the superfield formalism by noting that, after imposing the horizontality condition, we have the following:

$$\begin{array}{c}\hfill \tilde{A}=e^{-\vartheta c}A{e}^{\vartheta c}+e^{-\vartheta c}\tilde{d}{e}^{\vartheta c},\tilde{F}=e^{-\vartheta c}F{e}^{\vartheta c}.\end{array}$$

(9)

A comment is in order concerning the horizontality condition (HC). This condition is suggested by the analogy with the principal fiber bundle geometry. In the total space of a principal fiber bundle, one can define horizontal (or basic) forms. These are forms with no components in the vertical direction: for instance, given a connection, its curvature is horizontal. In our superfield approach, the $\vartheta$ coordinate mimics the vertical direction, as the curvature $\tilde{F}$ does not have components in that direction. This horizontality principle can be extended also to other quantities, for instance, to covariant derivatives of matter fields and, in general, to all quantities that are invariant under local gauge transformations.

2.1. Extension to Anti-BRST Transformations

The superfield representation of the BRST symmetry with one single anticommuting variable is, in general, not sufficient for ordinary Yang–Mills theories because gauge fixing requires, in general, other fields besides $A_{\mu }$ and c. For instance, in the Lorenz gauge, the Lagrangian density takes the following form:

$$\begin{array}{c}\hfill {\mathcal{L}}_{YM}=-\mathrm{t}r\left(\frac{1}{4g^2}{F}_{\mu \nu }{F}^{\mu \nu }+A_{\mu }{\partial }^{\mu }B-{\partial }^{\mu }\overline{c}{D}_{\mu }c+\frac{\alpha }{2}{B}^2\right),\end{array}$$

(10)

where two new fields are introduced, the antighost field $\overline{c}\left(x\right)$ and the Nakanishi–Lautrup field $B\left(x\right)$. It is necessary to enlarge the algebra (3) as follows:

$$\begin{array}{c}\hfill \mathfrak{s}\overline{c}=B,\mathfrak{s}B=0,\end{array}$$

(11)

in order to obtain a symmetry of (10). At this point, ${\mathcal{L}}_{YM}$ is invariant under a larger symmetry, whose transformations, besides (3) and (11), are the anti-BRST ones:

$$\begin{array}{c}\hfill \overline{\mathfrak{s}}A=d\overline{c}+[A,\overline{c}],\overline{\mathfrak{s}}\overline{c}=-\frac{1}{2}[\overline{c},\overline{c}],\overline{\mathfrak{s}}c=\overline{B},\overline{\mathfrak{s}}\overline{B}=0,\end{array}$$

(12)

provided the following:

$$\begin{array}{c}\hfill B+\overline{B}+[c,\overline{c}]=0.\end{array}$$

(13)

This is the Curci–Ferrari condition, ref. [12].

The BRST and anti-BRST transformation are nilpotent and anticommute:

$$\begin{array}{c}\hfill {\mathfrak{s}}^2=0,{\overline{\mathfrak{s}}}^2=0,\mathfrak{s}\overline{\mathfrak{s}}+\overline{\mathfrak{s}}\mathfrak{s}=0.\end{array}$$

(14)

The superfield formalism applies well to this enlarged symmetry, provided we introduce another anticommuting coordinate, $\overline{\vartheta }$: ${\overline{\vartheta }}^2=0,\vartheta \overline{\vartheta }+\overline{\vartheta }\vartheta =0$. Here, we do not repeat the full derivation as in the previous case but simply introduce the supergauge transformation [15,40,41,42,43]:

$$\begin{array}{c}\hfill U(x,\vartheta ,\overline{\vartheta })=exp[\vartheta \overline{c}\left(x\right)+\overline{\vartheta }c\left(x\right)+\vartheta \overline{\vartheta }(B\left(x\right)+[c\left(x\right),\overline{c}\left(x\right)])],\end{array}$$

(15)

and generate the following superconnection:

$$\begin{array}{c}\hfill \tilde{A}(x,\vartheta ,\overline{\vartheta })=U{(x,\vartheta ,\overline{\vartheta })}^{†}\left(\tilde{d}+A\left(x\right)\right)U(x,\vartheta ,\overline{\vartheta }),\end{array}$$

(16)

where $\tilde{d}=d+d\vartheta \frac{\partial }{\partial \vartheta }+d\overline{\vartheta }\frac{\partial }{\partial \overline{\vartheta }}$ and the hermitean operation is defined as follows:

$$\begin{array}{c}\hfill {\vartheta }^{†}=\vartheta ,{\overline{\vartheta }}^{†}=-\overline{\vartheta },{\left(c^a\right)}^{†}=c^a,{\left({\overline{c}}^a\right)}^{†}=-{\overline{c}}^a,\end{array}$$

while the $B^a\left(x\right),{\overline{B}}^a\left(x\right)$ are real. Then, the superconnection is the following:

$$\begin{array}{c}\hfill \tilde{A}(x,\vartheta ,\overline{\vartheta })=\Phi (x,\vartheta ,\overline{\vartheta })+\eta (x,\vartheta ,\overline{\vartheta })d\overline{\vartheta }+\overline{\eta }(x,\vartheta ,\overline{\vartheta })d\vartheta .\end{array}$$

(17)

The one-form $\Phi$ is the following:

$$\begin{array}{c}\hfill \Phi (x,\vartheta ,\overline{\vartheta })=A\left(x\right)+\vartheta D\overline{c}\left(x\right)+\overline{\vartheta }Dc\left(x\right)+\vartheta \overline{\vartheta }(DB\left(x\right)+[Dc\left(x\right),\overline{c}\left(x\right)]),\end{array}$$

(18)

where D denotes the covariant differential: $Dc=dc+[A,c]$, etc., and the anticommuting functions $\eta ,\overline{\eta }$ are the following:

$$\begin{array}{ccc}\hfill \eta (x,\vartheta ,\overline{\vartheta })& =& c\left(x\right)+\vartheta \overline{B}\left(x\right)-\frac{1}{2}\overline{\vartheta }[c\left(x\right),c\left(x\right)]+\vartheta \overline{\vartheta }[\overline{B}\left(x\right),c\left(x\right)],\hfill \end{array}$$

(19)

$$\begin{array}{ccc}\hfill \overline{\eta }(x,\vartheta ,\overline{\vartheta })& =& \overline{c}\left(x\right)-\frac{1}{2}\vartheta [\overline{c}\left(x\right),\overline{c}\left(x\right)]+\overline{\vartheta }B\left(x\right)+\vartheta \overline{\vartheta }[\overline{c}\left(x\right),B\left(x\right)],\hfill \end{array}$$

(20)

together with the condition (13). One can verify that the supercurvature $\tilde{F}$ satisfies the following horizontality condition:

$$\begin{array}{c}\hfill \tilde{F}(x,\vartheta ,\overline{\vartheta })=d\Phi (x,\vartheta ,\overline{\vartheta })+\frac{1}{2}[\Phi (x,\vartheta ,\overline{\vartheta }),\Phi (x,\vartheta ,\overline{\vartheta })].\end{array}$$

(21)

The BRST transformation correspond to $\overline{\vartheta }$ translations and the anti-BRST to $\vartheta$ ones:

$$\begin{array}{c}\hfill \mathfrak{s}={\left.\frac{\partial }{\partial \overline{\vartheta }}\right|}_{\vartheta =0},\overline{\mathfrak{s}}={\left.\frac{\partial }{\partial \vartheta }\right|}_{\overline{\vartheta }=0}.\end{array}$$

(22)

At the end of this short review, it is important to highlight an important fact. As anticipated, above the Lagrangian density, (10) is invariant under both the BRST and anti-BRST transformations—(3), (11) and (12)—provided that (13) is satisfied. However, while the Lagrangian density contains a specific gauge fixing, the BRST and anti-BRST algebras (when they hold) are independent of any gauge-fixing condition. We can change the gauge fixing, but the BRST and anti-BRST algebras (when they are present), as well as their superfield representation, are always the same. These algebras can be considered the quantum versions of the original classical gauge algebra. A classical geometrical approach based on fiber bundle geometry was originally proposed in [5,8]. Subsequently, the nature of the BRST transformations was clarified in [10,11]. In fact, it is possible to uncover the BRST algebra in the geometry of principal fiber bundles, particularly in terms of the evaluation map as shown in Appendix A. However, while classical geometry is certainly the base of classical gauge theories, it becomes very cumbersome and actually intractable for perturbative quantum gauge theories. On the other hand, in dealing with the latter, anticommuting ghost and antighost fields and (graded) BRST algebra seem to be the natural tools. Therefore, as noted previously, one may wonder whether the natural language for a quantum gauge field theory is, in fact, the superfield formalism. We leave this idea for future developments.

Here ends our short introduction of the superfield formalism in gauge field theories, which was historically the first application. Later on, we shall see a few of its applications. Now, we would like to explore the possibility to apply this formalism to other local symmetries. The first example, and probably the closest to the one presented in this section, is a theory of gerbes. A gerbe is a mathematical construct, which, in a sense, generalizes the idea of gauge theory. From the field theory point of view, the main difference with the latter is that it is not based on a single connection but, besides one-forms, it contains also other forms. Here, we consider the simplest case, an Abelian 1-gerbe; see [44,45].

3. 1-Gerbes

Let us recall a few basic definitions. A 1-gerbe [46,47,48,49,50,51,52,53] is a mathematical object that can be described with a triple $(B,A,f)$, formed by the 2-form B, 1-form A and 0-form f, respectively. These are related in the following way. Given a covering $\left\{U_i\right\}$ of the manifold $\mathsf{M}$, we associate to each $U_i$ a 2-form $B_i$. On a double intersection $U_i\cap U_j$, we have $B_i-B_j=d{A}_{ij}$. On the triple intersections $U_i\cap U_j\cap U_k$, we must have $A_{ij}+A_{jk}+A_{ki}=d{f}_{ijk}$ ($B_i$ denotes B in $U_i$, $A_{ij}$ denotes A in $U_i\cap U_j$, etc.). Finally, on the quadruple intersections $U_i\cap U_j\cap U_k\cap U_l$, the following integral cocycle condition must be satisfied by f:

$$\begin{array}{c}\hfill f_{ijl}-f_{ijk}+f_{jkl}-f_{ikl}=2\pi n,n=0,1,2,3\cdots \cdots \cdots \end{array}$$

(23)

This integrality condition does not concern us in our Lagrangian formulation but it has to be imposed as an external condition.

Two triples, represented by $(B,A,f)$ and $(B^{\prime },A^{\prime },f^{\prime })$, respectively, are said to be gauge equivalent if they satisfy the following relations:

$$\begin{array}{ccc}& & B_i^{\prime }=B_i+d{C}_i\mathrm{on}{U}_i,\hfill \end{array}$$

(24)

$$\begin{array}{ccc}& & A_{ij}^{\prime }=A_{ij}+C_i-C_j+d{\lambda }_{ij}\mathrm{on}{U}_i\cap U_j,\hfill \end{array}$$

(25)

$$\begin{array}{ccc}& & f_{ijk}^{\prime }=f_{ijk}+{\lambda }_{ij}+{\lambda }_{ki}+{\lambda }_{jk}\mathrm{on}{U}_i\cap U_j\cap U_k,\hfill \end{array}$$

(26)

for the 1-form C and the 0-form $\lambda$.

Let us now define the BRST and anti-BRST transformations corresponding to these geometrical transformations. It should be recalled that, while the above geometric transformations are defined on (multiple) neighborhood overlaps, the BRST and anti–BRST transformations, in quantum field theory, are defined on a single local coordinate patch. These (local, field-dependent) transformations are the means for QFT to record the underlying geometry.

The appropriate BRST and anti-BRST transformations are as follows:

$$\begin{array}{c}\hfill \begin{array}{ccc}\mathfrak{s}B=dC,& \mathfrak{s}A=C+d\lambda ,& \mathfrak{s}f=\lambda +\mu ,\\ \mathfrak{s}C=-dh,& \mathfrak{s}\lambda =h,& \mathfrak{s}\mu =-h,\\ \mathfrak{s}\overline{C}=-K,& \mathfrak{s}\overline{K}=d\rho ,& \mathfrak{s}\overline{\mu }=-g,\\ \mathfrak{s}\overline{\beta }=-\overline{\rho },& \mathfrak{s}\overline{\lambda }=g,& \mathfrak{s}\overline{g}=\rho ,\end{array}\end{array}$$

(27)

together with $\mathfrak{s}[\rho ,\overline{\rho },g,K_{\mu },\beta ]=0$, and

$$\begin{array}{ccc}& & \overline{\mathfrak{s}}B=d\overline{C},\overline{\mathfrak{s}}A=\overline{C}+d\overline{\lambda },\overline{\mathfrak{s}}f=\overline{\lambda }+\overline{\mu },\hfill \\ & & \overline{\mathfrak{s}}\overline{C}=+d\overline{h},\overline{\mathfrak{s}}\overline{\lambda }=-\overline{h},\overline{\mathfrak{s}}\overline{\mu }=-\overline{h},\hfill \\ & & \overline{\mathfrak{s}}C=+\overline{K},\overline{\mathfrak{s}}K=-d\overline{\rho },\overline{\mathfrak{s}}\overline{\mu }=\overline{g},\hfill \\ & & \overline{\mathfrak{s}}\beta =+\rho ,\overline{\mathfrak{s}}\lambda =-\overline{g},\overline{\mathfrak{s}}g=-\overline{\rho },\hfill \end{array}$$

(28)

while $\overline{\mathfrak{s}}[\overline{\beta },\overline{g},\overline{K},\mu ,\rho ,\overline{\rho }]=0$.

In these formulas, $C,\overline{C}$ are anticommuting 1-forms, and $K,\overline{K}$ are commuting 1-forms. The remaining fields are scalars, which are commuting if denoted by Latin letters and anticommuting if denoted by Greek letters.

It can be easily verified that ${(\mathfrak{s}+\overline{\mathfrak{s}})}^2=0$ if the following constraint is satisfied:

$$\overline{K}-K=d\overline{g}-dg.$$

(29)

This condition is both BRST and anti-BRST invariant. It is the analogue of the Curci–Ferrari condition in non-Abelian 1-form gauge theories, and we refer to it with the same name.

Before we proceed to the superfield method, we would like to note that the above realization of the BRST and anti-BRST algebra is not the only possibility. In general, it may be possible to augment it by the addition of a sub-algebra of elements that are all in the kernel of both s and $\overline{s}$, or, if it contains such a sub-algebra, the latter could be moded out. For instance, in Equations (27) and (28), $\rho$ and $\overline{\rho }$ form an example of this type of subalgebra. It is easy to see that $\rho$ and $\overline{\rho }$ can be consistently set equal to 0.

The Superfield Approach to Gerbes

We introduce superfields, whose lowest components are $B,A$ and f.

$$\begin{array}{ccc}\hfill \tilde{\mathcal{B}}={\tilde{\mathcal{B}}}_{MN}\left(X\right)d{X}^M\wedge d{X}^N& =& {\mathcal{B}}_{\mu \nu }\left(X\right)d{x}^{\mu }\wedge d{x}^{\nu }+{\mathcal{B}}_{\mu \vartheta }\left(X\right)d{x}^{\mu }\wedge d\vartheta +{\mathcal{B}}_{\mu \overline{\vartheta }}\left(X\right)d{x}^{\mu }\wedge d\overline{\vartheta },\hfill \\ & & +{\mathcal{B}}_{\vartheta \vartheta }\left(X\right)d\vartheta \wedge d\vartheta +{\mathcal{B}}_{\overline{\vartheta }\overline{\vartheta }}\left(X\right)d\overline{\vartheta }\wedge d\overline{\vartheta }+{\mathcal{B}}_{\vartheta \overline{\vartheta }}\left(X\right)d\vartheta \wedge \overline{\vartheta },\hfill \end{array}$$

(30)

$$\begin{array}{ccc}\hfill \tilde{\mathcal{A}}={\tilde{\mathcal{A}}}_M\left(X\right)d{X}^M& =& {\mathcal{A}}_{\mu }\left(X\right)d{x}^{\mu }+{\mathcal{A}}_{\vartheta }\left(X\right)d\vartheta +{\mathcal{A}}_{\overline{\vartheta }}\left(X\right)d\overline{\vartheta },\hfill \end{array}$$

(31)

$$\begin{array}{ccc}\hfill \tilde{\mathsf{f}}\left(X\right)& =& f\left(x\right)+\vartheta \overline{\varphi }\left(x\right)+\overline{\vartheta }\varphi \left(x\right)+\vartheta \overline{\vartheta }F\left(x\right).\hfill \end{array}$$

(32)

where X denotes the superspace point and $X^M=(x^{\mu },\vartheta ,\overline{\vartheta })$, the superspace coordinates. All the intrinsic components are to be expanded like (32). Then, we impose the horizontality conditions. There are two, which are as follows:

$$\begin{array}{ccc}\hfill \tilde{d}\tilde{\mathcal{B}}& =& d\mathcal{B},\mathcal{B}={\mathcal{B}}_{\mu \nu }\left(X\right)d{x}^{\mu }\wedge d{x}^{\nu },\hfill \end{array}$$

(33)

$$\begin{array}{ccc}\hfill \tilde{\mathcal{B}}-\tilde{d}\tilde{\mathcal{A}}& =& \mathcal{B}-d\mathcal{A},\mathcal{A}={\mathcal{A}}_{\mu }\left(X\right)d{x}^{\mu }.\hfill \end{array}$$

(34)

The first is suggested by the invariance of $H=dB$ under $B\to B+d\Lambda$, where $\Lambda$ is a 1-form, and the second by the invariance of $B-dA$ due to the transformations $B\to B+d\Sigma ,A\to A+\Sigma$, where $\Sigma$ is also a 1-form.

Using the second, we can eliminate many components of $\tilde{\mathcal{B}}$ in favor of the components of $\tilde{\mathcal{A}}$:

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{\mu }\left(X\right)& =& A_{\mu }\left(x\right)+\vartheta {\overline{\alpha }}_{\mu }\left(x\right)+\overline{\vartheta }{\alpha }_{\mu }\left(x\right)+\vartheta \overline{\vartheta }{A}_{\mu }\left(x\right),\hfill \end{array}$$

(35)

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{\vartheta }\left(X\right)& =& \gamma \left(x\right)+\vartheta \overline{e}\left(x\right)+\overline{\vartheta }e\left(x\right)+\vartheta \overline{\vartheta }\Gamma \left(x\right),\hfill \end{array}$$

(36)

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{\overline{\vartheta }}\left(X\right)& =& \overline{\gamma }\left(x\right)+\vartheta \overline{a}\left(x\right)+\overline{\vartheta }a\left(x\right)+\vartheta \overline{\vartheta }\overline{\Gamma }\left(x\right).\hfill \end{array}$$

(37)

Imposing (34), $\tilde{\mathcal{B}}$ takes the following form:

$$\begin{array}{ccc}\hfill {\mathcal{B}}_{\mu \nu }\left(X\right)& =& B_{\mu \nu }\left(x\right)+\vartheta {\overline{\beta }}_{\mu \nu }\left(x\right)+\overline{\vartheta }{\beta }_{\mu \nu }\left(x\right)+\vartheta \overline{\vartheta }{M}_{\mu \nu }\left(x\right),\hfill \end{array}$$

(38)

$$\begin{array}{ccc}\hfill {\mathcal{B}}_{\mu \vartheta }\left(X\right)& =& \left(-{\overline{\alpha }}_{\mu }\left(x\right)+{\partial }_{\mu }\gamma \left(x\right)\right)+\vartheta {\partial }_{\mu }\overline{e}\left(x\right)+\overline{\vartheta }\left({\partial }_{\mu }e\left(x\right)-{\mathcal{A}}_{\mu }\left(x\right)\right)+\vartheta \overline{\vartheta }{\partial }_{\mu }\Gamma \left(x\right),\hfill \end{array}$$

(39)

$$\begin{array}{ccc}\hfill {\mathcal{B}}_{\mu \overline{\vartheta }}\left(X\right)& =& \left(-{\alpha }_{\mu }\left(x\right)+{\partial }_{\mu }\overline{\gamma }\left(x\right)\right)+\vartheta {\partial }_{\mu }\left(\overline{a}\left(x\right)+{\mathcal{A}}_{\mu }\left(x\right)\right)+\overline{\vartheta }{\partial }_{\mu }a\left(x\right)+\vartheta \overline{\vartheta }{\partial }_{\mu }\overline{\Gamma }\left(x\right),\hfill \end{array}$$

(40)

$$\begin{array}{ccc}\hfill {\mathcal{B}}_{\vartheta \vartheta }\left(X\right)& =& \overline{e}\left(x\right)+\overline{\vartheta }\Gamma \left(x\right),\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill {\mathcal{B}}_{\overline{\vartheta }\overline{\vartheta }}\left(X\right)& =& a\left(x\right)-\vartheta \overline{\Gamma }\left(x\right),\hfill \end{array}$$

(42)

$$\begin{array}{ccc}\hfill {\mathcal{B}}_{\vartheta \overline{\vartheta }}\left(X\right)& =& (e\left(x\right)+\overline{a}\left(x\right))+\overline{\vartheta }\overline{\Gamma }\left(x\right)-\vartheta \Gamma \left(x\right),\hfill \end{array}$$

(43)

where all the component fields on the RHSs are so far unrestricted. If we now impose (33), we obtain the following further restrictions:

$$\begin{array}{ccc}\hfill {\overline{\beta }}_{\mu \nu }\left(x\right)& =& -{\left(d\beta \right)}_{\mu \nu }\left(x\right)={\left(d\overline{\alpha }\right)}_{\mu \nu }\left(x\right),\hfill \end{array}$$

(44)

$$\begin{array}{ccc}\hfill {\beta }_{\mu \nu }\left(x\right)& =& -{\left(d\overline{\beta }\right)}_{\mu \nu }\left(x\right)={\left(d\alpha \right)}_{\mu \nu }\left(x\right),\hfill \end{array}$$

(45)

$$\begin{array}{ccc}\hfill M_{\mu \nu }\left(x\right)& =& {\left(d\mathcal{A}\right)}_{\mu \nu }\left(x\right),\hfill \end{array}$$

(46)

where $\beta ,\overline{\beta },\mathcal{A}$ denote 1-forms with components ${\beta }_{\mu }\left(x\right),{\overline{\beta }}_{\mu }\left(x\right),{\mathcal{A}}_{\mu }\left(x\right)$, respectively.

We also consider, instead of $\tilde{\mathcal{A}}$, the superfield $\tilde{\mathcal{A}}-\tilde{d}\tilde{\mathsf{f}}$, and, in particular, we replace A with $A^{\prime }=A-df$.

From the previous equations, we can read off the BRST transformations of the independent component fields. Dropping the argument $\left(x\right)$ and using the form notation for the BRST transformations, we have the following:

$$\begin{array}{c}\hfill \begin{array}{ccc}\mathfrak{s}B=d\overline{\alpha },& \mathfrak{s}{A}^{\prime }=\overline{\alpha }-d\overline{\varphi },& \mathfrak{s}f=\overline{\varphi },\\ \mathfrak{s}\alpha =-d\mathcal{A},& \mathfrak{s}\gamma =\overline{e},& \mathfrak{s}e=-\Gamma ,\\ \mathfrak{s}\overline{\gamma }=\overline{a},& \mathfrak{s}a=-\overline{\Gamma },& \mathfrak{s}\varphi =-F,\end{array}\end{array}$$

(47)

all the other $\mathfrak{s}$ transformations being trivial. For the anti-BRST transformations, we have the following:

$$\begin{array}{c}\hfill \begin{array}{ccc}\overline{\mathfrak{s}}B=d\alpha ,& \overline{\mathfrak{s}}{A}^{\prime }=\alpha -d\varphi ,& \overline{\mathfrak{s}}f=\varphi ,\\ \overline{\mathfrak{s}}\overline{\alpha }=\mathcal{A},& \overline{\mathfrak{s}}\overline{\gamma }=a,& \overline{\mathfrak{s}}\overline{a}=\overline{\Gamma },\\ \overline{\mathfrak{s}}\gamma =e,& \overline{\mathfrak{s}}\overline{e}=\overline{\Gamma },& \overline{\mathfrak{s}}\overline{\varphi }=F,\end{array}\end{array}$$

(48)

All the other anti-BRST transformations are trivial.

The system (47) and (48) differs from (27) and (28) only by field redefinitions. Let us set the following:

$$\begin{array}{ccc}& & C=\overline{\alpha }+d\gamma ,\lambda =-\gamma -\overline{\varphi },\hfill \end{array}$$

(49)

$$\begin{array}{ccc}& & \overline{C}=\alpha -d\overline{\gamma },\overline{\lambda }=\overline{\gamma }-\varphi .\hfill \end{array}$$

(50)

Then, the first equation of (47) and the first of (48) become the following:

$$\begin{array}{c}\hfill \begin{array}{ccc}\mathfrak{s}B=dC,& \mathfrak{s}{A}^{\prime }=C+d\lambda ,& \mathfrak{s}f=\lambda +\gamma ,\\ \overline{\mathfrak{s}}\overline{B}=d\overline{C},& \overline{\mathfrak{s}}{A}^{\prime }=\overline{C}+d\overline{\lambda },& \overline{\mathfrak{s}}f=\overline{\lambda }-\overline{\gamma }.\end{array}\end{array}$$

(51)

Next, we define the following:

$$\begin{array}{c}\hfill K_{\mu }={\mathcal{A}}_{\mu }+{\partial }_{\mu }\overline{a},{\overline{K}}_{\mu }={\mathcal{A}}_{\mu }+{\partial }_{\mu }e.\end{array}$$

(52)

The remaining s and $\overline{s}$ transformations become the following:

$$\begin{array}{c}\hfill \begin{array}{ccc}\mathfrak{s}C=d\overline{e},& \mathfrak{s}\lambda =-\overline{e},& \mathfrak{s}\gamma =\overline{e},\\ \mathfrak{s}\overline{C}=-K,& \mathfrak{s}\overline{K}=-d\Gamma ,& \mathfrak{s}\overline{\gamma }=\overline{a},\\ \mathfrak{s}a=-\overline{\Gamma },& \mathfrak{s}\overline{\lambda }=\overline{a}+F,& \mathfrak{s}a=-\overline{\Gamma },\end{array}\end{array}$$

(53)

and

$$\begin{array}{c}\hfill \begin{array}{ccc}\overline{\mathfrak{s}}\overline{C}=-da,& \overline{\mathfrak{s}}\overline{\lambda }=a,& \overline{\mathfrak{s}}\overline{\gamma }=a,\\ \overline{\mathfrak{s}}C=\overline{K},& \overline{\mathfrak{s}}K=-d\overline{\Gamma },& \overline{\mathfrak{s}}\gamma =e,\\ \overline{\mathfrak{s}}\overline{e}=\Gamma ,& \overline{\mathfrak{s}}\lambda =-e-F,& \overline{\mathfrak{s}}\overline{a}=\overline{\Gamma }.\end{array}\end{array}$$

(54)

Moreover, we have the following CF-like condition:

$$\begin{array}{c}\hfill \overline{K}-K=d(e-\overline{a}).\end{array}$$

(55)

These relations coincide with those of the 1-gerbe, provided that we make the following replacements: $\gamma \to \mu ,\overline{\gamma }\to -\overline{\mu },a\to -\overline{\beta },\overline{a}\to g,e\to \overline{g},\overline{e}\to -\beta$ and $\Gamma \to -\rho ,\overline{\Gamma }\to \overline{\rho }$.

There is only one difference: the presence of F in two cases in the last lines of both (53) and (54). This is an irrelevant term, as it belongs to the kernel of both s and $\overline{s}$.

Remark 1.

One can also impose the horizontality condition $\tilde{\mathcal{A}}-\tilde{d}\tilde{\mathsf{f}}=\mathcal{A}-d\mathsf{f}$, but this does not change much the final result: in fact, the resulting 1-gerbe algebra is the same.

4. Diffeomorphisms and the Superfield Formalism

After the successful extension of the superfield formalism to gerbes, we wish to deal with an entirely different type of symmetry: the diffeomorphisms. Our aim is to answer a few questions:

• Is the superfield formalism applicable to diffeomorphisms?
• What are the horizontality conditions for the latter?
• What are the CF conditions?
• Can we generalize the Riemannian geometry to the superspace?

In the sequel, we will answer all these questions. The answer to the last question will be partly negative, because an inverse supermetric does not exist. Nevertheless, it is possible to develop a superfield formalism in the horizontal (commuting) directions.

The first proposal of a superfield formalism for diffeomorphisms was made by [54,55,56]. Here, we present another approach, presented in [57], closer in spirit to the standard (commutative) geometrical approach.

Diffeomorphisms, or general coordinate transformations, are given in terms of generic (smooth) functions of $x^{\mu }$:

$$\begin{array}{c}\hfill x^{\mu }\to x^{{}^{\prime }\mu }=f^{\mu }\left(x\right).\end{array}$$

An infinitesimal diffeomorphism is defined by means of a local parameter ${\xi }^{\mu }\left(x\right)$: $f^{\mu }\left(x\right)=x^{\mu }-{\xi }^{\mu }\left(x\right)$. In a quantized theory, this is promoted to an anticommuting field, and the BRST transformations for a scalar field, a vector field, the metric and $\xi$, respectively, are the following:

$$\begin{array}{ccc}& & {\delta }_{\xi }\phi ={\xi }^{\lambda }{\partial }_{\lambda }\phi ,\hfill \end{array}$$

(56)

$$\begin{array}{ccc}& & {\delta }_{\xi }{A}_{\mu }={\xi }^{\lambda }{\partial }_{\lambda }{A}_{\mu }+{\partial }_{\mu }{\xi }^{\lambda }{A}_{\lambda },\hfill \end{array}$$

(57)

$$\begin{array}{ccc}& & {\delta }_{\xi }{g}_{\mu \nu }={\xi }^{\lambda }{\partial }_{\lambda }{g}_{\mu \nu }+{\partial }_{\mu }{\xi }^{\lambda }{g}_{\lambda \nu }+{\partial }_{\nu }{\xi }^{\lambda }{g}_{\mu \lambda },\hfill \end{array}$$

(58)

$$\begin{array}{ccc}& & {\delta }_{\xi }{\xi }^{\mu }={\xi }^{\lambda }{\partial }_{\lambda }{\xi }^{\mu },\hfill \end{array}$$

(59)

It is easy to see that these transformations are nilpotent. We wish now to define the analogs of anti-BRST transformations. To this end, we introduce another anticommuting field, $\overline{\xi }$, and a ${\delta }_{\overline{\xi }}$ transformation, which transforms a scalar, vector, the metric and $\overline{\xi }$ in just the same way as ${\delta }_{\xi }$ (these transformations are not rewritten here). In addition, we have the following cross-transformations:

$$\begin{array}{ccc}& & {\delta }_{\xi }{\overline{\xi }}^{\mu }=b^{\mu },{\delta }_{\overline{\xi }}{\xi }^{\mu }={\overline{b}}^{\mu },\hfill \end{array}$$

(60)

$$\begin{array}{ccc}& & {\delta }_{\xi }{b}^{\mu }=0,{\delta }_{\xi }{\overline{b}}^{\mu }=-\overline{b}·\partial {\xi }^{\mu }+\xi ·\partial {\overline{b}}^{\mu },\hfill \end{array}$$

(61)

$$\begin{array}{ccc}& & {\delta }_{\overline{\xi }}{\overline{b}}^{\mu }=0,{\delta }_{\overline{\xi }}{b}^{\mu }=-b·\partial {\overline{\xi }}^{\mu }+\overline{\xi }·\partial b^{\mu },\hfill \end{array}$$

(62)

It follows that the overall transformation ${\delta }_{\xi }+{\delta }_{\overline{\xi }}$ is nilpotent:

$$\begin{array}{c}\hfill {({\delta }_{\xi }+{\delta }_{\overline{\xi }})}^2=0.\end{array}$$

4.1. The Superfield Formalism

Our aim now is to reproduce the above transformations by means of the superfield formalism. The superspace coordinates are $X^M=(x^{\mu },\vartheta ,\overline{\vartheta })$, where $\vartheta ,\overline{\vartheta }$ are the same anticommuting variables as above. A (super)diffeomorphism is represented by a superspace transformation $X^M=(x^{\mu },\vartheta ,\overline{\vartheta })\to {\tilde{X}}^M=(F^{\mu }\left(X^M\right),\vartheta ,\overline{\vartheta })$, where3,

$$\begin{array}{c}\hfill F^{\mu }\left(X^M\right)=f^{\mu }\left(x\right)-\vartheta {\overline{\xi }}^{\mu }-\overline{\vartheta }{\xi }^{\mu }\left(x\right)+\vartheta \overline{\vartheta }{h}^{\mu }\left(x\right).\end{array}$$

(63)

Here, $f^{\mu }\left(x\right)$ is an ordinary diffeomorphism, $\xi ,\overline{\xi }$ are the generic anticommuting functions introduced before, and $h^{\mu }$ is a generic commuting one.

The horizontality condition is formulated by selecting appropriate invariant geometric expressions in ordinary spacetime and identifying them with the same expressions extended to the superspace. To start, we work out explicitly the case of a scalar field.

4.2. The Scalar

The diffeomorphism transformation properties of an ordinary scalar field are as follows:

$$\begin{array}{c}\hfill \tilde{\phi }\left(f^{\mu }\left(x\right)\right)=\phi \left(x^{\mu }\right).\end{array}$$

(64)

Now, we embed the scalar field $\phi$ in a superfield4

$$\begin{array}{c}\hfill \Phi \left(X\right)=\phi \left(x\right)+\vartheta \overline{\beta }\left(x\right)+\overline{\vartheta }\beta \left(x\right)+\vartheta \overline{\vartheta }C\left(x\right),\end{array}$$

(65)

The BRST interpretation is ${\delta }_{\xi }=\frac{\partial }{\partial \overline{\vartheta }}{|}_{\vartheta =0},{\delta }_{\overline{\xi }}=\frac{\partial }{\partial \vartheta }{|}_{\overline{\vartheta }=0}$. The horizontality condition, suggested by (64), is the following:

$$\begin{array}{c}\hfill \Phi \left(F\right(X\left)\right)=\phi \left(x\right).\end{array}$$

(66)

Using (63) with $f\left(x^{\mu }\right)=x^{\mu }$, this becomes the following:

$$\begin{array}{ccc}\hfill \Phi \left(F\right(X\left)\right)& =& \phi \left(x\right)-(\vartheta \overline{\xi }\left(x\right)+\overline{\vartheta }\xi \left(x\right)-\vartheta \overline{\vartheta }h\left(x\right))·\partial \phi \left(x\right)+\vartheta \left(\overline{\beta }\left(x\right)-\overline{\vartheta }\xi \left(x\right)·\partial \overline{\beta }\left(x\right)\right)\hfill \\ & & +\overline{\vartheta }\left(\beta \left(x\right)-\vartheta \overline{\xi }·\partial \beta \left(x\right)\right)+\vartheta \overline{\vartheta }\left(C\left(x\right)-{\overline{\xi }}^{\mu }{\xi }^{\nu }{\partial }_{\mu }{\partial }_{\nu }\phi \left(x\right)\right)\hfill \\ & =& \phi \left(x\right)+\vartheta \left(\overline{\beta }\left(x\right)-\overline{\xi }·\partial \phi \left(x\right)\right)+\overline{\vartheta }\left(\beta \left(x\right)-\xi ·\partial \phi \left(x\right)\right)\hfill \\ & & +\vartheta \overline{\vartheta }\left(C\left(x\right)-\xi ·\partial \overline{\beta }\left(x\right)+\overline{\xi }·\partial \beta \left(x\right)+h\left(x\right)·\partial \phi \left(x\right)-{\overline{\xi }}^{\mu }{\xi }^{\nu }{\partial }_{\mu }{\partial }_{\nu }\phi \left(x\right)\right),\hfill \end{array}$$

(67)

where · denotes index contraction. Then, (66) implies the following:

$$\begin{array}{ccc}\hfill \beta \left(x\right)& =& \xi ·\partial \phi \left(x\right),\overline{\beta }\left(x\right)=\overline{\xi }·\partial \phi \left(x\right),\hfill \\ \hfill C\left(x\right)& =& \xi ·\partial \overline{\beta }\left(x\right)-\overline{\xi }·\partial \beta \left(x\right)-\xi \overline{\xi }{\partial }^2\phi \left(x\right)-h\left(x\right)·\partial \phi \left(x\right),\hfill \end{array}$$

(68)

where $\xi \overline{\xi }{\partial }^2\phi \left(x\right)={\xi }^{\mu }{\overline{\xi }}^{\nu }{\partial }_{\mu }{\partial }_{\nu }\phi \left(x\right)$.

Now, the BRST interpretation implies the following:

$$\begin{array}{c}\hfill {\delta }_{\overline{\xi }}\phi \left(x\right)=\overline{\beta }\left(x\right)=\overline{\xi }·\partial \phi \left(x\right),{\delta }_{\xi }\phi \left(x\right)=\beta \left(x\right)=\xi ·\partial \phi \left(x\right),\end{array}$$

(69)

and ${\delta }_{\xi }\overline{\beta }\left(x\right)=C\left(x\right),{\delta }_{\overline{\xi }}\beta \left(x\right)=-C\left(x\right)$.

Inserting $\beta$ and $\overline{\beta }$ into C in (68), we obtain the following:

$$\begin{array}{c}\hfill {\delta }_{\xi }{\delta }_{\overline{\xi }}\phi =b·\partial \phi -\overline{\xi }·\partial \xi ·\partial \phi -\overline{\xi }\xi {\partial }^2\phi .\end{array}$$

(70)

This coincides with the expression of C, (68), if

$$\begin{array}{c}\hfill h^{\mu }=-b^{\mu }+\xi ·\partial {\overline{\xi }}^{\mu }.\end{array}$$

(71)

Likewise,

$$\begin{array}{c}\hfill {\delta }_{\overline{\xi }}{\delta }_{\xi }\phi =\overline{b}·\partial \phi -\xi ·\partial \overline{\xi }·\partial \phi -\xi \overline{\xi }{\partial }^2\phi ,\end{array}$$

(72)

which coincides with the expression of $-C$, (68), if the following holds:

$$\begin{array}{c}\hfill h^{\mu }={\overline{b}}^{\mu }-\overline{\xi }·\partial {\xi }^{\mu }.\end{array}$$

(73)

Equating (71) with (73) we obtain the following:

$$\begin{array}{c}\hfill h^{\mu }\left(x\right)=-b^{\mu }\left(x\right)+\xi \left(x\right)·\partial {\overline{\xi }}^{\mu }\left(x\right)={\overline{b}}^{\mu }\left(x\right)-\overline{\xi }\left(x\right)·\partial {\xi }^{\mu }\left(x\right),\end{array}$$

(74)

which is possible if and only if the following CF condition is satisfied:

$$\begin{array}{c}\hfill b^{\mu }+{\overline{b}}^{\mu }={\xi }^{\lambda }{\partial }_{\lambda }{\overline{\xi }}^{\mu }+{\overline{\xi }}^{\lambda }{\partial }_{\lambda }{\xi }^{\mu },\end{array}$$

This condition is consistent, for applying ${\delta }_{\xi }$ and ${\delta }_{\overline{\xi }}$ to both sides produces the same result. As we shall see, this condition is, so to speak, universal: it appears whenever BRST and anti-BRST diffeomorphisms are involved, and it is the only required condition.

4.3. The Vector

We now extend the previous approach to a vector field. In order to apply the horizontality condition, we must first identify the appropriate expression. This is a 1-superform:

$$\begin{array}{c}\hfill \mathbb{A}\equiv {\mathcal{A}}_M\left(X\right)d{X}^M=A_{\mu }\left(X\right)d{x}^{\mu }+A_{\vartheta }\left(X\right)d\vartheta +A_{\overline{\vartheta }}\left(X\right)d\overline{\vartheta },\end{array}$$

(75)

where

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{\mu }\left(X\right)& =& A_{\mu }\left(x\right)+\vartheta {\overline{\varphi }}_{\mu }\left(x\right)+\overline{\vartheta }{\varphi }_{\mu }\left(x\right)+\vartheta \overline{\vartheta }{B}_{\mu }\left(x\right),\hfill \end{array}$$

(76)

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{\vartheta }\left(X\right)& =& \chi \left(x\right)+\vartheta \overline{C}\left(x\right)+\overline{\vartheta }C\left(x\right)+\vartheta \overline{\vartheta }\psi \left(x\right),\hfill \end{array}$$

(77)

$$\begin{array}{ccc}\hfill {\mathcal{A}}_{\overline{\vartheta }}\left(X\right)& =& \omega \left(x\right)+\vartheta \overline{D}\left(x\right)+\overline{\vartheta }D\left(x\right)+\vartheta \overline{\vartheta }\rho \left(x\right).\hfill \end{array}$$

(78)

According to our prescription, horizontality means the following:

$$\begin{array}{c}\hfill {\mathcal{A}}_M\left(\tilde{X}\right)\tilde{d}{\tilde{X}}^M=A_{\mu }\left(x\right)d{x}^{\mu },\end{array}$$

(79)

where $\tilde{d}=\frac{\partial }{\partial x^{\mu }}d{x}^{\mu }+\frac{\partial }{\partial \vartheta }d\vartheta +\frac{\partial }{\partial \overline{\vartheta }}d\overline{\vartheta }$. Thus, we obtain the following:

$$\begin{array}{ccc}\hfill \tilde{d}{\tilde{X}}^M& =& (d{x}^{\mu }-\vartheta {\partial }_{\lambda }{\overline{\xi }}^{\mu }d{x}^{\lambda }-\overline{\vartheta }{\partial }_{\lambda }{\xi }^{\mu }d{x}^{\lambda }+\vartheta \overline{\vartheta }{\partial }_{\lambda }{h}^{\mu }d{x}^{\lambda }\hfill \\ & & -({\overline{\xi }}^{\mu }-\overline{\vartheta }{h}^{\mu })d\vartheta -({\xi }^{\mu }+\vartheta h^{\mu })d\overline{\vartheta },d\vartheta ,d\overline{\vartheta }).\hfill \end{array}$$

(80)

It remains for us to expand the LHS of (79). The explicit expression can be found in Appendix B. The commutation prescriptions are the following: $x^{\mu },\vartheta ,\overline{\vartheta },{\xi }^{\mu }$ commute with $d{x}^{\mu }$ and $d\vartheta ,d\overline{\vartheta }$; and ${\xi }^{\mu },{\overline{\xi }}^{\mu }$ anticommute with $\vartheta ,\overline{\vartheta }$. From (A21), we obtain the following identifications:

$$\begin{array}{ccc}\hfill {\varphi }_{\mu }& =& \xi ·\partial A_{\mu }+{\partial }_{\mu }{\xi }^{\lambda }{A}_{\lambda },\hfill \end{array}$$

(81)

$$\begin{array}{ccc}\hfill {\overline{\varphi }}_{\mu }& =& \overline{\xi }·\partial A_{\mu }+{\partial }_{\mu }{\overline{\xi }}^{\lambda }{A}_{\lambda },\hfill \end{array}$$

(82)

$$\begin{array}{ccc}\hfill B_{\mu }& =& \xi ·\partial {\overline{\varphi }}_{\mu }-\overline{\xi }·\partial {\varphi }_{\mu }-\xi \overline{\xi }·{\partial }^2{A}_{\mu }+{\partial }_{\mu }{\overline{\xi }}^{\lambda }\xi ·\partial A_{\lambda }-{\partial }_{\mu }{\xi }^{\lambda }\overline{\xi }·\partial A_{\lambda }\hfill \\ & & -{\partial }_{\mu }{\overline{\xi }}^{\lambda }{\varphi }_{\lambda }+{\partial }_{\mu }{\xi }^{\lambda }{\overline{\varphi }}_{\lambda }-h·\partial A_{\mu }-{\partial }_{\mu }h·A,\hfill \end{array}$$

(83)

$$\begin{array}{ccc}\hfill \chi & =& A_{\mu }{\overline{\xi }}^{\mu },\hfill \end{array}$$

(84)

$$\begin{array}{ccc}\hfill C& =& -\xi ·\partial A_{\mu }{\overline{\xi }}^{\mu }+{\varphi }_{\mu }{\overline{\xi }}^{\mu }+\xi ·\partial \chi -h·A,\hfill \end{array}$$

(85)

$$\begin{array}{ccc}\hfill \overline{C}& =& -\overline{\xi }·\partial A_{\mu }{\overline{\xi }}^{\mu }+{\overline{\varphi }}_{\mu }{\overline{\xi }}^{\mu }+\overline{\xi }·\partial \chi ,\hfill \end{array}$$

(86)

$$\begin{array}{ccc}\hfill \psi & =& \xi \overline{\xi }·{\partial }^2{A}_{\mu }{\overline{\xi }}^{\mu }-\xi ·\partial {\overline{\varphi }}_{\mu }{\overline{\xi }}^{\mu }+\overline{\xi }·\partial {\varphi }_{\mu }{\overline{\xi }}^{\mu }+B_{\mu }{\overline{\xi }}^{\mu }-\xi \overline{\xi }·{\partial }^2\chi +\xi ·\partial \overline{C}-\overline{\xi }·\partial C\hfill \\ & & -\overline{\xi }·\partial A·h+\overline{\varphi }·h-h·\partial \chi +h·\partial A_{\mu }{\overline{\xi }}^{\mu },\hfill \end{array}$$

(87)

and

$$\begin{array}{ccc}\hfill \omega & =& A_{\mu }{\xi }^{\mu },\hfill \end{array}$$

(88)

$$\begin{array}{ccc}\hfill D& =& -\xi ·\partial A_{\mu }{\xi }^{\mu }+{\varphi }_{\mu }{\xi }^{\mu }+\xi ·\partial \omega ,\hfill \end{array}$$

(89)

$$\begin{array}{ccc}\hfill \overline{D}& =& -\overline{\xi }·\partial A_{\mu }{\xi }^{\mu }+{\overline{\varphi }}_{\mu }{\xi }^{\mu }+\overline{\xi }·\partial \omega +h·A,\hfill \end{array}$$

(90)

$$\begin{array}{ccc}\hfill \rho & =& \xi \overline{\xi }·{\partial }^2{A}_{\mu }{\xi }^{\mu }-\xi ·\partial {\overline{\varphi }}_{\mu }{\xi }^{\mu }+\overline{\xi }·\partial {\varphi }_{\mu }{\xi }^{\mu }+B_{\mu }{\xi }^{\mu }-\xi \overline{\xi }·{\partial }^2\omega +\xi ·\partial \overline{D}-\overline{\xi }·\partial D.\hfill \\ & & -\xi ·\partial A·h+\varphi ·h-h·\partial \omega +h·\partial A_{\mu }{\xi }^{\mu }.\hfill \end{array}$$

(91)

One can see that

$$\begin{array}{c}\hfill {\varphi }_{\mu }={\delta }_{\xi }{A}_{\mu },{\overline{\varphi }}_{\mu }={\delta }_{\overline{\xi }}{A}_{\mu },B_{\mu }={\delta }_{\overline{\xi }}{\varphi }_{\mu }=-{\delta }_{\xi }{\overline{\varphi }}_{\mu }.\end{array}$$

(92)

$$\begin{array}{c}\hfill D={\delta }_{\xi }\omega ,\overline{D}={\delta }_{\overline{\xi }}\omega ,\rho =-{\delta }_{\overline{\xi }}D={\delta }_{\xi }\overline{D},\end{array}$$

(93)

and

$$\begin{array}{c}\hfill C={\delta }_{\xi }\chi ,\overline{C}={\delta }_{\overline{\xi }}\chi ,\psi =-{\delta }_{\overline{\xi }}C={\delta }_{\xi }\overline{C},\end{array}$$

(94)

provided

$$\begin{array}{c}\hfill h^{\mu }\left(x\right)=-b^{\mu }\left(x\right)+\xi \left(x\right)·\partial {\overline{\xi }}^{\mu }\left(x\right)={\overline{b}}^{\mu }\left(x\right)-\overline{\xi }\left(x\right)·\partial {\xi }^{\mu }\left(x\right),\end{array}$$

(95)

which is possible if and only if the following CF condition is satisfied:

$$\begin{array}{c}\hfill b^{\mu }+{\overline{b}}^{\mu }={\xi }^{\lambda }{\partial }_{\lambda }{\overline{\xi }}^{\mu }+{\overline{\xi }}^{\lambda }{\partial }_{\lambda }{\xi }^{\mu },\end{array}$$

In particular, $\rho$ can be rewritten as follows:

$$\begin{array}{ccc}\hfill \rho & =& \xi ·\partial \overline{\xi }·\partial A_{\mu }{\xi }^{\mu }-\overline{\xi }·\partial \xi ·\partial A_{\mu }{\xi }^{\mu }+\overline{\xi }·\partial {\xi }^{\mu }\xi ·\partial A_{\mu }{\xi }^{\mu }\hfill \\ & & -\xi ·\partial {\overline{\xi }}^{\mu }\xi ·\partial A_{\mu }{\xi }^{\mu }+\xi \overline{\xi }·{\partial }^2{A}_{\mu }{\xi }^{\mu }-\xi ·\partial A·h+\varphi ·h-h·\partial \omega .\hfill \end{array}$$

4.4. The Metric

The most important field for theories invariant under diffeomorphisms is the metric $g_{\mu \nu }\left(x\right)$. To represent its BRST transformation properties in the superfield formalism, we embed it in a supermetric $G_{MN}\left(X\right)$ and form the symmetric 2-superdifferential as follows:

$$\begin{array}{c}\hfill \mathbb{G}=G_{MN}\left(X\right)\tilde{d}{X}^M\vee \tilde{d}{X}^N,\end{array}$$

(96)

where ∨ denotes the symmetric tensor product and

$$\begin{array}{ccc}\hfill G_{\mu \nu }\left(X\right)& =& g_{\mu \nu }\left(x\right)+\vartheta {\overline{\Gamma }}_{\mu \nu }\left(x\right)+\overline{\vartheta }{\Gamma }_{\mu \nu }\left(x\right)+\vartheta \overline{\vartheta }{V}_{\mu \nu }\left(x\right),\hfill \\ \hfill G_{\mu \vartheta }\left(X\right)& =& {\gamma }_{\mu }\left(x\right)+\vartheta {\overline{g}}_{\mu }\left(x\right)+\overline{\vartheta }{g}_{\mu }\left(x\right)+\vartheta \overline{\vartheta }{\Gamma }_{\mu }\left(x\right)=G_{\vartheta \mu }\left(X\right),\hfill \\ \hfill G_{\mu \overline{\vartheta }}\left(X\right)& =& {\overline{\gamma }}_{\mu }\left(x\right)+\vartheta {\overline{f}}_{\mu }\left(x\right)+\overline{\vartheta }{f}_{\mu }\left(x\right)+\vartheta \overline{\vartheta }{\overline{\Gamma }}_{\mu }\left(x\right)=G_{\overline{\vartheta }\mu }\left(X\right),\hfill \end{array}$$

(97)

$$\begin{array}{ccc}\hfill \frac{1}{2}{G}_{\vartheta \overline{\vartheta }}\left(X\right)& =& g\left(x\right)+\vartheta \overline{\gamma }\left(x\right)+\overline{\vartheta }\gamma \left(x\right)+\vartheta \overline{\vartheta }G\left(x\right)=-\frac{1}{2}{G}_{\overline{\vartheta }\vartheta }\left(X\right),\hfill \end{array}$$

(98)

while $G_{\vartheta \vartheta }\left(X\right)=0=G_{\overline{\vartheta }\overline{\vartheta }}\left(X\right)$, because the symmetric tensor product becomes antisymmetric for anticommuting variables: $d\vartheta \vee d\vartheta =0=d\overline{\vartheta }\vee d\overline{\vartheta },d\vartheta \vee d\overline{\vartheta }=-d\overline{\vartheta }\vee d\vartheta$.

The horizontality condition is obtained by requiring the following:

$$\begin{array}{c}\hfill {\tilde{G}}_{MN}\left(\tilde{X}\right)\tilde{d}{\tilde{X}}^M\vee \tilde{d}{\tilde{X}}^N=g_{\mu \nu }\left(x\right)d{x}^{\mu }\vee d{x}^{\nu }.\end{array}$$

(99)

The explicit expression of the LHS of this equation can be found again in Appendix B, from which the following identification follows:

$$\begin{array}{ccc}\hfill {\Gamma }_{\mu \nu }& =& \xi ·\partial g_{\mu \nu }+{\partial }_{\mu }{\xi }^{\lambda }{g}_{\lambda \nu }+{\partial }_{\nu }{\xi }^{\lambda }{g}_{\mu \lambda }={\delta }_{\xi }{g}_{\mu \nu },\hfill \end{array}$$

(100)

$$\begin{array}{ccc}\hfill {\overline{\Gamma }}_{\mu \nu }& =& \overline{\xi }·\partial g_{\mu \nu }+{\partial }_{\mu }{\overline{\xi }}^{\lambda }{g}_{\lambda \nu }+{\partial }_{\nu }{\overline{\xi }}^{\lambda }{g}_{\mu \lambda }={\delta }_{\overline{\xi }}{g}_{\mu \nu },\hfill \end{array}$$

(101)

$$\begin{array}{ccc}\hfill V_{\mu \nu }& =& -\xi \overline{\xi }·{\partial }^2{g}_{\mu \nu }+\xi ·\partial {\overline{\Gamma }}_{\mu \nu }+{\partial }_{\mu }{\xi }^{\lambda }{\overline{\Gamma }}_{\lambda \nu }+{\partial }_{\nu }{\xi }^{\lambda }{\overline{\Gamma }}_{\mu \lambda }-\overline{\xi }·\partial {\Gamma }_{\mu \nu }-{\partial }_{\mu }{\overline{\xi }}^{\lambda }{\Gamma }_{\lambda \nu }-{\partial }_{\nu }{\overline{\xi }}^{\lambda }{\Gamma }_{\mu \lambda }\hfill \\ & & +{\partial }_{\mu }{\overline{\xi }}^{\lambda }\xi ·\partial g_{\lambda \nu }+{\partial }_{\nu }{\overline{\xi }}^{\lambda }\xi ·\partial g_{\mu \lambda }-{\partial }_{\mu }{\xi }^{\lambda }\overline{\xi }·\partial g_{\lambda \nu }-{\partial }_{\nu }{\xi }^{\lambda }\overline{\xi }·\partial g_{\mu \lambda }+{\partial }_{\mu }{\overline{\xi }}^{\lambda }{\partial }_{\nu }{\xi }^{\rho }{g}_{\lambda \rho }\hfill \\ & & +{\partial }_{\nu }{\overline{\xi }}^{\lambda }{\partial }_{\mu }{\xi }^{\rho }{g}_{\lambda \rho }-h·\partial g_{\mu \nu }-{\partial }_{\mu }{h}^{\lambda }{g}_{\lambda \nu }-{\partial }_{\nu }{h}^{\lambda }{g}_{\mu \lambda }\hfill \\ & =& {\delta }_{\xi }{\overline{\Gamma }}_{\mu \nu }=-{\delta }_{\overline{\xi }}{\Gamma }_{\mu \nu }.\hfill \end{array}$$

(102)

Moreover,

$$\begin{array}{ccc}\hfill {\gamma }_{\mu }& =& g_{\mu \nu }{\overline{\xi }}^{\nu },\hfill \end{array}$$

(103)

$$\begin{array}{ccc}\hfill g_{\mu }& =& {\partial }_{\mu }{\xi }^{\lambda }{g}_{\lambda \nu }{\overline{\xi }}^{\nu }+{\partial }_{\nu }{\xi }^{\lambda }{g}_{\mu \lambda }{\overline{\xi }}^{\nu }+\xi ·\partial g_{\mu \nu }{\overline{\xi }}^{\nu }+g_{\mu \nu }\xi ·\partial {\overline{\xi }}^{\nu }-g_{\mu \nu }{h}^{\nu }={\delta }_{\xi }{\gamma }_{\mu },\hfill \end{array}$$

(104)

$$\begin{array}{ccc}\hfill {\overline{g}}_{\mu }& =& \overline{\xi }·\partial g_{\mu \nu }{\overline{\xi }}^{\nu }+{\partial }_{\mu }{\overline{\xi }}^{\lambda }{g}_{\lambda \nu }{\overline{\xi }}^{\nu }={\delta }_{\overline{\xi }}{\gamma }_{\mu },\hfill \\ \hfill {\Gamma }_{\mu }& =& -\xi \overline{\xi }·{\partial }^2{\gamma }_{\mu }+\xi ·\partial {\overline{g}}_{\mu }-{\partial }_{\mu }{\overline{\xi }}^{\lambda }{g}_{\lambda }-\overline{\xi }·\partial g_{\mu }+{\partial }_{\mu }{\xi }^{\lambda }{\overline{g}}_{\lambda }+{\partial }_{\mu }{\overline{\xi }}^{\lambda }\xi ·\partial {\gamma }_{\lambda }-{\partial }_{\mu }{\xi }^{\lambda }\overline{\xi }·\partial {\gamma }_{\lambda }\hfill \\ & & -\overline{\xi }·\partial h^{\nu }{g}_{\mu \nu }-{\partial }_{\mu }{h}^{\lambda }{g}_{\lambda \nu }{\overline{\xi }}^{\nu }+{\overline{\Gamma }}_{\mu \nu }{h}^{\nu }-h·\partial {\gamma }_{\mu }-g_{\nu \lambda }{h}^{\nu }{\partial }_{\mu }{\overline{\xi }}^{\lambda }\hfill \\ & & +{\partial }_{\nu }{\xi }^{\lambda }{\overline{\Gamma }}_{\mu \lambda }{\overline{\xi }}^{\nu }-{\partial }_{\nu }{\overline{\xi }}^{\lambda }{\Gamma }_{\mu \lambda }{\overline{\xi }}^{\nu }+\overline{\xi }·\partial {\overline{\xi }}^{\lambda }\xi ·\partial g_{\mu \lambda }\hfill \end{array}$$

(105)

$$\begin{array}{ccc}& & -\overline{\xi }·\partial {\xi }^{\lambda }\overline{\xi }·\partial g_{\mu \lambda }-\overline{\xi }·\partial {\xi }^{\lambda }{g}_{\lambda \rho }{\partial }_{\mu }{\overline{\xi }}^{\rho }+\overline{\xi }·\partial {\overline{\xi }}^{\lambda }{g}_{\lambda \rho }{\partial }_{\mu }{\xi }^{\rho }=-{\delta }_{\overline{\xi }}{g}_{\mu }={\delta }_{\xi }{\overline{g}}_{\mu },\hfill \end{array}$$

(106)

and

$$\begin{array}{ccc}\hfill {\overline{\gamma }}_{\mu }& =& g_{\mu \nu }{\xi }^{\nu },\hfill \end{array}$$

(107)

$$\begin{array}{ccc}\hfill {\overline{f}}_{\mu }& =& {\partial }_{\mu }{\overline{\xi }}^{\lambda }{g}_{\lambda \nu }{\xi }^{\nu }+{\partial }_{\nu }{\overline{\xi }}^{\lambda }{g}_{\mu \lambda }{\xi }^{\nu }+\overline{\xi }·\partial g_{\mu \nu }{\xi }^{\nu }+g_{\mu \nu }\overline{\xi }·\partial {\xi }^{\nu }+g_{\mu \nu }{h}^{\nu }={\delta }_{\overline{\xi }}{\overline{\gamma }}_{\mu },\hfill \end{array}$$

(108)

$$\begin{array}{ccc}\hfill f_{\mu }& =& \xi ·\partial g_{\mu \nu }{\xi }^{\nu }+{\partial }_{\mu }{\xi }^{\lambda }{g}_{\lambda \nu }{\xi }^{\nu }={\delta }_{\xi }{\overline{\gamma }}_{\mu },\hfill \\ \hfill {\overline{\Gamma }}_{\mu }& =& -\xi ·\partial g_{\mu \nu }{h}^{\nu }-{\partial }_{\mu }{h}^{\lambda }{g}_{\lambda \nu }{\xi }^{\nu }+{\Gamma }_{\mu \nu }{h}^{\nu }-h·\partial {\overline{\gamma }}_{\mu }-{\partial }_{\mu }{h}^{\lambda }{\overline{\gamma }}_{\lambda }-\xi ·\partial h^{\lambda }{g}_{\lambda \mu }\hfill \\ & & +{\partial }_{\mu }{\xi }_{\lambda }{\overline{f}}^{\lambda }+\xi ·\partial {\overline{f}}_{\mu }-\overline{\xi }·\partial f_{\mu }-{\xi }^{\lambda }{\overline{\xi }}^{\rho }{\partial }_{\lambda }{\partial }_{\rho }{\overline{\gamma }}_{\mu }-{\partial }_{\mu }{\overline{\xi }}^{\lambda }{\partial }_{\lambda }{\xi }^{\rho }{\overline{\gamma }}_{\rho }\hfill \\ & & -\overline{\xi }·\partial {\overline{\gamma }}_{\lambda }{\partial }_{\mu }{\xi }^{\lambda }+{\partial }_{\mu }{\overline{\xi }}^{\lambda }\xi ·\partial {\xi }^{\rho }{g}_{\lambda \rho }-\xi ·\partial {\overline{\xi }}^{\rho }{\partial }_{\rho }{\xi }^{\lambda }{g}_{\lambda \mu }+\xi ·\partial {\xi }^{\rho }{\partial }_{\rho }{\overline{\xi }}^{\lambda }{g}_{\lambda \mu }\hfill \end{array}$$

(109)

$$\begin{array}{ccc}& =& {\delta }_{\xi }{\overline{f}}_{\mu }=-{\delta }_{\overline{\xi }}{f}_{\mu }.\hfill \end{array}$$

(110)

Finally, we obtain the following:

$$\begin{array}{ccc}\hfill g& =& g_{\mu \nu }\left({\overline{\xi }}^{\mu }{\xi }^{\nu }-{\xi }^{\mu }{\overline{\xi }}^{\nu }\right)=2g_{\mu \nu }{\overline{\xi }}^{\mu }{\xi }^{\nu },\hfill \end{array}$$

(111)

$$\begin{array}{ccc}\hfill \gamma & =& 2\xi ·\partial g_{\mu \nu }{\overline{\xi }}^{\mu }{\xi }^{\nu }+2g_{\mu \nu }\left(\xi ·\partial {\overline{\xi }}^{\mu }-\overline{\xi }·\partial {\xi }^{\mu }\right){\xi }^{\nu }-2{\overline{\gamma }}_{\mu }{h}^{\mu }={\delta }_{\xi }g,\hfill \end{array}$$

(112)

$$\begin{array}{ccc}\hfill \overline{\gamma }& =& 2\overline{\xi }·\partial g_{\mu \nu }{\overline{\xi }}^{\mu }{\xi }^{\nu }+2g_{\mu \nu }\left(\xi ·\partial {\overline{\xi }}^{\mu }-\overline{\xi }·\partial {\xi }^{\mu }\right){\overline{\xi }}^{\nu }-2{\gamma }_{\mu }{h}^{\mu }={\delta }_{\overline{\xi }}g,\hfill \end{array}$$

(113)

$$\begin{array}{ccc}\hfill G& =& 2b^{\mu }{\partial }_{\mu }g+2b^{\mu }{g}_{\mu }-2\overline{\xi }·\partial \gamma -2\overline{\xi }·\partial {\overline{\xi }}^{\mu }{f}_{\mu }-2b·\partial {\overline{\xi }}^{\rho }{\overline{\gamma }}_{\rho }-2\overline{\xi }·\partial b^{\rho }{\overline{\gamma }}_{\rho }={\delta }_{\xi }\overline{\gamma }=-{\delta }_{\overline{\xi }}\gamma .\hfill \end{array}$$

(114)

This completes the verification of the horizontality condition. As expected, it leads to identifying the $\overline{\vartheta }$-and $\vartheta$-superpartners of the metric as BRST and anti-BRST transforms.

4.5. Inverse of $G_{\mu \nu }\left(X\right)$

A fundamental ingredient of Riemannian geometry is the inverse metric. Therefore, in order to see whether a super-Riemannian geometry can be introduced in the supermanifold, we have to verify whether an inverse supermetric exists. We start by the inverse of $G_{\mu \nu }\left(X\right)$, which is defined by first writing it as follows:

$$\begin{array}{ccc}\hfill G_{\mu \nu }\left(X\right)& =& g_{\mu \lambda }\left(x\right)\left[{\delta }_{\nu }^{\lambda }+g^{\lambda \rho }\left(\vartheta {\overline{\Gamma }}_{\rho \nu }\left(x\right)+\overline{\vartheta }{\Gamma }_{\rho \nu }\left(x\right)+\vartheta \overline{\vartheta }{V}_{\rho \nu }\left(x\right)\right)\right]\equiv g_{\mu \lambda }\left(x\right){\left(1+X\right)}^{\lambda }{}_{\nu },\hfill \end{array}$$

then in matrix terms as follows:

$$\begin{array}{c}\hfill {\widehat{G}}^{-1}=\left(1-X+X^2\right){\widehat{g}}^{-1},\end{array}$$

where ${\widehat{g}}^{-1}$ is the inverse of g, i.e., the following:

$$\begin{array}{ccc}\hfill {\widehat{G}}^{\mu \nu }& =& {\left(1-X+X^2\right)}^{\mu }{}_{\lambda }{\widehat{g}}^{\lambda \nu }\hfill \\ & =& \left({\delta }_{\lambda }^{\mu }-{\widehat{g}}^{\mu \rho }\left(\vartheta {\overline{\Gamma }}_{\rho \lambda }\left(x\right)+\overline{\vartheta }{\Gamma }_{\rho \lambda }\left(x\right)+\vartheta \overline{\vartheta }{V}_{\rho \lambda }\left(x\right)\right)+\vartheta \overline{\vartheta }{\widehat{g}}^{\mu \rho }\left({\Gamma }_{\rho \sigma }{g}^{\sigma \tau }{\overline{\Gamma }}_{\tau \lambda }-{\overline{\Gamma }}_{\rho \sigma }{g}^{\sigma \tau }{\Gamma }_{\tau \lambda }\right)\right){\widehat{g}}^{\lambda \nu }\hfill \\ & \equiv & {\widehat{g}}^{\mu \nu }\left(x\right)+\vartheta {\widehat{\overline{\Gamma }}}^{\mu \nu }\left(x\right)+\overline{\vartheta }{\widehat{\Gamma }}^{\mu \nu }\left(x\right)+\vartheta \overline{\vartheta }{\widehat{V}}^{\mu \nu }\left(x\right),\hfill \end{array}$$

(115)

and ${\widehat{g}}^{\mu \nu }$ is the ordinary metric inverse. Moreover,

$$\begin{array}{ccc}\hfill {\widehat{\Gamma }}^{\mu \nu }& =& -{\widehat{g}}^{\mu \lambda }{\Gamma }_{\lambda \rho }{\widehat{g}}^{\rho \nu },\hfill \end{array}$$

(116)

$$\begin{array}{ccc}\hfill {\widehat{\overline{\Gamma }}}^{\mu \nu }& =& -{\widehat{g}}^{\mu \lambda }{\overline{\Gamma }}_{\lambda \rho }{\widehat{g}}^{\rho \nu },\hfill \end{array}$$

(117)

$$\begin{array}{ccc}\hfill {\widehat{V}}^{\mu \nu }& =& {\widehat{g}}^{\mu \lambda }\left(-V_{\lambda \rho }+{\Gamma }_{\lambda \sigma }{\widehat{g}}^{\sigma \tau }{\overline{\Gamma }}_{\tau \rho }-{\overline{\Gamma }}_{\lambda \sigma }{\widehat{g}}^{\sigma \tau }{\Gamma }_{\tau \rho }\right){\widehat{g}}^{\rho \nu }.\hfill \end{array}$$

(118)

This contains the correct BRST transformation properties. For instance, we have the following:

$$\begin{array}{ccc}\hfill {\widehat{\Gamma }}^{\mu \nu }& =& -{\widehat{g}}^{\mu \lambda }\left(\xi ·\partial g_{\lambda \rho }+{\partial }_{\lambda }{\xi }^{\tau }{g}_{\tau \rho }+{\partial }_{\rho }{\xi }^{\tau }{g}_{\lambda \tau }\right){\widehat{g}}^{\rho \nu }\hfill \\ & =& \xi ·\partial {\widehat{g}}^{\mu \nu }-{\partial }_{\lambda }{\xi }^{\mu }{\widehat{g}}^{\lambda \nu }-{\partial }_{\lambda }{\xi }^{\nu }{\widehat{g}}^{\nu \lambda }={\delta }_{\xi }{\widehat{g}}^{\mu \nu }.\hfill \end{array}$$

(119)

Similarly, we obtain

$$\begin{array}{ccc}\hfill {\widehat{\overline{\Gamma }}}^{\mu \nu }& =& {\delta }_{\overline{\xi }}{\widehat{g}}^{\mu \nu },\hfill \end{array}$$

(120)

$$\begin{array}{ccc}\hfill {\widehat{V}}^{\mu \nu }& =& {\delta }_{\xi }{\delta }_{\overline{\xi }}{\widehat{g}}^{\mu \nu }.\hfill \end{array}$$

(121)

The simplest way to obtain (121) is to proceed as follows:

$$\begin{array}{ccc}\hfill {\delta }_{\xi }{\delta }_{\overline{\xi }}{\widehat{g}}^{\mu \nu }& =& {\delta }_{\xi }{\delta }_{\overline{\xi }}\left({\widehat{g}}^{\mu \lambda }{g}_{\lambda \rho }{\widehat{g}}^{\rho \nu }\right)\hfill \\ & =& {\delta }_{\xi }\left(-{\widehat{g}}^{\mu \lambda }{\delta }_{\overline{\xi }}{g}_{\lambda \rho }{\widehat{g}}^{\rho \nu }\right)\hfill \\ & =& -{\widehat{g}}^{\mu \lambda }{\delta }_{\xi }{\delta }_{\overline{\xi }}{g}_{\lambda \rho }{\widehat{g}}^{\rho \nu }+{\widehat{g}}^{\mu \lambda }{\delta }_{\xi }{g}_{\lambda \rho }{\widehat{g}}^{\rho \sigma }{\delta }_{\overline{\xi }}{g}_{\sigma \tau }{\widehat{g}}^{\tau \nu }-{\widehat{g}}^{\mu \lambda }{\delta }_{\overline{\xi }}{g}_{\lambda \rho }{\widehat{g}}^{\rho \sigma }{\delta }_{\xi }{g}_{\sigma \tau }{\widehat{g}}^{\tau \nu }\hfill \\ & =& -{\widehat{g}}^{\mu \lambda }{V}_{\lambda \rho }{\widehat{g}}^{\rho \nu }+{\widehat{g}}^{\mu \lambda }{\Gamma }_{\lambda \rho }{\widehat{g}}^{\rho \sigma }{\overline{\Gamma }}_{\sigma \tau }{\widehat{g}}^{\tau \nu }-{\widehat{g}}^{\mu \lambda }{\overline{\Gamma }}_{\lambda \rho }{\widehat{g}}^{\rho \sigma }{\Gamma }_{\sigma \tau }{\widehat{g}}^{\tau \nu }.\hfill \end{array}$$

(122)

4.6. ${\widehat{G}}^{MN}$

Now, we are ready to tackle the problem of the supermetric inverse. In ordinary Riemannian geometry, the inverse ${\widehat{g}}^{\mu \nu }$ of the metric is defined by the following: ${\widehat{g}}^{\mu \lambda }{g}_{\lambda \nu }={\delta }_{\nu }^{\mu }$. However, ${\widehat{g}}^{\mu \nu }$ can also be considered as a bi-vector such that

$$\begin{array}{c}\hfill \widehat{\mathfrak{g}}={\widehat{g}}^{\mu \nu }\frac{\partial }{\partial x^{\mu }}\vee \frac{\partial }{\partial x^{\nu }},\end{array}$$

(123)

is invariant under diffeomorphisms.

We can try to define the analog of (123) in the superspace, i.e.,

$$\begin{array}{c}\hfill \widehat{\mathbb{G}}={\widehat{G}}^{MN}\left(X\right)\frac{\tilde{\partial }}{\tilde{\partial }{X}^M}\vee \frac{\tilde{\partial }}{\tilde{\partial }{X}^N},\end{array}$$

(124)

where $\frac{\tilde{\partial }}{\tilde{\partial }{X}^M}=\left(\frac{\partial }{\partial x^{\mu }},\frac{\partial }{\partial \vartheta },\frac{\partial }{\partial \overline{\vartheta }}\right)$, and

$$\begin{array}{ccc}\hfill {\widehat{G}}^{\mu \nu }\left(X\right)& =& {\widehat{g}}^{\mu \nu }\left(x\right)+\vartheta {\widehat{\overline{\Gamma }}}^{\mu \nu }\left(x\right)+\overline{\vartheta }{\widehat{\Gamma }}^{\mu \nu }\left(x\right)+\vartheta \overline{\vartheta }{\widehat{V}}^{\mu \nu }\left(x\right),\hfill \\ \hfill {\widehat{G}}^{\mu \vartheta }\left(X\right)& =& {\widehat{\gamma }}^{\mu }\left(x\right)+\vartheta {\widehat{\overline{g}}}^{\mu }\left(x\right)+\overline{\vartheta }{\widehat{g}}^{\mu }\left(x\right)+\vartheta \overline{\vartheta }{\widehat{\Gamma }}^{\mu }\left(x\right)={\widehat{G}}^{\vartheta \mu }\left(X\right),\hfill \\ \hfill {\widehat{G}}^{\mu \overline{\vartheta }}\left(X\right)& =& {\widehat{\overline{\gamma }}}^{\mu }\left(x\right)+\vartheta {\widehat{\overline{f}}}^{\mu }\left(x\right)+\overline{\vartheta }{\widehat{f}}^{\mu }\left(x\right)+\vartheta \overline{\vartheta }{\widehat{\overline{\Gamma }}}^{\mu }\left(x\right)={\widehat{G}}^{\overline{\vartheta }\mu }\left(X\right),\hfill \\ \hfill \frac{1}{2}{\widehat{G}}^{\vartheta \overline{\vartheta }}\left(X\right)& =& \widehat{g}\left(x\right)+\vartheta \widehat{\overline{\gamma }}\left(x\right)+\overline{\vartheta }\widehat{\gamma }\left(x\right)+\vartheta \overline{\vartheta }\widehat{G}\left(x\right)=-\frac{1}{2}{\widehat{G}}^{\overline{\vartheta }\vartheta }\left(X\right).\hfill \end{array}$$

(125)

This suggests immediately the horizontality condition $\widehat{\mathfrak{G}}=\widehat{\mathfrak{g}}$, i.e.,

$$\begin{array}{c}\hfill {\tilde{\widehat{G}}}^{MN}\left(\tilde{X}\right)\frac{\partial }{\tilde{\partial }{\tilde{X}}^M}\vee \frac{\tilde{\partial }}{\tilde{\partial }{\tilde{X}}^N}={\widehat{g}}^{\mu \nu }\left(x\right)\frac{\partial }{\partial x^{\mu }}\vee \frac{\partial }{\partial x^{\nu }}.\end{array}$$

(126)

The partial derivative $\frac{\partial }{\tilde{\partial }{\tilde{X}}^M}$ can be derived from $\tilde{d}\tilde{X}$ by inverting the relation as follows:

$$\begin{array}{c}\hfill \tilde{d}{\tilde{X}}^M=\left(\begin{array}{ccc}{\delta }_{\nu }^{\mu }-\vartheta {\partial }_{\nu }{\overline{\xi }}^{\mu }-\overline{\vartheta }{\partial }_{\nu }{\xi }^{\mu }+\vartheta \overline{\vartheta }{\partial }_{\nu }{h}^{\mu }& -{\overline{\xi }}^{\mu }+\overline{\vartheta }{h}^{\mu }& -{\xi }^{\mu }-\vartheta h^{\mu }\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)\left(\begin{array}{c}d{x}^{\nu }\\ d\vartheta \\ d\overline{\vartheta }\end{array}\right).\end{array}$$

(127)

The matrix has the structure $\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)$, where $A,D$ are commuting square matrices, while $B,C$ are anticommuting rectangular ones (in this case $C=0$ and $D=1$). Its inverse is $\left(\begin{array}{cc}{A}^{-1}& -A^{-1}B\\ 0& 1\end{array}\right)$. Therefore, we have the following:

$$\begin{array}{ccc}\hfill \frac{\tilde{\partial }}{\tilde{\partial }{\tilde{x}}^{\mu }}& =& \frac{\partial }{\partial x^{\mu }}+\left(\vartheta {\partial }_{\mu }{\overline{\xi }}^{\nu }+\overline{\vartheta }{\partial }_{\mu }{\xi }^{\nu }-\vartheta \overline{\vartheta }\left({\partial }_{\mu }{h}^{\nu }+{\partial }_{\mu }{\overline{\xi }}^{\lambda }{\partial }_{\lambda }{\xi }^{\nu }-{\partial }_{\mu }{\xi }^{\lambda }{\partial }_{\lambda }{\overline{\xi }}^{\nu }\right)\right)\frac{\partial }{\partial x^{\nu }},\hfill \\ \hfill \frac{\tilde{\partial }}{\tilde{\partial }\tilde{\vartheta }}& =& \frac{\partial }{\partial \vartheta }+\left(-{\overline{\xi }}^{\nu }+\vartheta \overline{\xi }·\partial {\overline{\xi }}^{\nu }+\overline{\vartheta }\left(-h^{\nu }+\overline{\xi }·\partial {\xi }^{\nu }\right)+\vartheta \overline{\vartheta }\left(h·\partial {\overline{\xi }}^{\nu }-\overline{\xi }·\partial h^{\nu }-\overline{\xi }·\partial \overline{\xi }·\partial {\xi }^{\nu }+\overline{\xi }·\partial \xi ·\partial {\overline{\xi }}^{\nu }\right)\right)\frac{\partial }{\partial x^{\nu }},\hfill \\ \hfill \frac{\tilde{\partial }}{\tilde{\partial }\tilde{\overline{\vartheta }}}& =& \frac{\partial }{\partial \overline{\vartheta }}+\left(-{\xi }^{\nu }+\overline{\vartheta }\xi ·\partial {\xi }^{\nu }+\vartheta \left(h^{\nu }+\xi ·\partial {\overline{\xi }}^{\nu }\right)+\vartheta \overline{\vartheta }\left(h·\partial {\xi }^{\nu }-\xi ·\partial h^{\nu }-\xi ·\partial \overline{\xi }·\partial {\xi }^{\nu }+\xi ·\partial \xi ·\partial {\overline{\xi }}^{\nu }\right)\right)\frac{\partial }{\partial x^{\nu }}.\hfill \end{array}$$

(128)

The explicit form of the RHS can be found on Appendix B (see (A23)) from which we can now proceed to identify the various fields in (125).

From the $\frac{\partial }{\partial \vartheta }\vee \frac{\partial }{\partial \overline{\vartheta }}$ term, we obtain the following equation:

$$\begin{array}{c}\hfill \widehat{g}-\left(\vartheta \overline{\xi }+\overline{\vartheta }\xi -\vartheta \overline{\vartheta }h\right)·\partial \widehat{g}+\vartheta \overline{\vartheta }\xi \overline{\xi }·{\partial }^2\widehat{g}+\vartheta \left(\widehat{\overline{\gamma }}-\overline{\vartheta }\xi ·\partial \widehat{\overline{\gamma }}\right)+\overline{\vartheta }\left(\widehat{\gamma }-\vartheta \xi ·\widehat{\gamma }\right)+\vartheta \overline{\vartheta }\widehat{G}=0.\end{array}$$

(129)

from which we deduce the following:

$$\begin{array}{c}\hfill \widehat{g}=0,\widehat{\gamma }=0,\widehat{\overline{\gamma }}=0,\widehat{G}=0.\end{array}$$

(130)

Similarly, from the $\frac{\partial }{\partial x^{\mu }}\vee \frac{\partial }{\partial \vartheta }$ term we deduce the following:

$$\begin{array}{c}\hfill {\widehat{\gamma }}^{\mu }=0,{\widehat{g}}^{\mu }=0,{\widehat{\overline{g}}}^{\mu }=0,{\widehat{\Gamma }}^{\mu }=0,\end{array}$$

(131)

and from $\frac{\partial }{\partial x^{\mu }}\vee \frac{\partial }{\partial \overline{\vartheta }}$

$$\begin{array}{c}\hfill {\widehat{\overline{\gamma }}}^{\mu }=0,{\widehat{f}}^{\mu }=0,{\widehat{\overline{f}}}^{\mu }=0,{\widehat{\overline{\Gamma }}}^{\mu }=0.\end{array}$$

(132)

Therefore, only the components of ${\widehat{G}}^{\mu \nu }\left(X\right)$ do not vanish. Equation (A23) becomes the following:

$$\begin{array}{ccc}& & {\widehat{g}}^{\mu \nu }\left(x\right)\frac{\partial }{\partial x^{\mu }}\vee \frac{\partial }{\partial x^{\nu }}={\tilde{\widehat{G}}}^{MN}\left(\tilde{X}\right)\frac{\partial }{\tilde{\partial }{\tilde{X}}^M}\vee \frac{\tilde{\partial }}{\tilde{\partial }{\tilde{X}}^N}\hfill \\ & =& ({\widehat{g}}^{\mu \nu }-\left(\vartheta \overline{\xi }+\overline{\vartheta }\xi -\vartheta \overline{\vartheta }h\right)·\partial {\widehat{g}}^{\mu \nu }+\vartheta \overline{\vartheta }\xi \overline{\xi }·{\partial }^2{\widehat{g}}^{\mu \nu }+\vartheta \left({\widehat{\overline{\Gamma }}}^{\mu \nu }-\overline{\vartheta }\xi ·\partial {\widehat{\overline{\Gamma }}}^{\mu \nu }\right)\hfill \\ & & +\overline{\vartheta }\left({\widehat{\Gamma }}^{\mu \nu }-\vartheta \overline{\xi }·\partial {\widehat{\Gamma }}^{\mu \nu }\right)+\vartheta \overline{\vartheta }{\widehat{V}}^{\mu \nu }\left(x\right))\hfill \\ & & ·\left(\frac{\partial }{\partial x^{\mu }}+\left(\vartheta {\partial }_{\mu }{\overline{\xi }}^{\lambda }+\overline{\vartheta }{\partial }_{\mu }{\xi }^{\lambda }-\vartheta \overline{\vartheta }\left({\partial }_{\mu }{h}^{\lambda }+{\partial }_{\mu }{\overline{\xi }}^{\sigma }{\partial }_{\sigma }{\xi }^{\lambda }-{\partial }_{\mu }{\xi }^{\sigma }{\partial }_{\sigma }{\overline{\xi }}^{\lambda }\right)\right)\frac{\partial }{\partial x^{\lambda }}\right)\hfill \\ & & \vee \left(\frac{\partial }{\partial x^{\nu }}+\left(\vartheta {\partial }_{\nu }{\overline{\xi }}^{\rho }+\overline{\vartheta }{\partial }_{\nu }{\xi }^{\rho }-\vartheta \overline{\vartheta }\left({\partial }_{\nu }{h}^{\rho }+{\partial }_{\nu }{\overline{\xi }}^{\tau }{\partial }_{\tau }{\xi }^{\rho }-{\partial }_{\nu }{\xi }^{\tau }{\partial }_{\tau }{\overline{\xi }}^{\rho }\right)\right)\frac{\partial }{\partial x^{\rho }}\right).\hfill \end{array}$$

(133)

This implies the following:

$$\begin{array}{ccc}\hfill {\widehat{\Gamma }}^{\mu \nu }& =& \xi ·\partial {\widehat{g}}^{\mu \nu }-{\partial }_{\lambda }{\xi }^{\mu }{\widehat{g}}^{\lambda \nu }-{\partial }_{\lambda }{\xi }^{\nu }{\widehat{g}}^{\mu \lambda }={\delta }_{\xi }{\widehat{g}}^{\mu \nu },\hfill \\ \hfill {\widehat{\overline{\Gamma }}}^{\mu \nu }& =& \overline{\xi }·\partial {\widehat{g}}^{\mu \nu }-{\partial }_{\lambda }{\overline{\xi }}^{\mu }{\widehat{g}}^{\lambda \nu }-{\partial }_{\lambda }{\overline{\xi }}^{\nu }{\widehat{g}}^{\mu \lambda }={\delta }_{\overline{\xi }}{\widehat{g}}^{\mu \nu },\hfill \\ \hfill {\widehat{V}}^{\mu \nu }& =& {\delta }_{\xi }{\delta }_{\overline{\xi }}{\widehat{g}}^{\mu \nu }.\hfill \end{array}$$

(134)

If we impose ${\widehat{g}}^{\mu \nu }\left(x\right)$ to be the inverse of $g_{\mu \nu }\left(x\right)$, these are identical to Equations (116)–(118).

4.7. Super-Christoffel Symbols and Super-Riemann Tensor

From the previous results and from Appendix B.5, it is clear that we cannot define an inverse of $G_{MN}\left(X\right)$; therefore, we must give up the idea of mimicking Riemannian geometry in the superspace. However, no obstacles exist if we limit ourselves to $G_{\mu \nu }\left(X\right)$. We have seen that its inverse exists. Therefore, we can introduce a horizontal Riemannian geometry in the superspace, that is, a Riemannian geometry where the involved tensors are horizontal, i.e., they do not have components in the anticommuting directions. To start with, we can define the super-Christoffel symbol as follows:

$$\begin{array}{ccc}\hfill {\Gamma }_{\mu \nu }^{\lambda }& =& \frac{1}{2}{\widehat{G}}^{\lambda \kappa }\left({\partial }_{\mu }{G}_{\nu \kappa }+{\partial }_{\nu }{G}_{\mu \kappa }-{\partial }_{\kappa }{G}_{\mu \nu }\right)\hfill \end{array}$$

(135)

$$\begin{array}{ccc}& =& {\Gamma }_{\mu \nu }^{\lambda }+\vartheta {\overline{K}}_{\mu \nu }^{\lambda }+\overline{\vartheta }{K}_{\mu \nu }^{\lambda }+\vartheta \overline{\vartheta }{H}_{\mu \nu }^{\lambda },\hfill \end{array}$$

(136)

where

$$\begin{array}{ccc}\hfill {\overline{K}}_{\mu \nu }^{\lambda }& =& \frac{1}{2}\left({\widehat{\overline{\Gamma }}}^{\lambda \rho }\left({\partial }_{\mu }{g}_{\rho \nu }+{\partial }_{\nu }{g}_{\mu \rho }-{\partial }_{\rho }{g}_{\mu \nu }\right)+{\widehat{g}}^{\lambda \rho }\left({\partial }_{\mu }{\overline{\Gamma }}_{\rho \nu }+{\partial }_{\nu }{\overline{\Gamma }}_{\mu \rho }-{\partial }_{\rho }{\overline{\Gamma }}_{\mu \nu }\right)\right)\hfill \\ & =& \frac{1}{2}\left({\delta }_{\overline{\xi }}{\widehat{g}}^{\lambda \rho }\left({\partial }_{\mu }{g}_{\rho \nu }+{\partial }_{\nu }{g}_{\mu \rho }-{\partial }_{\rho }{g}_{\mu \nu }\right)+{\widehat{g}}^{\lambda \rho }\left({\partial }_{\mu }{\delta }_{\overline{\xi }}{g}_{\rho \nu }+{\partial }_{\nu }{\delta }_{\overline{\xi }}{g}_{\mu \rho }-{\partial }_{\rho }{\delta }_{\overline{\xi }}{g}_{\mu \nu }\right)\right)\hfill \\ & =& {\delta }_{\overline{\xi }}{\Gamma }_{\mu \nu }^{\lambda }.\hfill \end{array}$$

(137)

Similarly, we note the following:

$$\begin{array}{c}\hfill K_{\mu \nu }^{\lambda }={\delta }_{\xi }{\Gamma }_{\mu \nu }^{\lambda },\end{array}$$

(138)

and

$$\begin{array}{ccc}\hfill H_{\mu \nu }^{\lambda }& =& \frac{1}{2}\left({\widehat{V}}^{\lambda \rho }\left({\partial }_{\mu }{g}_{\rho \nu }+{\partial }_{\nu }{g}_{\mu \rho }-{\partial }_{\rho }{g}_{\mu \nu }\right)+{\widehat{g}}^{\lambda \rho }\left({\partial }_{\mu }{V}_{\rho \nu }+{\partial }_{\nu }{V}_{\mu \rho }-{\partial }_{\rho }{V}_{\mu \nu }\right)\right.\hfill \\ & & +\left.{\widehat{\Gamma }}^{\lambda \rho }\left({\partial }_{\mu }{\overline{\Gamma }}_{\rho \nu }+{\partial }_{\nu }{\overline{\Gamma }}_{\mu \rho }-{\partial }_{\rho }{\overline{\Gamma }}_{\mu \nu }\right)-{\widehat{\overline{\Gamma }}}^{\lambda \rho }\left({\partial }_{\mu }{\Gamma }_{\rho \nu }+{\partial }_{\nu }{\Gamma }_{\mu \rho }-{\partial }_{\rho }{\Gamma }_{\mu \nu }\right)\right)\hfill \\ & =& \frac{1}{2}\left({\delta }_{\xi }{\delta }_{\overline{\xi }}{\widehat{g}}^{\lambda \rho }\left({\partial }_{\mu }{g}_{\rho \nu }+{\partial }_{\nu }{g}_{\mu \rho }-{\partial }_{\rho }{g}_{\mu \nu }\right)+{\widehat{g}}^{\lambda \rho }\left({\partial }_{\mu }{\delta }_{\xi }{\delta }_{\overline{\xi }}{g}_{\rho \nu }+{\partial }_{\nu }{\delta }_{\xi }{\delta }_{\overline{\xi }}{g}_{\mu \rho }-{\partial }_{\rho }{\delta }_{\xi }{\delta }_{\overline{\xi }}{g}_{\mu \nu }\right)\right.\hfill \\ & & +\left.{\delta }_{\xi }{\widehat{g}}^{\lambda \rho }\left({\partial }_{\mu }{\delta }_{\overline{\xi }}{g}_{\rho \nu }+{\partial }_{\nu }{\delta }_{\overline{\xi }}{g}_{\mu \rho }-{\partial }_{\rho }{\delta }_{\overline{\xi }}{g}_{\mu \nu }\right)-{\delta }_{\overline{\xi }}{\widehat{g}}^{\lambda \rho }\left({\partial }_{\mu }{\delta }_{\xi }{g}_{\rho \nu }+{\partial }_{\nu }{\delta }_{\xi }{g}_{\mu \rho }-{\partial }_{\rho }{\delta }_{\xi }{g}_{\mu \nu }\right)\right)\hfill \\ & =& {\delta }_{\xi }{\delta }_{\overline{\xi }}{\Gamma }_{\mu \nu }^{\lambda },\hfill \end{array}$$

(139)

in agreement with (136).

The super-Riemann curvature is

$$\begin{array}{ccc}\hfill {\mathbf{R}}_{\mu \nu \lambda }{}^{\rho }& =& -{\partial }_{\mu }{\Gamma }_{\nu \lambda }^{\rho }+{\partial }_{\nu }{\Gamma }_{\mu \lambda }^{\rho }-{\Gamma }_{\mu \sigma }^{\rho }{\Gamma }_{\nu \lambda }^{\sigma }+{\Gamma }_{\nu \sigma }^{\rho }{\Gamma }_{\mu \lambda }^{\sigma }\hfill \\ & =& R_{\mu \nu \lambda }{}^{\rho }+\vartheta {\overline{\Omega }}_{\mu \nu \lambda }{}^{\rho }+\overline{\vartheta }{\Omega }_{\mu \nu \lambda }{}^{\rho }+\vartheta \overline{\vartheta }{S}_{\mu \nu \lambda }{}^{\rho },\hfill \end{array}$$

(140)

where

$$\begin{array}{ccc}\hfill {\overline{\Omega }}_{\mu \nu \lambda }{}^{\rho }& =& -{\partial }_{\mu }{\overline{K}}_{\nu \lambda }^{\rho }+{\partial }_{\nu }{\overline{K}}_{\mu \lambda }^{\rho }-{\Gamma }_{\mu \sigma }^{\rho }{\overline{K}}_{\nu \lambda }^{\sigma }+{\Gamma }_{\nu \sigma }^{\rho }{\overline{K}}_{\mu \lambda }^{\sigma }-{\overline{K}}_{\mu \sigma }^{\rho }{\Gamma }_{\nu \lambda }^{\sigma }+{\overline{K}}_{\nu \sigma }^{\rho }{\Gamma }_{\mu \lambda }^{\sigma }\hfill \\ & =& -{\partial }_{\mu }{{\delta }_{\overline{\xi }}\Gamma }_{\nu \lambda }^{\rho }+{\partial }_{\nu }{{\delta }_{\overline{\xi }}\Gamma }_{\mu \lambda }^{\rho }-{\Gamma }_{\mu \sigma }^{\rho }{{\delta }_{\overline{\xi }}\Gamma }_{\nu \lambda }^{\sigma }+{\Gamma }_{\nu \sigma }^{\rho }{{\delta }_{\overline{\xi }}\Gamma }_{\mu \lambda }^{\sigma }-{{\delta }_{\overline{\xi }}\Gamma }_{\mu \sigma }^{\rho }{\Gamma }_{\nu \lambda }^{\sigma }+{{\delta }_{\overline{\xi }}\Gamma }_{\nu \sigma }^{\rho }{\Gamma }_{\mu \lambda }^{\sigma }\hfill \\ & =& {\delta }_{\overline{\xi }}{R}_{\mu \nu \lambda }{}^{\rho }.\hfill \end{array}$$

(141)

Likewise,

$$\begin{array}{c}\hfill {\Omega }_{\mu \nu \lambda }{}^{\rho }={\delta }_{\xi }{R}_{\mu \nu \lambda }{}^{\rho },\end{array}$$

(142)

and

$$\begin{array}{ccc}\hfill S_{\mu \nu \lambda }{}^{\rho }& =& {\partial }_{\mu }{H}_{\nu \lambda }^{\rho }+{\partial }_{\nu }{H}_{\mu \lambda }^{\rho }-H_{\mu \sigma }^{\rho }{\Gamma }_{\nu \lambda }^{\sigma }+H_{\nu \sigma }^{\rho }{\Gamma }_{\mu \lambda }^{\sigma }-{\Gamma }_{\mu \sigma }^{\rho }{H}_{\nu \lambda }^{\sigma }+{\Gamma }_{\nu \sigma }^{\rho }{H}_{\mu \lambda }^{\sigma }\hfill \\ & & +{\overline{K}}_{\mu \sigma }^{\rho }{K}_{\nu \lambda }^{\sigma }-K_{\mu \sigma }^{\rho }{\overline{K}}_{\nu \lambda }^{\sigma }-{\overline{K}}_{\nu \sigma }^{\rho }{K}_{\mu \lambda }^{\sigma }+K_{\nu \sigma }^{\rho }{\overline{K}}_{\mu \lambda }^{\sigma }\hfill \\ & =& -{\partial }_{\mu }{{\delta }_{\xi }{\delta }_{\overline{\xi }}\Gamma }_{\nu \lambda }^{\rho }+{\partial }_{\nu }{{\delta }_{\xi }{\delta }_{\overline{\xi }}\Gamma }_{\mu \lambda }^{\rho }-{{\delta }_{\xi }{\delta }_{\overline{\xi }}\Gamma }_{\mu \sigma }^{\rho }{\Gamma }_{\nu \lambda }^{\sigma }+{{\delta }_{\xi }{\delta }_{\overline{\xi }}\Gamma }_{\nu \sigma }^{\rho }{\Gamma }_{\mu \lambda }^{\sigma }-{\Gamma }_{\mu \sigma }^{\rho }{{\delta }_{\xi }{\delta }_{\overline{\xi }}\Gamma }_{\nu \lambda }^{\sigma }+{\Gamma }_{\nu \sigma }^{\rho }{{\delta }_{\xi }{\delta }_{\overline{\xi }}\Gamma }_{\mu \lambda }^{\sigma }\hfill \\ & & +{{\delta }_{\overline{\xi }}\Gamma }_{\mu \sigma }^{\rho }{{\delta }_{\xi }\Gamma }_{\nu \lambda }^{\sigma }-{{\delta }_{\xi }\Gamma }_{\mu \sigma }^{\rho }{{\delta }_{\overline{\xi }}\Gamma }_{\nu \lambda }^{\sigma }-{{\delta }_{\overline{\xi }}\Gamma }_{\nu \sigma }^{\rho }{{\delta }_{\xi }\Gamma }_{\mu \lambda }^{\sigma }+{{\delta }_{\xi }\Gamma }_{\nu \sigma }^{\rho }{{\delta }_{\overline{\xi }}\Gamma }_{\mu \lambda }^{\sigma }\hfill \\ & =& {\delta }_{\xi }{\delta }_{\overline{\xi }}{R}_{\mu \nu \lambda }{}^{\rho }.\hfill \end{array}$$

(143)

This gives immediately the super-Ricci tensor

$$\begin{array}{ccc}\hfill {\mathbf{R}}_{\mu \lambda }\equiv {\mathbf{R}}_{\mu \nu \lambda }{}^{\nu }& =& R_{\mu \lambda }+\vartheta {\overline{\Omega }}_{\mu \lambda }+\overline{\vartheta }{\Omega }_{\mu \lambda }+\vartheta \overline{\vartheta }{S}_{\mu \lambda },\hfill \end{array}$$

(144)

with ${\overline{\Omega }}_{\mu \lambda }={\overline{\Omega }}_{\mu \nu \lambda }{}^{\nu },{\Omega }_{\mu \lambda }={\Omega }_{\mu \nu \lambda }{}^{\nu }$ and $S_{\mu \lambda }=S_{\mu \nu \lambda }{}^{\nu }$. Of course,

$$\begin{array}{c}\hfill {\overline{\Omega }}_{\mu \lambda }={\delta }_{\overline{\xi }}{R}_{\mu \lambda },{\Omega }_{\mu \lambda }={\delta }_{\xi }{R}_{\mu \lambda },\mathrm{and}{S}_{\mu \lambda }={\delta }_{\xi }{\delta }_{\overline{\xi }}{R}_{\mu \lambda }.\end{array}$$

(145)

The super-Ricci scalar is the following:

$$\begin{array}{c}\hfill \mathbf{R}\equiv {\widehat{G}}^{\mu \nu }{\mathbf{R}}_{\mu \nu }=R+\vartheta \overline{\Omega }+\overline{\vartheta }\Omega +\vartheta \overline{\vartheta }S.\end{array}$$

(146)

It is easy to show the following:

$$\begin{array}{c}\hfill \overline{\Omega }={\delta }_{\overline{\xi }}R,\Omega ={\delta }_{\xi }R,S={\delta }_{\xi }{\delta }_{\overline{\xi }}R.\end{array}$$

(147)

5. The Vielbein

If we want to include fermions in a theory in curved spacetime, we need frame fields. This section is devoted to introducing vielbein in the superspace. We define the supervierbein as the following d-vector 1-form:

$$\begin{array}{c}\hfill {\mathbb{E}}^a=E_M^a\left(X\right)\tilde{d}{X}^M,\end{array}$$

(148)

where

$$\begin{array}{ccc}\hfill E_{\mu }^a\left(X\right)& =& e_{\mu }^a\left(x\right)+\vartheta {\overline{\varphi }}_{\mu }^a\left(x\right)+\overline{\vartheta }{\varphi }_{\mu }^a\left(x\right)+\vartheta \overline{\vartheta }{f}_{\mu }^a\left(x\right),\hfill \\ \hfill E_{\vartheta }^a\left(X\right)& =& {\chi }^a\left(x\right)+\vartheta {\overline{C}}^a\left(x\right)+\overline{\vartheta }{C}^a\left(x\right)+\vartheta \overline{\vartheta }{\psi }^a\left(x\right),\hfill \\ \hfill E_{\overline{\vartheta }}^a\left(X\right)& =& {\lambda }^a\left(x\right)+\vartheta {\overline{D}}^a\left(x\right)+\overline{\vartheta }{D}^a\left(x\right)+\vartheta \overline{\vartheta }{\rho }^a\left(x\right).\hfill \end{array}$$

(149)

The natural horizontality condition is the following:

$$\begin{array}{c}\hfill {\tilde{E}}_M^a\left(\tilde{X}\right)\tilde{d}{\tilde{X}}^M=e_{\mu }^a\left(x\right)d{x}^{\mu }.\end{array}$$

(150)

This is the same condition as for a vector field. So, we immediately obtain the following results:

$$\begin{array}{ccc}\hfill {\varphi }_{\mu }^a& =& \xi ·\partial e_{\mu }^a+{\partial }_{\mu }{\xi }^{\lambda }{e}_{\lambda }^a={\delta }_{\xi }{e}_{\mu }^a,\hfill \end{array}$$

(151)

$$\begin{array}{ccc}\hfill {\overline{\varphi }}_{\mu }^a& =& \overline{\xi }·\partial e_{\mu }^a+{\partial }_{\mu }{\overline{\xi }}^{\lambda }{e}_{\lambda }^a={\delta }_{\overline{\xi }}{e}_{\mu }^a,\hfill \\ \hfill f_{\mu }^a& =& \xi ·\partial {\overline{\varphi }}_{\mu }^a-\overline{\xi }·\partial {\varphi }_{\mu }^a-\xi \overline{\xi }·{\partial }^2{e}_{\mu }^a+{\partial }_{\mu }{\overline{\xi }}^{\lambda }\xi ·\partial e_{\lambda }^a-{\partial }_{\mu }{\xi }^{\lambda }\overline{\xi }·\partial e_{\lambda }^a\hfill \end{array}$$

(152)

$$\begin{array}{ccc}& & -{\partial }_{\mu }{\overline{\xi }}^{\lambda }{\varphi }_{\lambda }^a+{\partial }_{\mu }{\xi }^{\lambda }{\overline{\varphi }}_{\lambda }^a-h·\partial e_{\mu }^a-{\partial }_{\mu }h·e^a=-{\delta }_{\overline{\xi }}{\varphi }_{\mu }^a={\delta }_{\xi }{\overline{\varphi }}_{\mu }^a,\hfill \end{array}$$

(153)

$$\begin{array}{ccc}\hfill {\chi }^a& =& e_{\mu }^a{\overline{\xi }}^{\mu },\hfill \end{array}$$

(154)

$$\begin{array}{ccc}\hfill C^a& =& -\xi ·\partial e_{\mu }^a{\overline{\xi }}^{\mu }+{\varphi }_{\mu }^a{\overline{\xi }}^{\mu }+\xi ·\partial {\chi }^a-h·e^a={\delta }_{\xi }{\chi }^a,\hfill \end{array}$$

(155)

$$\begin{array}{ccc}\hfill {\overline{C}}^a& =& -\overline{\xi }·\partial e_{\mu }^a{\overline{\xi }}^{\mu }+{\overline{\varphi }}_{\mu }^a{\overline{\xi }}^{\mu }+\overline{\xi }·\partial {\chi }^a={\delta }_{\overline{\xi }}{\chi }^a,\hfill \end{array}$$

(156)

$$\begin{array}{ccc}\hfill {\psi }^a& =& \xi \overline{\xi }·{\partial }^2{e}_{\mu }^a{\overline{\xi }}^{\mu }-\xi ·\partial {\overline{\varphi }}_{\mu }^a{\overline{\xi }}^{\mu }+\overline{\xi }·\partial {\varphi }_{\mu }^a{\overline{\xi }}^{\mu }+f_{\mu }^a{\overline{\xi }}^{\mu }-\xi \overline{\xi }·{\partial }^2{\chi }^a+\xi ·\partial {\overline{C}}^a-\overline{\xi }·\partial C^a\hfill \\ & & -\overline{\xi }·\partial e^a·h+{\overline{\varphi }}^a·h-h·\partial {\chi }^a+h·\partial e_{\mu }^a{\overline{\xi }}^{\mu }={\delta }_{\xi }{\overline{C}}^a=-{\delta }_{\overline{\xi }}{C}^a,\hfill \end{array}$$

(157)

and

$$\begin{array}{ccc}\hfill {\lambda }^a& =& e_{\mu }^a{\xi }^{\mu },\hfill \end{array}$$

(158)

$$\begin{array}{ccc}\hfill D^a& =& -\xi ·\partial e_{\mu }^a{\xi }^{\mu }+{\varphi }_{\mu }^a{\xi }^{\mu }+\xi ·\partial {\lambda }^a={\delta }_{\xi }{\lambda }^a,\hfill \end{array}$$

(159)

$$\begin{array}{ccc}\hfill {\overline{D}}^a& =& -\overline{\xi }·\partial e_{\mu }^a{\xi }^{\mu }+{\overline{\varphi }}_{\mu }^a{\xi }^{\mu }+\overline{\xi }·\partial {\lambda }^a+h·e^a={\delta }_{\overline{\xi }}{\lambda }^a,\hfill \end{array}$$

(160)

$$\begin{array}{ccc}\hfill {\rho }^a& =& \xi \overline{\xi }·{\partial }^2{e}_{\mu }^a{\xi }^{\mu }-\xi ·\partial {\overline{\varphi }}_{\mu }^a{\xi }^{\mu }+\overline{\xi }·\partial {\varphi }_{\mu }^a{\xi }^{\mu }+f_{\mu }^a{\xi }^{\mu }-\xi \overline{\xi }·{\partial }^2{\lambda }^a+\xi ·\partial {\overline{D}}^a-\overline{\xi }·\partial D^a\hfill \\ & & -\xi ·\partial e^a·h+{\varphi }^a·h-h·\partial {\lambda }^a+h·\partial e_{\mu }^a{\xi }^{\mu }={\delta }_{\xi }{\overline{D}}^a=-{\delta }_{\overline{\xi }}{D}^a.\hfill \end{array}$$

(161)

5.1. The Inverse Vielbein ${\widehat{E}}_a^{\mu }$

Here, we introduce the inverse vielbein ${\widehat{E}}_a^{\mu }$. Let us write it as the following:

$$\begin{array}{c}\hfill E_{\mu }^a\left(X\right)=e_{\lambda }^a\left(x\right)\left({\delta }_{\mu }^{\lambda }+\vartheta {\overline{\varphi }}_{\mu }^{\lambda }\left(x\right)+\overline{\vartheta }{\varphi }_{\mu }^{\lambda }\left(x\right)+\vartheta \overline{\vartheta }{f}_{\mu }^{\lambda }\left(x\right)\right)=e_{\lambda }^a{(1+X)}_{\mu }^{\lambda },\end{array}$$

(162)

where ${\varphi }_{\mu }^{\lambda }=e_a^{\lambda }{\varphi }_{\mu }^a,{\overline{\varphi }}_{\mu }^{\lambda }=e_a^{\lambda }{\overline{\varphi }}_{\mu }^a$ and $f_{\mu }^{\lambda }=e_a^{\lambda }{f}_{\lambda }^a$, and $e_a^{\lambda }\left(x\right)e_{\lambda }^b\left(x\right)={\delta }_a^b$. Then we define

$$\begin{array}{c}\hfill {\widehat{E}}_a^{\mu }={(1-X+X^2)}_{\lambda }^{\mu }{e}_a^{\lambda }.\end{array}$$

(163)

The following is evident:

$$\begin{array}{c}\hfill {\widehat{E}}_a^{\mu }{E}_{\mu }^b={\delta }_a^b.\end{array}$$

(164)

In terms of components, we have the following:

$$\begin{array}{ccc}\hfill {\widehat{\varphi }}_a^{\mu }& =& -{\widehat{e}}_b^{\mu }{\varphi }_{\lambda }^b{\widehat{e}}_a^{\lambda }=-{\widehat{e}}_b^{\mu }{\delta }_{\xi }{e}_{\lambda }^b{\widehat{e}}_a^{\lambda }={\delta }_{\xi }{\widehat{e}}_a^{\mu },\hfill \end{array}$$

(165)

$$\begin{array}{ccc}\hfill {\widehat{\overline{\varphi }}}_a^{\mu }& =& -{\widehat{e}}_b^{\mu }{\overline{\varphi }}_{\lambda }^b{\widehat{e}}_a^{\lambda }=-{\widehat{e}}_b^{\mu }{\delta }_{\overline{\xi }}{e}_{\lambda }^b{\widehat{e}}_a^{\lambda }={\delta }_{\overline{\xi }}{\widehat{e}}_a^{\mu },\hfill \end{array}$$

(166)

and

$$\begin{array}{c}\hfill {\widehat{f}}_a^{\mu }={\widehat{e}}_b^{\mu }\left(-f_{\lambda }^b-{\overline{\varphi }}_{\rho }^b{e}_c^{\rho }{\varphi }_{\lambda }^c+{\varphi }_{\rho }^b{e}_c^{\rho }{\overline{\varphi }}_{\lambda }^c\right){\widehat{e}}_a^{\lambda }={\delta }_{\xi }{\delta }_{\overline{\xi }}{\widehat{e}}_a^{\mu }.\end{array}$$

(167)

A simple way to prove the last step is as follows:

$$\begin{array}{ccc}\hfill {\delta }_{\xi }{\delta }_{\overline{\xi }}{\widehat{e}}_a^{\mu }& =& {\delta }_{\xi }\left(-{\widehat{e}}_b^{\mu }{\delta }_{\overline{\xi }}{e}_{\lambda }^b{\widehat{e}}_a^{\lambda }\right)=-{\delta }_{\xi }{\widehat{e}}_b^{\mu }{\delta }_{\overline{\xi }}{e}_{\lambda }^b{\widehat{e}}_a^{\lambda }-{\widehat{e}}_b^{\mu }{\delta }_{\xi }{\delta }_{\overline{\xi }}{e}_{\lambda }^b{\widehat{e}}_a^{\lambda }+{\widehat{e}}_b^{\mu }{\delta }_{\overline{\xi }}{e}_{\lambda }^b{\delta }_{\overline{\xi }}{\widehat{e}}_a^{\lambda }\hfill \\ & =& {\widehat{e}}_b^{\mu }\left(-f_{\lambda }^b-{\overline{\varphi }}_{\rho }^b{e}_c^{\rho }{\varphi }_{\lambda }^c+{\varphi }_{\rho }^b{e}_c^{\rho }{\overline{\varphi }}_{\lambda }^c\right){\widehat{e}}_a^{\lambda }={\widehat{f}}_a^{\mu }.\hfill \end{array}$$

(168)

5.2. The Inverse Supervielbein ${\widehat{E}}_a^M$

In this subsection, as we did for the supermetric, we try to define the inverse of the supervielbein. Analogous with what we did for the metric, we define the following:

$$\begin{array}{c}\hfill {\widehat{\mathbb{E}}}_a={\widehat{E}}_a^M\left(X\right)\frac{\partial }{\partial X^M},\end{array}$$

(169)

where

$$\begin{array}{ccc}\hfill {\widehat{E}}_a^{\mu }\left(X\right)& =& {\widehat{e}}_a^{\mu }\left(x\right)+\vartheta {\widehat{\overline{\varphi }}}_a^{\mu }\left(x\right)+\overline{\vartheta }{\widehat{\varphi }}_a^{\mu }\left(x\right)+\vartheta \overline{\vartheta }{\widehat{f}}_a^{\mu }\left(x\right),\hfill \\ \hfill {\widehat{E}}_a^{\vartheta }\left(X\right)& =& {\widehat{\chi }}_a\left(x\right)+\vartheta {\widehat{\overline{C}}}_a\left(x\right)+\overline{\vartheta }{\widehat{C}}_a\left(x\right)+\vartheta \overline{\vartheta }{\widehat{\psi }}_a\left(x\right),\hfill \\ \hfill {\widehat{E}}_a^{\overline{\vartheta }}\left(X\right)& =& {\widehat{\lambda }}_a\left(x\right)+\vartheta {\widehat{\overline{D}}}_a\left(x\right)+\overline{\vartheta }{\widehat{D}}_a\left(x\right)+\vartheta \overline{\vartheta }{\widehat{\rho }}_a\left(x\right),\hfill \end{array}$$

(170)

and impose the following horizontality condition:

$$\begin{array}{c}\hfill {\tilde{\widehat{E}}}_a^M\left(X\right)\frac{\partial }{\partial {\tilde{X}}^M}={\widehat{e}}_a^{\mu }\frac{\partial }{\partial x^{\mu }}.\end{array}$$

(171)

The explicit form of the LHS of this equation can be found in Appendix B, see Equation (A24). In the latter, the coefficient of $\frac{\partial }{\partial \vartheta }$ leads to the following:

$$\begin{array}{c}\hfill {\widehat{\chi }}_a={\widehat{\overline{C}}}_a={\widehat{C}}_a={\widehat{\psi }}_a=0,\end{array}$$

(172)

and the equation proportional to $\frac{\partial }{\partial \overline{\vartheta }}$ to the following:

$$\begin{array}{c}\hfill {\widehat{\lambda }}_a={\widehat{\overline{D}}}_a={\widehat{D}}_a={\widehat{\rho }}_a=0.\end{array}$$

(173)

Therefore, only the components of ${\widehat{E}}_a^{\mu }\left(X\right)$ are nonvanishing. The remaining equations give the following:

$$\begin{array}{ccc}\hfill {\widehat{\varphi }}_a^{\mu }& =& \xi ·{\widehat{e}}_a^{\mu }-{\partial }_{\lambda }{\xi }^{\mu }{\widehat{e}}_a^{\lambda }={\delta }_{\xi }{\widehat{e}}_a^{\mu },\hfill \\ \hfill {\widehat{\overline{\varphi }}}_a^{\mu }& =& \overline{\xi }·{\widehat{e}}_a^{\mu }-{\partial }_{\lambda }{\overline{\xi }}^{\mu }{\widehat{e}}_a^{\lambda }={\delta }_{\overline{\xi }}{\widehat{e}}_a^{\mu },\hfill \\ \hfill {\widehat{f}}_a^{\mu }& =& -\xi \overline{\xi }·{\partial }^2{\widehat{e}}_a^{\mu }-h·\partial e_a^{\mu }+{\partial }_{\lambda }{h}^{\mu }{e}_a^{\lambda }+\xi ·\partial {\widehat{\overline{\varphi }}}_a^{\mu }+{\partial }_{\lambda }{\xi }^{\mu }{\widehat{\overline{\varphi }}}_a^{\lambda }-\overline{\xi }·\partial {\widehat{\varphi }}_a^{\mu }-{\partial }_{\lambda }{\overline{\xi }}^{\mu }{\widehat{\varphi }}_a^{\lambda },\hfill \\ & & +{\widehat{e}}_a^{\lambda }{\partial }_{\lambda }\overline{\xi }·\partial {\xi }^{\mu }+\xi ·\partial {\widehat{e}}_a^{\lambda }{\partial }_{\lambda }{\overline{\xi }}^{\mu }-{\widehat{e}}_a^{\lambda }{\partial }_{\lambda }\xi ·\partial {\overline{\xi }}^{\mu }-\overline{\xi }·\partial {\widehat{e}}_a^{\lambda }{\partial }_{\lambda }{\xi }^{\mu }={\delta }_{\xi }{\delta }_{\overline{\xi }}{\widehat{e}}_a^{\mu }.\hfill \end{array}$$

(174)

If ${\widehat{e}}_a^{\mu }$ is the inverse of $e_{\mu }^a$, these formulas coincide with those of the previous subsection.

The results of this section confirm what was found in the previous subsection. In the superspace, it makes sense to consider only horizontal tensors, i.e., tensors whose components in the anticommuting directions vanish. We continue, therefore, to define a frame geometry with this characteristic.

5.3. The Spin Superconnection

The spin superconnection is defined as follows:

$$\begin{array}{ccc}\hfill {\Omega }_{\mu }^{ab}& =& \frac{1}{2}\left[{\widehat{E}}^{a\nu }\left({\partial }_{\mu }{E}_{\nu }^b-{\partial }_{\nu }{E}_{\mu }^b\right)-{\widehat{E}}^{b\nu }\left({\partial }_{\mu }{E}_{\nu }^a-{\partial }_{\nu }{E}_{\mu }^a\right)-{\widehat{E}}^{a\nu }{\widehat{E}}^{b\lambda }\left({\partial }_{\nu }{E}_{\lambda }^c-{\partial }_{\lambda }{E}_{\nu }^c\right)E_{c\mu }\right]\hfill \\ & =& {\omega }_{\mu }^{ab}+\vartheta {\overline{P}}_{\mu }^{ab}+\overline{\vartheta }{P}_{\mu }^{ab}+\vartheta \overline{\vartheta }{Q}_{\mu }^{ab},\hfill \end{array}$$

(175)

where ${\omega }_{\mu }^{ab}$ is the usual spin connection and the following holds:

$$\begin{array}{ccc}\hfill P_{\mu }^{ab}& =& \frac{1}{2}[{\widehat{e}}^{a\nu }{\partial }_{\mu }{\varphi }_{\nu }^b+{\widehat{\varphi }}^{a\nu }{\partial }_{\mu }{e}_{\nu }^b-{\widehat{e}}^{b\nu }{\partial }_{\mu }{\varphi }_{\nu }^a-{\widehat{\varphi }}^{b\nu }{\partial }_{\mu }{e}_{\nu }^a-\left({\widehat{e}}^{a\nu }{\widehat{\varphi }}^{b\lambda }+{\widehat{\varphi }}^{a\nu }{\widehat{e}}^{b\lambda }\right)\left({\partial }_{\nu }{e}_{\lambda }^c-{\partial }_{\lambda }{e}_{\nu }^c\right)e_{c\mu }\hfill \\ & & -{\widehat{e}}^{a\nu }{e}^{b\lambda }\left({\partial }_{\nu }{\varphi }_{\lambda }^c-{\partial }_{\lambda }{\varphi }_{\nu }^c\right)e_{c\mu }-{\widehat{e}}^{a\nu }{e}^{b\lambda }\left({\partial }_{\nu }{e}_{\lambda }^c-{\partial }_{\lambda }{e}_{\nu }^c\right){\varphi }_{c\mu }]\hfill \\ & =& \frac{1}{2}[{\widehat{e}}^{a\nu }{\partial }_{\mu }{\delta }_{\xi }{e}_{\nu }^b+{\delta }_{\xi }{\widehat{e}}^{a\nu }{\partial }_{\mu }{e}_{\nu }^b-{\widehat{e}}^{b\nu }{\partial }_{\mu }{\delta }_{\xi }{e}_{\nu }^a-{\delta }_{\xi }{\widehat{e}}^{b\nu }{\partial }_{\mu }{e}_{\nu }^a-\left({\widehat{e}}^{a\nu }{\delta }_{\xi }{\widehat{e}}^{b\lambda }+{\delta }_{\xi }{\widehat{e}}^{a\nu }{\widehat{e}}^{b\lambda }\right)\left({\partial }_{\nu }{e}_{\lambda }^c-{\partial }_{\lambda }{e}_{\nu }^c\right)e_{c\mu }\hfill \\ & & -{\widehat{e}}^{a\nu }{e}^{b\lambda }\left({\partial }_{\nu }{\delta }_{\xi }{e}_{\lambda }^c-{\partial }_{\lambda }{\delta }_{\xi }{e}_{\nu }^c\right)e_{c\mu }-{\widehat{e}}^{a\nu }{e}^{b\lambda }\left({\partial }_{\nu }{e}_{\lambda }^c-{\partial }_{\lambda }{e}_{\nu }^c\right){\delta }_{\xi }{e}_{c\mu }]={\delta }_{\xi }{\omega }_{\mu }^{ab}.\hfill \end{array}$$

(176)

where $e_{\mu }^a,{\varphi }_{\mu }^a,{\widehat{e}}^{a\mu },{\widehat{\varphi }}^{a\mu }$ were explained earlier in Section 5.1 and Section 5.2.

Similarly, we have the following:

$$\begin{array}{ccc}\hfill {\overline{P}}_{\mu }^{ab}& =& {\delta }_{\overline{\xi }}{\omega }_{\mu }^{ab},\hfill \end{array}$$

(177)

$$\begin{array}{ccc}\hfill {\overline{Q}}_{\mu }^{ab}& =& {\delta }_{\xi }{\delta }_{\overline{\xi }}{\omega }_{\mu }^{ab}.\hfill \end{array}$$

(178)

Thus, we have recovered the complete set of (anti)BRST transformations for the spin superconnection.

5.4. The Curvature

The 2-form supercurvature is the following:

$$\begin{array}{c}\hfill {\mathbb{R}}^{ab}={\mathbb{R}}_{\mu \nu }^{ab}d{x}^{\mu }\wedge d{x}^{\nu },\end{array}$$

(179)

where

$$\begin{array}{ccc}\hfill {\mathbb{R}}_{\mu \nu }^{ab}& =& {\partial }_{\mu }{\Omega }_{\nu }^{ab}-{\partial }_{\nu }{\Omega }_{\mu }^{ab}+{\Omega }_{\mu c}^a{\Omega }_{\nu }^{cb}-{\Omega }_{\nu c}^a{\Omega }_{\mu }^{cb}\hfill \\ & =& R_{\mu \nu }^{ab}+\vartheta {\overline{\Sigma }}_{\mu \nu }^{ab}+\overline{\vartheta }{\Sigma }_{\mu \nu }^{ab}+\vartheta \overline{\vartheta }{S}_{\mu \nu }^{ab}.\hfill \end{array}$$

(180)

$R_{\mu \nu }^{ab}$ is the usual spin connection curvature. Next, we have the following:

$$\begin{array}{ccc}\hfill {\Sigma }_{\mu \nu }^{ab}& =& {\partial }_{\mu }{P}_{\nu }^{ab}-{\partial }_{\nu }{P}_{\mu }^{ab}+P_{\mu }^{ac}{\omega }_{\nu c}{}^b+{\omega }_{\mu }^{ac}{P}_{\nu c}{}^b-P_{\nu }^{ac}{\omega }_{\mu c}{}^b-{\omega }_{\nu }^{ac}{P}_{\mu c}{}^b\hfill \\ & =& {\partial }_{\mu }{\delta }_{\xi }{\omega }_{\nu }^{ab}-{\partial }_{\nu }{\delta }_{\xi }{\omega }_{\mu }^{ab}+{\delta }_{\xi }{\omega }_{\mu }^{ac}{\omega }_{\nu c}{}^b+{\omega }_{\mu }^{ac}{\delta }_{\xi }{\omega }_{\nu c}{}^b-{\delta }_{\xi }{\omega }_{\nu }^{ac}{\omega }_{\mu c}{}^b-{\omega }_{\nu }^{ac}{\delta }_{\xi }{\omega }_{\mu c}{}^b\hfill \\ & =& {\delta }_{\xi }{R}_{\mu \nu }^{ab}.\hfill \end{array}$$

(181)

At the same time, we have the following identifications:

$$\begin{array}{ccc}\hfill {\overline{\Sigma }}_{\mu \nu }^{ab}& =& {\delta }_{\overline{\xi }}{R}_{\mu \nu }^{ab},\hfill \end{array}$$

(182)

$$\begin{array}{ccc}\hfill S_{\mu \nu }^{ab}& =& {\delta }_{\xi }{\delta }_{\overline{\xi }}{R}_{\mu \nu }^{ab}.\hfill \end{array}$$

(183)

5.5. Fermions

Fermion fields, under diffeomorphisms, behave like scalars. A Dirac fermion superfield has the following expansion:

$$\begin{array}{c}\hfill \Psi \left(X\right)=\psi \left(x\right)+\vartheta \overline{F}\left(x\right)+\overline{\vartheta }F\left(x\right)+\vartheta \overline{\vartheta }\Theta (x,\end{array}$$

(184)

$\psi ,F,\overline{F}$ and $\Theta$ are four-component complex column vector fields. The horizontality condition is the following:

$$\begin{array}{c}\hfill \tilde{\Psi }\left(\tilde{X}\right)=\psi \left(x\right).\end{array}$$

(185)

Repeating the analysis of the scalar superfield, we obtain the following:

$$\begin{array}{c}\hfill F\left(x\right)={\delta }_{\xi }\psi \left(x\right),\overline{F}\left(x\right)={\delta }_{\overline{\xi }}\psi \left(x\right),\Theta \left(x\right)={\delta }_{\xi }{\delta }_{\overline{\xi }}\psi \left(x\right).\end{array}$$

(186)

The covariant derivative of a vector superfield is as follows:

$$\begin{array}{c}\hfill {\mathbf{D}}_{\mu }\Psi =\left({\partial }_{\mu }+\frac{1}{2}{\Omega }_{\mu }\right)\Psi ,\end{array}$$

where ${\Omega }_{\mu }={\Omega }_{\mu }^{ab}{\Sigma }_{ab}$, and ${\Sigma }_{ab}=\frac{1}{4}[{\gamma }_a,{\gamma }_b]$ are the Lorentz generators.

The Lagrangian density for a Dirac superfield is the following:

$$\begin{array}{c}\hfill L=\sqrt{g}i\overline{\Psi }{\widehat{\gamma }}^{\mu }{\mathbf{D}}_{\mu }\Psi ,\end{array}$$

(187)

with ${\widehat{\gamma }}^{\mu }={\widehat{E}}_a^{\mu }{\gamma }_a$. The Lagrangian density L is invariant under Lorentz transformations and, up to total derivatives, under diffeomorphisms.

5.6. The Super-LORENTZ Transformations

The fermion superfield transforms under local Lorentz transformations as follows:

$$\begin{array}{c}\hfill {\delta }_{\Lambda }\Psi =\frac{1}{2}\Lambda \Psi ,\Lambda ={\Lambda }^{ab}\left(X\right){\Sigma }_{ab},\end{array}$$

(188)

where ${\Lambda }^{ab}\left(X\right)$ is an infinitesimal antisymmetric supermatrix with arbitrary entries:

$$\begin{array}{c}\hfill {\Lambda }^{ab}\left(X\right)={\lambda }^{ab}\left(x\right)+\vartheta {\overline{h}}^{ab}\left(x\right)+\overline{\vartheta }{h}^{ab}\left(x\right)+\vartheta \overline{\vartheta }{\Lambda }^{ab}\left(x\right).\end{array}$$

(189)

Local Lorentz transformations act on the supervielbein as follows:

$$\begin{array}{c}\hfill {\delta }_{\Lambda }{E}_{\mu }^a=E_{\mu }^b{\Lambda }_b{}^a,{\delta }_{\Lambda }{\widehat{E}}_a^{\mu }={\widehat{E}}_b^{\mu }{\Lambda }^b{}_a.\end{array}$$

(190)

Using the definition (175), one finds the following:

$$\begin{array}{c}\hfill {\delta }_{\Lambda }{\Omega }_{\mu }={\partial }_{\mu }\Lambda +\frac{1}{2}[{\Omega }_{\mu },\Lambda ],\end{array}$$

(191)

or

$$\begin{array}{c}\hfill {\delta }_{\Lambda }{\Omega }_{\mu }^{ab}={\partial }_{\mu }{\Lambda }^{ab}+{\Omega }_{\mu }^{cb}{\Lambda }_c{}^a+{\Omega }_{\mu }^{ac}{\Lambda }_c{}^b.\end{array}$$

(192)

With this and

$$\begin{array}{c}\hfill [{\widehat{\gamma }}^{\mu },\Lambda ]=2E_a^{\mu }{\Lambda }^{ab}{\gamma }_b,\end{array}$$

(193)

one can easily prove that (187) is invariant under local Lorentz transformations.

6. Superfield Formalism and Consistent Anomalies

The superfield formalism nicely applies to the description of consistent anomalies. In this section, we first summarize the definitions and properties of gauge anomalies and then we apply to them the superfield description. Basic material for the following algebraic approach to anomalies can be found in [11,38,39,57,58,59].

Here, formulas refer to a $\mathsf{d}$-dimensional spacetime $\mathsf{M}$ without boundary.

6.1. BRST, Descent Equations and Consistent Gauge Anomalies

The BRST operation $\mathfrak{s}$ in (3) is nilpotent. We represent with the same symbol $\mathfrak{s}$ the corresponding functional operator, i.e.,

$$\begin{array}{c}\hfill \mathfrak{s}=\int d^{d}x\left(\mathfrak{s}{A}_{\mu }^a\left(x\right)\frac{\partial }{\partial A_{\mu }^a\left(x\right)}+\mathfrak{s}{c}^a\left(x\right)\frac{\partial }{\partial c^a\left(x\right)}\right).\end{array}$$

(194)

To construct the descent equations, we start from a symmetric polynomial in the Lie algebra of degree n, i.e., $P_n(T^{a_1},...,T^{a_n})$, invariant under the adjoint transformations:

$$\begin{array}{c}\hfill P_n([X,T^{a_1}],...,T^{a_n})+\dots +P_n(T^{a_1},...,[X,T^{a_n}])=0,\end{array}$$

(195)

for any element X of the Lie algebra $\mathfrak{g}$. In many cases, these polynomials are symmetric traces of the generators in the corresponding representation as follows:

$$\begin{array}{c}\hfill P_n(T^{a_1},...,T^{a_n})=Str(T^{a_1}...T^{a_n}),\end{array}$$

(196)

($Str$ denotes the symmetric trace). With this, one can construct the 2n-form

$$\begin{array}{c}\hfill {\Delta }_{2n}\left(A\right)=P_n(F,F,\dots F),\end{array}$$

(197)

where $F=dA+\frac{1}{2}[A,A]$. It is easy to prove the following:

$$\begin{array}{c}\hfill P_n(F,F,\dots F)=d\left(n{\int }_0^{1}dt{P}_n(A,F_t,\dots ,F_t)\right)\equiv d{\Delta }_{2n-1}^{\left(0\right)}\left(A\right),\end{array}$$

(198)

where we have introduced the symbols $A_t=tA$ and its curvature $F_t=d{A}_t+\frac{1}{2}[A_t,A_t]$, where $0\le t\le 1$. In the above expressions, the product of the forms is understood to be the exterior product. It is important to recall that in order to prove Equation (198), one uses in an essential way the symmetry of $P_n$ and the graded commutativity of the exterior product of forms.

${\Delta }_{2n-1}^{\left(0\right)}\left(A\right)$ is often denoted also as $T{P}_n\left(A\right)$. The following is known as the transgression formula:

$$\begin{array}{c}\hfill T{P}_n\left(A\right)=n{\int }_0^{1}dt{P}_n(A,F_t,\dots ,F_t),\end{array}$$

(199)

Equation (198) is the first of a sequence of equations that can be proven:

$$\begin{array}{ccc}& & {\Delta }_{2n}\left(A\right)-d{\Delta }_{2n-1}^{\left(0\right)}\left(A\right)=0,\hfill \end{array}$$

(200)

$$\begin{array}{ccc}& & s{\Delta }_{2n-1}^{\left(0\right)}\left(A\right)+d{\Delta }_{2n-2}^{\left(1\right)}(A,c)=0,\hfill \end{array}$$

(201)

$$\begin{array}{ccc}& & s{\Delta }_{2n-2}^{\left(1\right)}(A,c)+d{\Delta }_{2n-3}^{\left(2\right)}(A,c)=0,\hfill \\ & & \cdots \cdots \hfill \end{array}$$

(202)

$$\begin{array}{ccc}& & s{\Delta }_0^{(2n-1)}\left(c\right)=0.\hfill \end{array}$$

(203)

All the expressions ${\Delta }_k^{\left(p\right)}(A,c)$ are polynomials of and $A,c,cdA,cdc$ and their commutators. The lower index k is the form degree, and the upper one p is the ghost number, i.e., the number of c factors. The last polynomial ${\Delta }_0^{(2n-1)}\left(c\right)$ is a 0-form and clearly a function only of c. All these polynomials have an explicit compact form. For instance, the next interesting case after Equation (200) is the following:

$$\begin{array}{c}\hfill s{\Delta }_{2n-1}\left(A\right)=-d\left(n(n-1){\int }_0^{1}dt(1-t)P_n(dc,A,F_t,\dots F_t)\right).\end{array}$$

(204)

This means, in particular, that integrating ${\Delta }_{2n-1}\left(A\right)$ over spacetime in $\mathsf{d}=2n-1$ dimensions, we obtain an invariant local expression. This gives the gauge CS action in any odd dimension. What matters here is that the RHS contains the general expression of the consistent gauge anomaly in $\mathsf{d}=2n-2$ dimension. Integrating (202) over spacetime, one obtains the following:

$$\begin{array}{ccc}\hfill s\mathcal{A}[c,A]& =& 0,\hfill \\ \hfill \mathcal{A}[c,A]& =& \int d^{d}x{\Delta }_d^{\left(1\right)}(c,A),\mathrm{where}\hfill \end{array}$$

(205)

$$\begin{array}{ccc}\hfill {\Delta }_d^1(c,A)& =& n(n-1){\int }_0^{1}dt(t-1)P_n(dc,A,F_t,\dots F_t),\hfill \end{array}$$

(206)

where $\mathcal{A}[c,A]$ identifies the anomaly up to an overall numerical coefficient.

Thus, the existence of chiral gauge anomalies relies on the existence of the adjoint–invariant polynomials $P_n$. One can prove that the so-obtained cocycles are non-trivial.

Although the above formulas are formally correct, one should remark that, in order to describe a consistent anomaly in a $\mathsf{d}=2n-2$ dimensional spacetime, we need two forms, $P_n(F,\dots ,F)$ and ${\Delta }_{2n-1}^{\left(0\right)}\left(A\right)$, which are identically vanishing. This is an unsatisfactory aspect of the previous purely algebraic approach. The superfield formalism overcomes this difficulty and gives automatically the anomaly as well as its descendants [60].

6.2. Superfield Formalism, BRST Transformations and Anomalies

For simplicity, we introduce only one anticommuting variable $\vartheta$ and consider the superconnection (where $\tilde{d}=d+\frac{\partial }{\partial \vartheta }d\vartheta$) as follows:

$$\begin{array}{c}\hfill \mathcal{A}=e^{-\vartheta c}(\tilde{d}+A)e^{\vartheta c}=A+\vartheta (dc+[A,c])+\left(c-\vartheta \frac{1}{2}[c,c]\right)d\vartheta \equiv \varphi +\eta d\vartheta .\end{array}$$

(207)

The supercurvature is the following:

$$\begin{array}{c}\hfill \mathcal{F}=e^{-\vartheta c}F{e}^{\vartheta c}=F+\vartheta [F,c].\end{array}$$

(208)

From these formulas, it is immediately visible that the derivative with respect to $\vartheta$ corresponds to the BRST transformation:

$$\begin{array}{ccc}\hfill \frac{\partial }{\partial \vartheta }\varphi & =& dc+[A,c]=D_{A}c=\mathfrak{s}A,\hfill \\ \hfill \frac{\partial }{\partial \vartheta }\eta & =& -\frac{1}{2}[c,c]=\mathfrak{s}c,\hfill \\ \hfill \frac{\partial }{\partial \vartheta }\mathcal{F}& =& [F,c]=\mathfrak{s}F.\hfill \end{array}$$

Let us now consider the transgression formula as follows:

$$\begin{array}{c}\hfill T{P}_n\left(\mathcal{A}\right)=n{\int }_0^{1}dt{P}_n(\mathcal{A},{\mathcal{F}}_t,\dots ,{\mathcal{F}}_t),\end{array}$$

(209)

in which we have replaced A everywhere with $\mathcal{A}$, and $F_t$ with ${\mathcal{F}}_t=t\tilde{d}\mathcal{A}+\frac{t^2}{2}[\mathcal{A},\mathcal{A}]$ (t is an auxiliary parameter varying from 0 to 1).

The claim is that (209) contains all the information about the gauge anomaly, including the explicit form of all its descendants. To see this, it is enough to expand the polyform $T{P}_n\left(\mathcal{A}\right)$ in component forms as follows:

$$\begin{array}{c}\hfill T{P}_n\left(\mathcal{A}\right)=\sum _{i=0}^{2n-1}{\tilde{\Delta }}_{2n-i-1}^{\left(i\right)}(\varphi ,\eta )\underset{i\mathrm{factors}}{\underbrace{d\vartheta \wedge \dots \wedge d\vartheta }},\end{array}$$

(210)

where $2n-i-1$ is the spacetime form degree. Notice that the wedge product is commutative for the $d\vartheta$ factors. Of course, both ${\tilde{\Delta }}_{2n-1}^{\left(0\right)}{\left(\varphi \right)|}_{\vartheta =0}=T{P}_n\left(A\right)$ and ${\tilde{\Delta }}_{2n-1}^{\left(0\right)}\left(\varphi \right)$ vanish in dimension $\mathsf{d}=2n-2$. However, the remaining forms are nonvanishing.

Let us extract the term ${\tilde{\Delta }}_{2n-2}^{\left(1\right)}(A,c)$:

$$\begin{array}{ccc}\hfill {\tilde{\Delta }}_{2n-2}^{\left(1\right)}(\varphi ,\eta )& =& n{\int }_0^{1}dt\left(P_n(\eta ,{\tilde{F}}_t,\dots ,{\tilde{F}}_t)+(n-1)P_n(\varphi ,t(d\eta -{\partial }_{\vartheta }\varphi )+t^2[\varphi ,\eta ],{\tilde{F}}_t,\dots ,{\tilde{F}}_t)\right),\hfill \end{array}$$

(211)

where ${\tilde{F}}_t=td\varphi +\frac{t^2}{2}[\varphi ,\varphi ]$. Let us take the $\vartheta =0$ part of this.

$$\begin{array}{c}\hfill {\tilde{\Delta }}_{2n-2}^{\left(1\right)}(A,c)=n{\int }_0^{1}dt\left(P_n(c,F_t,\dots ,F_t)+(n-1)P_n(A,(t^2-t)[A,c],F_t,\dots ,F_t)\right).\end{array}$$

(212)

Using ${\int }_0^{1}dt(1-t)\frac{d}{dt}f\left(t\right)={\int }_0^{1}dtf\left(t\right)$ when $f\left(0\right)=0$, we can rewrite this as follows:

$$\begin{array}{c}\hfill {\tilde{\Delta }}_{2n-2}^{\left(1\right)}(A,c)=n(n-1){\int }_0^{1}dt(1-t)\left(P_n(c,\frac{d{F}_t}{dt},\dots ,F_t)-P_n(A,[tA,c],F_t,\dots ,F_t)\right).\end{array}$$

(213)

Using $\frac{d{F}_t}{dt}=D_{tA}A$ and the ad invariance of $P_n$, we obtain finally the following:

$$\begin{array}{ccc}\hfill {\tilde{\Delta }}_{2n-2}^{\left(1\right)}(A,c)& =& d\left(n(n-1){\int }_0^{1}dt(1-t)(P_n(c,A,\dots ,F_t)\right)\hfill \\ & & -n(n-1){\int }_0^{1}dt(1-t)P_n(dc,A,F_t,\dots ,F_t).\hfill \end{array}$$

(214)

Therefore ${\tilde{\Delta }}_{2n-2}^{\left(1\right)}(A,c)$ coincides with the opposite of ${\Delta }_{2n-2}^{\left(1\right)}(A,c)$ up to a total spacetime derivative, which is irrelevant for the integrated anomaly.

The $\vartheta$ derivative of ${\tilde{\Delta }}_{2n-2}^{\left(1\right)}(\varphi ,\eta )$ is the BRST transformation of ${\tilde{\Delta }}_{2n-2}^{\left(1\right)}(A,c)$, and turns out to be a total spacetime derivative. This can be checked with an explicit calculation.

Remark 2.

The cocycles Δ and $\tilde{\Delta }$ may differ. For instance, in the case $\mathsf{d}=2,n=2$, we obtain the following:

$$\begin{array}{c}\hfill {\tilde{\Delta }}_2^{\left(1\right)}(A,c)=P_2(c,dA),{\tilde{\Delta }}_1^{\left(2\right)}(A,c)=\frac{1}{2}{P}_2(A,[c,c]),{\tilde{\Delta }}_0^{\left(3\right)}\left(c\right)=\frac{1}{6}{P}_2(c,[c,c]),\end{array}$$

(215)

while

$$\begin{array}{c}\hfill {\Delta }_2^{\left(1\right)}(A,c)=P_2(dc,A),{\Delta }_1^{\left(2\right)}\left(c\right)=P_2(dc,c),{\Delta }_0^{\left(3\right)}\left(c\right)=\frac{1}{6}{P}_2(c,[c,c]).\end{array}$$

(216)

This originates from the difference of a total derivative between ${\tilde{\Delta }}_2^{\left(1\right)}(A,c)$ and ${\Delta }_2^{\left(1\right)}(A,c)$.

6.3. Anomalies with Background Connection

The expressions of anomalies introduced so far are generally well defined in a local patch of spacetime, but they may not be globally well defined on the whole spacetime (they may not be basic forms in the language of fiber bundles, i.e., they may be well-defined forms in the total space but not in the base spacetime). To obtain globally well defined anomalies, we need to introduce a background connection $A_0$, which is invariant under BRST transformations: $\mathfrak{s}{A}_0=0$. The dynamical connection transforms, instead, in the usual way; see (3). We call F and $F_0$ the curvatures of A and $A_0$. Since the space of connections is affine, also ${\widehat{A}}_t=tA+(1-t)A_0$, with $0\le t\le 1$ is a connection. We call ${\widehat{F}}_t$ its curvature, which takes the values F and $F_0$ for $t=1$ and $t=0$, respectively.

The relevant connection is now the following:

$$\begin{array}{ccc}\hfill {\widehat{\mathcal{A}}}_t& =& t{e}^{-\vartheta c}(\tilde{d}+A)e^{\vartheta c}+(1-t)A_0=t\mathcal{A}+(1-t)A_0\hfill \\ & =& t\left(A+\vartheta (dc+[A,c])+\left(c-\vartheta \frac{1}{2}[c,c]\right)d\vartheta \right)+(1-t)A_0\equiv {\widehat{\varphi }}_t+{\widehat{\eta }}_{t}d\vartheta ,\hfill \end{array}$$

(217)

where ${\widehat{\varphi }}_t=t\varphi +(1-t)A_0$ and ${\widehat{\eta }}_t=t\eta$. We call ${\widehat{\mathcal{F}}}_t$ the curvature of ${\widehat{\mathcal{A}}}_t$. Notice that ${\widehat{\mathcal{F}}}_1=\mathcal{F}$ and ${\widehat{\mathcal{F}}}_0=F_0$, which is straightforward to be checked.

We start again from the transgression formula as follows:

$$\begin{array}{c}\hfill \mathcal{T}{P}_n(\mathcal{A},A_0)=n{\int }_0^{1}dt{P}_n(\mathcal{A}-A_0,{\widehat{\mathcal{F}}}_t,\dots ,{\widehat{\mathcal{F}}}_t).\end{array}$$

(218)

In the same way as before, one can prove that, if we assume the spacetime dimension is $\mathsf{d}=2n-2$, we have the following:

$$\begin{array}{c}\hfill \tilde{d}\mathcal{T}{P}_n(\mathcal{A},A_0)=P_n(\mathcal{F},\cdots ,\mathcal{F})-P_n(F_0,\dots ,F_0)=P_n(F,\cdots ,F)-P_n(F_0,\dots ,F_0)=0.\end{array}$$

(219)

As before, we decompose the following:

$$\begin{array}{c}\hfill \mathcal{T}{P}_n(\mathcal{A},A_0)=\sum _{i=0}^{2n-1}{\widehat{\Delta }}_{2n-i-1}^{\left(i\right)}(\varphi ,\eta ,A_0)\underset{i\mathrm{factors}}{\underbrace{d\vartheta \wedge \dots \wedge d\vartheta }}.\end{array}$$

(220)

The relevant term for the anomaly is ${\Delta }_{2n-2}^{\left(1\right)}(\varphi ,\eta ,A_0)$, i.e.,

$$\begin{array}{ccc}\hfill {\widehat{\Delta }}_{2n-2}^{\left(1\right)}(\varphi ,\eta ,A_0)& =& n{\int }_0^{1}dt\left[P_n\left(\varphi ,{\widehat{\Phi }}_t,\dots ,{\widehat{\Phi }}_t\right)+P_n\left(\mathcal{A}-A_0,\left(d_{{\widehat{\varphi }}_t}{\widehat{\eta }}_t-\frac{\partial }{\partial \vartheta }{\widehat{\varphi }}_t\right),{\widehat{\Phi }}_t,\dots ,{\widehat{\Phi }}_t\right)\right.\hfill \\ & & +\left.\dots +P_n\left(\mathcal{A}-A_0,{\widehat{\Phi }}_t,\dots ,{\widehat{\Phi }}_t,\left(d_{{\widehat{\varphi }}_t}{\widehat{\eta }}_t-\frac{\partial }{\partial \vartheta }{\widehat{\varphi }}_t\right)\right)\right],\hfill \end{array}$$

(221)

where $d_{{\widehat{\varphi }}_t}=d+[{\widehat{\varphi }}_t,·]$, and ${\widehat{\Phi }}_t=d{\widehat{\varphi }}_t+\frac{1}{2}[{\widehat{\varphi }}_t,{\widehat{\varphi }}_t]$. We have to select the $\vartheta =0$ part

$$\begin{array}{ccc}\hfill {\left.{\widehat{\Delta }}_{2n-2}^{\left(1\right)}(\varphi ,\eta ,A_0)\right|}_{\vartheta =0}& =& n{\int }_0^{1}dt\left[P_n\left(c,{\widehat{F}}_t,\dots ,{\widehat{F}}_t\right)-t(1-t)P_n\left(A-A_0,[A-A_0,c],{\widehat{F}}_t,\dots ,{\widehat{F}}_t\right)\right.\hfill \\ & & -\left.\dots -t(1-t)P_n\left(A-A_0,{\widehat{F}}_t,\dots ,{\widehat{F}}_t,[A-A_0,c]\right)\right].\hfill \end{array}$$

(222)

This is the chiral anomaly with background connection. It can be written in a more familiar form by rewriting the first term on the RHS, using ${\int }_0^{1}dt(1-t)\frac{d}{dt}f\left(t\right)+f\left(0\right)={\int }_0^{1}dtf\left(t\right)$:

$$\begin{array}{ccc}\hfill {\left.{\widehat{\Delta }}_{2n-2}^{\left(1\right)}(\varphi ,\eta ,A_0)\right|}_{\vartheta =0}& =& n{P}_n(c,F_0,\dots ,F_0)-n(n-1){\int }_0^{1}dt(1-t)P_n(d_{A_0}c,A-A_0,{\widehat{F}}_t,\dots ,{\widehat{F}}_t)\hfill \\ & & +d\left(n(n-1){\int }_0^{1}dt(1-t)P_n(c,A-A_0,{\widehat{F}}_t,\dots ,{\widehat{F}}_t)\right).\hfill \end{array}$$

(223)

Integrating over the spacetime $\mathsf{M}$, the last term drops out. If we set $A_0=0$, we recover the formula (214). We notice that, as expected, the RHS of (223) is a basic quantity.

6.4. Wess-Zumino Terms in Field Theories with the Superfield Method

In a gauge field theory with connection A, valued in a Lie algebra with generators $T^a$ and structure constants $f^{abc}$, an anomaly ${\mathcal{A}}^a$ must satisfy the WZ consistency conditions [61]:

$$\begin{array}{c}\hfill X^a\left(x\right){\mathcal{A}}^b\left(y\right)-X^b\left(y\right){\mathcal{A}}^a\left(x\right)+f^{abc}{\mathcal{A}}^c\left(x\right)\delta (x-y)=0,\end{array}$$

(224)

where

$$\begin{array}{c}\hfill X^a\left(x\right)={\partial }_{\mu }\frac{\delta }{\delta A_{\mu }^a\left(x\right)}+f^{abc}{A}_{\mu }^b\left(x\right)\frac{\delta }{\delta A_{\mu }^c\left(x\right)}.\end{array}$$

(225)

Equation (224) is integrability conditions. This means that one can find a functional of the fields ${\mathcal{B}}_{WZ}$ such that the following holds:

$$\begin{array}{c}\hfill X^a\left(x\right){\mathcal{B}}_{WZ}={\mathcal{A}}^a\left(x\right).\end{array}$$

(226)

In this section, we show how to construct a term ${\mathcal{B}}_{WZ}$ which, upon BRST variation, generates the following anomaly:

$$\begin{array}{c}\hfill \mathcal{A}=-{\int }_{\mathsf{M}}{c}^a\left(x\right){\mathcal{A}}^a\left(x\right)=n(n-1){\int }_{\mathsf{M}}{\int }_0^{1}dt(1-t)P_n(dc,A,F_t,\dots ,F_t),\end{array}$$

(227)

where $\mathsf{M}$ is the spacetime of dimension $\mathsf{d}=2n-2$, and $F_t=tdA+\frac{t^2}{2}[A,A]$.

This is possible provided we enlarge the set of fields of the theory by adding new fields as follows. Let us introduce a set of auxiliary fields $\sigma \left(x\right)={\sigma }^a\left(x\right)T^a$, which under a gauge transformations with parameters $\lambda \left(x\right)={\lambda }^a\left(x\right)T^a$, transform as the following:

$$\begin{array}{c}\hfill e^{\sigma \left(x\right)}⟶e^{{\sigma }^{\prime }\left(x\right)}=e^{-\lambda \left(x\right)}{e}^{\sigma \left(x\right)}.\end{array}$$

(228)

Using the Campbell–Hausdorff formula, this means $\delta \sigma \left(x\right)=-\lambda \left(x\right)-\frac{1}{2}[\lambda \left(x\right),\sigma \left(x\right)]+\dots$. Next, we pass to the infinitesimal transformations and replace the infinitesimal parameter $\lambda \left(x\right)$ with anticommuting fields $c\left(x\right)=c^a\left(x\right)T^a$. We have the following BRST transformations:

$$\begin{array}{c}\hfill \mathfrak{s}{e}^{\sigma \left(x\right)}=-c\left(x\right)e^{\sigma \left(x\right)},\mathfrak{s}\sigma \left(x\right)=-c\left(x\right)+\frac{1}{2}[\sigma \left(x\right),c\left(x\right)]-\frac{1}{12}[\sigma \left(x\right),[\sigma \left(x\right),c\left(x\right)]]+\dots \end{array}$$

(229)

Now, we use a superspace technique by adding to the spacetime coordinates $x^{\mu }$ and the anticommuting one $\vartheta$, but, simultaneously, we enlarge the spacetime with the addition of an auxiliary commuting parameter, s, $0\le s\le 1$. So the local coordinates in the superspace are $(x^{\mu },s,\vartheta )$. In particular, we have the following:

$$\begin{array}{c}\hfill e^{-s\left(\sigma +\vartheta \mathfrak{s}\sigma \right)}=e^{-s\sigma }+\vartheta \mathfrak{s}{e}^{-s\sigma }.\end{array}$$

(230)

On this superspace, the superconnection is the following:

$$\begin{array}{ccc}\hfill \tilde{\mathcal{A}}(x,s,\vartheta )& =& e^{-s\left(\sigma +\vartheta \mathfrak{s}\sigma \right)}{e}^{-\vartheta c}\left(\tilde{d}+A\right)e^{\vartheta c}{e}^{s\left(\sigma +\vartheta \mathfrak{s}\sigma \right)}\hfill \\ & =& e^{-s\left(\sigma +\vartheta \mathfrak{s}\sigma \right)}\left(\mathcal{A}+d+\frac{\partial }{\partial s}ds\right)e^{s\left(\sigma +\vartheta \mathfrak{s}\sigma \right)},\hfill \end{array}$$

(231)

where $\tilde{d}=d+\frac{\partial }{\partial s}ds+\frac{\partial }{\partial \vartheta }d\vartheta$, and $\mathcal{A}=A+\vartheta \left(dc+[A,c]\right)+\left(c-\vartheta cc\right)d\vartheta$. We decompose $\tilde{\mathcal{A}}$ as follows:

$$\begin{array}{ccc}\hfill \tilde{\mathcal{A}}(x,s,\vartheta )& =& \varphi (x,s,\vartheta )+{\varphi }_s(x,s,\vartheta )ds+{\varphi }_{\vartheta }(x,s,\vartheta )d\vartheta ,\hfill \\ \hfill \varphi (x,s,\vartheta )& =& A_s+\vartheta \left(d{c}_s+[A_s,c_s]\right),\hfill \\ \hfill {\varphi }_s(x,s,\vartheta )& =& \sigma +\vartheta \mathfrak{s}\sigma ,\hfill \\ \hfill {\varphi }_{\vartheta }(x,s,\vartheta )& =& c_s-\vartheta c_s{c}_s,\hfill \end{array}$$

(232)

where

$$\begin{array}{ccc}\hfill A_s& =& e^{-s\sigma }A{e}^{s\sigma }+e^{-s\sigma }d{e}^{s\sigma },\hfill \\ \hfill c_s& =& e^{-s\sigma }c{e}^{s\sigma }+e^{-s\sigma }\mathfrak{s}{e}^{s\sigma }.\hfill \end{array}$$

(233)

In particular, $A_0=A$ and $c_s$ interpolates between c and 0. Since the derivative with respect to $\vartheta$ corresponds to the BRST transformation, we deduce the following:

$$\begin{array}{c}\hfill \mathfrak{s}{A}_s=d{c}_s+[A_s,c_s],\mathfrak{s}{c}_s=-c_s{c}_s.\end{array}$$

(234)

Moreover, if we denote by $\tilde{\mathcal{F}},\mathcal{F}$ and F the curvatures of $\tilde{\mathcal{A}},\mathcal{A}$ and A, respectively, we have the following:

$$\begin{array}{c}\hfill \tilde{\mathcal{F}}=e^{-s\left(\sigma +\vartheta \mathfrak{s}\sigma \right)}\mathcal{F}{e}^{s\left(\sigma +\vartheta \mathfrak{s}\sigma \right)}=e^{-s\left(\sigma +\vartheta \mathfrak{s}\sigma \right)}{e}^{-\vartheta c}A{e}^{s\left(\sigma +\vartheta \mathfrak{s}\sigma \right)}{e}^{\vartheta c}.\end{array}$$

(235)

Now, suppose the spacetime $\mathsf{M}$ has dimension $\mathsf{d}$; choose any ad-invariant polynomial $P_n$ with $n=\frac{\mathsf{d}}{2}-1$. Then, the following holds:

$$\begin{array}{c}\hfill P_n(\tilde{\mathcal{F}},\dots ,\tilde{\mathcal{F}})=P_n(\mathcal{F},\dots ,\mathcal{F})=P_n(F,\dots ,F)=0,\end{array}$$

(236)

where the last equality holds for dimensional reasons. Now, we can write the following:

$$\begin{array}{c}\hfill P_n(\tilde{\mathcal{F}},\dots ,\tilde{\mathcal{F}})=\tilde{d}\left(n{\int }_0^{1}dt{P}_n(\tilde{\mathcal{A}},{\tilde{\mathcal{F}}}_t,\dots ,{\tilde{\mathcal{F}}}_t)\right)\equiv \tilde{d}\left(T{P}_n\left(\tilde{\mathcal{A}}\right)\right).\end{array}$$

(237)

For notational simplicity, let us set $\tilde{\mathcal{Q}}\left(\tilde{A}\right)\equiv T{P}_n\left(\tilde{A}\right)$ and decompose it in the various components according to the form degree as follows:

$$\begin{array}{c}\hfill \tilde{\mathcal{Q}}=\sum _{\stackrel{k,i,j}{k+i+j=2n-1}}{\tilde{\mathcal{Q}}}_{(k,i)}^j,{\tilde{\mathcal{Q}}}_{(k,i)}^j=\left(Q_{(k,i)}^j+\vartheta {\widehat{Q}}_{(k,i)}^j\right){\left(d\vartheta \right)}^j{\left(ds\right)}^i,\end{array}$$

(238)

where k denotes the form degree in spacetime, j is the ghost number, and i is either 0 or 1. Next, let us decompose the equation $\tilde{d}\tilde{\mathcal{Q}}=0$ into components, and select, in particular, the component $(2n-2,1,1)$, i.e., the following:

$$\begin{array}{c}\hfill 0=d{\tilde{\mathcal{Q}}}_{(2n-3,1)}^1+\frac{\partial }{\partial \vartheta }{\tilde{\mathcal{Q}}}_{(2n-2,1)}^{0}d\vartheta +\frac{\partial }{\partial s}{\tilde{\mathcal{Q}}}_{(2n-2,0)}^{1}ds,\end{array}$$

(239)

and let us integrate it over $\mathsf{M}$ and s. We obtain the following:

$$\begin{array}{c}\hfill 0={\int }_{\mathsf{M}}{\int }_0^{1}ds\mathfrak{s}{Q}_{(2n-2,1)}^0+{\int }_{\mathsf{M}}{\int }_0^{1}ds\frac{\partial }{\partial s}{Q}_{(2n-2,0)}^1.\end{array}$$

(240)

Since $Q_{(2n-2,0)}^1$ is linear in $c_s$ and $c_0=c,c_1=0$, we obtain finally the following:

$$\begin{array}{c}\hfill {\int }_{\mathsf{M}}{Q}_{(2n-2,0)}^1=\mathfrak{s}{\int }_{\mathsf{M}}{\int }_0^{1}ds{Q}_{(2n-2,1)}^0.\end{array}$$

(241)

Now, we remark that ${\int }_{\mathsf{M}}{Q}_{(2n-2,0)}^1$ is linear in c and coincides precisely with the anomaly. On the other hand, $Q_{(2n-2,1)}^0$ has the same expression as the anomaly with c replaced by $\sigma$ and A by $A_s$, i.e., the following:

$$\begin{array}{c}\hfill Q_{(2n-2,1)}^0(\sigma ,A_s)=n(n-1){\int }_0^{1}dt(1-t)P_n(d\sigma ,A_s,F_{s,t},\dots ,F_{s,t}),\end{array}$$

(242)

where $F_{s,t}=td{A}_s+\frac{t^2}{2}[A_s,A_s]$. We call

$$\begin{array}{c}\hfill {\mathcal{B}}_{WZ}={\int }_{\mathsf{M}}{\int }_0^{1}ds{Q}_{(2n-2,1)}^0,\end{array}$$

(243)

the Wess–Zumino term.

The existence of ${\mathcal{B}}_{WZ}$ for any anomaly seems to contradict the non-triviality of anomalies. This is not so because the price we have to pay to construct the term (243) is the introduction of the new fields ${\sigma }^a$, which are not present in the initial theory. The proof of the non-triviality of anomalies is based on a definite differential space formed by $c,A$ and their exterior derivatives and commutators, which constrains the anomaly to be a polynomial in these fields. Of course, in principle, it is not forbidden to enlarge the theory by adding new fields plus the WZ term. However, the resulting theory is different from the initial one. Moreover, the ${\sigma }^a$ fields have zero canonical dimension. This means that, except in $2\mathsf{d}$, it is possible to construct new invariant action terms with more than two derivatives, which renders renormalization problematic.

7. HS-YM Models and Superfield Method

In this section, we apply the superfield method to higher-spin5 Yang–Mills (HS-YM) models. These models are characterized by a local gauge symmetry, the higher spin symmetry, with infinite parameters, encompassing, in particular, both ordinary gauge transformations and diffeomorphisms. In a sense, they unify ordinary gauge and gravitational theories. This makes them interesting in themselves but particularly for the superfield method, to whose bases they seem to perfectly adhere. These models were only recently introduced in the literature, and they are largely unexplored. For this reason, we devote a rather long and hopefully sufficiently detailed introduction.

HS-YM models in Minkowski spacetime are formulated in terms of master fields $h_a(x,p)$, which are local in the phase space $(x,p)$, with $[{\widehat{x}}^{\mu },{\widehat{p}}_{\nu }]=i\hslash {\delta }_{\nu }^{\mu }$ (ℏ will be set to the value 1), where $\widehat{x},\widehat{p}$ are the operators whose classical symbols are $x,p$, according to the Weyl–Wignar quantization. The master field can be expanded in powers of p as follows:

$$\begin{array}{ccc}\hfill h_a(x,p)& =& \sum _{n=0}^{\infty }\frac{1}{n!}{h}_a^{{\mu }_1\dots ,{\mu }_n}\left(x\right)p_{{\mu }_1}\dots p_{{\mu }_n}\hfill \\ & =& A_a\left(x\right)+{\chi }_a^{\mu }\left(x\right)p_{\mu }+\frac{1}{2}{b}_a^{\mu \nu }\left(x\right)p_{\mu }{p}_{\nu }+\frac{1}{6}{c}_a^{\mu \nu \lambda }\left(x\right)p_{\mu }{p}_{\nu }{p}_{\lambda }+\dots ,\hfill \end{array}$$

(244)

where $h_a^{{\mu }_1\dots {\mu }_n}\left(x\right)$ are ordinary tensor fields, symmetric in ${\mu }_1,\dots ,{\mu }_n$. The indices ${\mu }_1,\dots ,{\mu }_n$ are upper (contravariant) Lorentz indices, ${\mu }_i=0,\dots ,d-1$. The index a is also a vector index, but it is of a different nature. In fact, it will be interpreted as a flat index and $h_a$ will be referred to as a frame-like master field. Of course, when the background metric is flat, all indices are on the same footing, but it is preferable to keep them distinct to facilitate the correct interpretation.

The master field $h^a(x,p)$ can undergo the following (HS) gauge transformations, whose infinitesimal parameter $\epsilon (x,p)$ is itself a master field:

$$\begin{array}{c}\hfill {\delta }_{\epsilon }{h}_a(x,p)={\partial }_a^x\epsilon (x,p)-i\left[h_a(x,p)\stackrel{\ast }{,}\epsilon (x,p)\right]\equiv {\mathcal{D}}_a^{\ast x}\epsilon (x,p),\end{array}$$

(245)

where we have introduced the following covariant derivative:

$$\begin{array}{c}\hfill {\mathcal{D}}_a^{\ast x}={\partial }_a^x-i\left[h_a(x,p)\stackrel{\ast }{,}\right].\end{array}$$

(246)

The * product is the Moyal product, defined by the following:

$$\begin{array}{c}\hfill f(x,p)\ast g(x,p)=f(x,p)e^{\frac{i}{2}\left({\overleftarrow{\partial }}_x{\overrightarrow{\partial }}_p-{\overleftarrow{\partial }}_p{\overrightarrow{\partial }}_x\right)}g(x,p),\end{array}$$

(247)

between two regular phase-space functions $f(x,p)$ and $g(x,p)$.

Like in ordinary gauge theories, we use the compact notation $d={\partial }_{a}d{x}^a,h=h_{a}d{x}^a$ and write (245) as the following:

$$\begin{array}{c}\hfill {\delta }_{\epsilon }h(x,p)=d\epsilon (x,p)-i\left[h(x,p)\stackrel{\ast }{,}\epsilon (x,p)\right]\equiv D\epsilon (x,p),\end{array}$$

(248)

where it is understood that $\left[h(x,p)\stackrel{\ast }{,}\epsilon (x,p)\right]=\left[h_a(x,p)\stackrel{\ast }{,}\epsilon (x,p)\right]d{x}^a$.

Next, we introduce the curvature notation as follows:

$$\begin{array}{c}\hfill G=dh-\frac{i}{2}\left[h\stackrel{\ast }{,}h\right],\end{array}$$

(249)

with the transformation property ${\delta }_{\epsilon }G=-i\left[G\stackrel{\ast }{,}\epsilon \right]$.

The action functionals we consider are integrated polynomials of G or of its components $G_{ab}$. To imitate ordinary non-Abelian gauge theories, we need a ‘trace property’, similar to the trace of polynomials of Lie algebra generators. In this framework, we have the following:

$$\begin{array}{c}\hfill \langle \langle f\ast g\rangle \rangle \equiv \int d^{d}x\int \frac{d^{d}p}{{\left(2\pi \right)}^d}f(x,p)\ast g(x,p)=\int d^{d}x\int \frac{d^{d}p}{{\left(2\pi \right)}^d}f(x,p)g(x,p)=\langle \langle g\ast f\rangle \rangle .\end{array}$$

(250)

From this, plus associativity, it follows that

$$\begin{array}{ccc}& & \langle \langle f_1\ast f_2\ast \dots \ast f_n\rangle \rangle =\langle \langle f_1\ast (f_2\ast \dots \ast f_n)\rangle \rangle \hfill \\ & & ={(-1)}^{{ϵ}_1({ϵ}_2+\dots +{ϵ}_n)}\langle \langle (f_2\ast \dots \ast f_n)\ast f_1\rangle \rangle ={(-1)}^{{ϵ}_1({ϵ}_2+\dots +{ϵ}_n)}\langle \langle f_2\ast \dots \ast f_n\ast f_1\rangle \rangle ,\hfill \end{array}$$

(251)

where ${ϵ}_i$ is the Grassmann degree of $f_i$ (this is usually referred to as cyclicity property).

This property holds also when the $f_i$ is valued in a (finite dimensional) Lie algebra, provided that the symbol $\langle \langle \rangle \rangle$ includes also the trace over the Lie algebra generators.

The HS Yang-Mills action. The curvature components, see (249), are as follows:

$$\begin{array}{c}\hfill G_{ab}={\partial }_a{h}_b-{\partial }_b{h}_a-i\left[h_a\stackrel{\ast }{,}{h}_b\right],\end{array}$$

(252)

with the following transformation rule:

$$\begin{array}{c}\hfill {\delta }_{\epsilon }{G}_{ab}=-i\left[G_{ab}\stackrel{\ast }{,}\epsilon \right].\end{array}$$

(253)

If we consider the functional $\langle \langle G^{ab}\ast G_{ab}\rangle \rangle$, it follows from the above that

$$\begin{array}{c}\hfill {\delta }_{\epsilon }\langle \langle G^{ab}\ast G_{ab}\rangle \rangle =-i\langle \langle G^{ab}\ast G_{ab}\ast \epsilon -\epsilon \ast G^{ab}\ast G_{ab}\rangle \rangle =0.\end{array}$$

(254)

Therefore,

$$\begin{array}{c}\hfill \mathcal{YM}\left(h\right)=-\frac{1}{4g^2}\langle \langle G^{ab}\ast G_{ab}\rangle \rangle ,\end{array}$$

(255)

is invariant under HS gauge transformations and it is a well-defined functional. This is the HS-YM-like action.

From (255), we obtain the following eom:

$$\begin{array}{c}\hfill {\partial }_b{G}^{ab}-i\left[h_b\stackrel{\ast }{,}{G}^{ab}\right]\equiv {\mathcal{D}}_b^{\ast }{G}^{ab}=0,\end{array}$$

(256)

which is covariant by construction under HS gauge transformations

$$\begin{array}{c}\hfill {\delta }_{\epsilon }\left({\mathcal{D}}_b^{\ast }{G}^{ab}\right)=-i[{\mathcal{D}}_b^{\ast }{G}^{ab},\epsilon ].\end{array}$$

(257)

We recall that ${\mathcal{D}}_b^{\ast }$ is the covariant *-derivative and $\epsilon$, the HS gauge parameter.

All that has been said so far can be repeated for non-Abelian models with minor changes. For simplicity, here, we limit ourselves to the Abelian case.

Gravitational interpretation. The novel property of HS YM-like theories is that, nothwithstanding their evident similarity with ordinary YM theories, they can describe also gravity. To see this, let us expand the gauge master parameter $\epsilon (x,p)$:

$$\begin{array}{ccc}\hfill \epsilon (x,p)& =& ϵ\left(x\right)+{\xi }^{\mu }\left(x\right)p_{\mu }+\frac{1}{2}{\Lambda }^{\mu \nu }\left(x\right)p_{\mu }{p}_{\nu }+\frac{1}{3!}{\Sigma }^{\mu \nu \lambda }\left(x\right)p_{\mu }{p}_{\nu }{p}_{\lambda }+\dots \hfill \end{array}$$

(258)

In Appendix C, we show that the parameter $ϵ\left(x\right)$ is the usual $U\left(1\right)$ gauge parameter for the field $A_a\left(x\right)$, while ${\xi }^{\mu }\left(x\right)$ is the parameter for general coordinate transformations, and ${\chi }_a^{\mu }\left(x\right)$ can be interpreted as the fluctuating inverse vielbein field.

Scalar and spinor master fields. To HS YM-like theories, we can couple matter-type fields of any spin, for instance, a complex multi-index boson field,

$$\begin{array}{c}\hfill \Phi (x,p)=\sum _{n=0}^{\infty }\frac{1}{n!}{\Phi }^{{\mu }_1{\mu }_2\dots {\mu }_n}\left(x\right)p_{{\mu }_1}{p}_{{\mu }_2}\dots p_{{\mu }_n},\end{array}$$

(259)

which, under a master gauge transformation (245), transforms like ${\delta }_{\epsilon }\Phi =i\epsilon \ast \Phi$. The covariant derivative is $D_a^{\ast }\Phi ={\partial }_a\Phi -i{h}_a\ast \Phi$ with the property ${\delta }_{\epsilon }{D}_a^{\ast }\Phi =i\epsilon \ast D_a^{\ast }\Phi$. With these properties, the kinetic action term $\frac{1}{2}\langle \langle {(D_{\ast }^a\Phi )}^{†}\ast D_a^{\ast }\Phi \rangle \rangle$ and the potential terms $\langle \langle {({\Phi }^{†}\ast \Phi )}_{\ast }^n\rangle \rangle$ are HS-gauge invariant.

In a quite similar manner, we can introduce master spinor fields,

$$\begin{array}{c}\hfill \Psi (x,p)=\sum _{n=0}^{\infty }\frac{1}{n!}{\Psi }_{\left(n\right)}^{{\mu }_1\dots {\mu }_n}\left(x\right)p_{{\mu }_1}\dots p_{{\mu }_n},\end{array}$$

(260)

where ${\Psi }_{\left(0\right)}$ is a Dirac field. The HS gauge transformations are ${\delta }_{\epsilon }\Psi =i\epsilon \ast \Psi$, and the covariant derivative is $D_a^{\ast }\Psi ={\partial }_a\Psi -i{h}_a\ast \Psi$ with ${\delta }_{\epsilon }(D_a^{\ast }\Psi )=i\epsilon \ast (D_a^{\ast }\Psi )$.