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 stage
1. 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,
and
.
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
SYM theory in 4D can be easily enlarged by extending the superspace with the addition of
, 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
, where
are ordinary commuting coordinates, while
and
are anticommuting:
, but commute with
. We make use of a generalized differential geometric notation: the exterior differential
is generalized to
. Correspondingly, mimicking the ordinary differential geometry, we introduce super-forms; for instance,
, where
are ordinary commuting intrinsic components, while
anticommute with each other and commute with
. 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
, on the other constitute separate, mutually commuting sets. When
acts on a super-function
, it is understood that the derivatives act on it from the left to form the components of a 1-super-form:
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 becomes juxtaposed to the analogous symbols of the super-form from the left to form the combinations , with the usual rule for the spacetime symbols, and , but , and and 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
dimensional Minkowski spacetime
, with connection
(
), valued in a Lie algebra
with anti-hermitean generators
, such that
. In the following, it is convenient to use the more compact form notation and represent the connection as a one-form
. The curvature and gauge transformation are as follows:
with
and
. 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
to an anticommuting ghost field
and define the BRST transform as follows
2:
As a consequence of this, we have the following:
which is sometime referred to as the
Russian formula [
7,
15,
38,
39]. Equation (
A10) is true, provided we assume the following:
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
. 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
, where
is anticommuting, and promoting the connection
A to a one-form superconnection
with the following expansions:
and two-form supercurvature
with
and
. Notice that since
,
, while
. Then, we impose the ‘horizontality’ condition:
The last two conditions imply the following:
Moreover, .
This means that we can identify
,
,
, and the
translation with the BRST transformation
, i.e.,
. 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:
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 coordinate mimics the vertical direction, as the curvature 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
and
c. For instance, in the Lorenz gauge, the Lagrangian density takes the following form:
where two new fields are introduced, the antighost field
and the Nakanishi–Lautrup field
. It is necessary to enlarge the algebra (
3) as follows:
in order to obtain a symmetry of (
10). At this point,
is invariant under a larger symmetry, whose transformations, besides (
3) and (
11), are the anti-BRST ones:
provided the following:
This is the Curci–Ferrari condition, ref. [
12].
The BRST and anti-BRST transformation are nilpotent and anticommute:
The superfield formalism applies well to this enlarged symmetry, provided we introduce another anticommuting coordinate,
:
. Here, we do not repeat the full derivation as in the previous case but simply introduce the supergauge transformation [
15,
40,
41,
42,
43]:
and generate the following superconnection:
where
and the hermitean operation is defined as follows:
while the
are real. Then, the superconnection is the following:
The one-form
is the following:
where
D denotes the covariant differential:
, etc., and the anticommuting functions
are the following:
together with the condition (
13). One can verify that the supercurvature
satisfies the following horizontality condition:
The BRST transformation correspond to
translations and the anti-BRST to
ones:
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
, formed by the 2-form
B, 1-form
A and 0-form
f, respectively. These are related in the following way. Given a covering
of the manifold
, we associate to each
a 2-form
. On a double intersection
, we have
. On the triple intersections
, we must have
(
denotes
B in
,
denotes
A in
, etc.). Finally, on the quadruple intersections
, the following integral cocycle condition must be satisfied by
f:
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
and
, respectively, are said to be gauge equivalent if they satisfy the following relations:
for the 1-form
C and the 0-form
.
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:
together with
, and
while
.
In these formulas, are anticommuting 1-forms, and 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
if the following constraint is satisfied:
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
, or, if it contains such a sub-algebra, the latter could be moded out. For instance, in Equations (
27) and (
28),
and
form an example of this type of subalgebra. It is easy to see that
and
can be consistently set equal to 0.
The Superfield Approach to Gerbes
We introduce superfields, whose lowest components are
and
f.
where
X denotes the superspace point and
, 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:
The first is suggested by the invariance of under , where is a 1-form, and the second by the invariance of due to the transformations , where is also a 1-form.
Using the second, we can eliminate many components of
in favor of the components of
:
Imposing (
34),
takes the following form:
where all the component fields on the RHSs are so far unrestricted. If we now impose (
33), we obtain the following further restrictions:
where
denote 1-forms with components
, respectively.
We also consider, instead of , the superfield , and, in particular, we replace A with .
From the previous equations, we can read off the BRST transformations of the independent component fields. Dropping the argument
and using the form notation for the BRST transformations, we have the following:
all the other
transformations being
trivial. For the anti-BRST transformations, we have the following:
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:
Then, the first equation of (
47) and the first of (
48) become the following:
Next, we define the following:
The remaining
s and
transformations become the following:
and
Moreover, we have the following CF-like condition:
These relations coincide with those of the 1-gerbe, provided that we make the following replacements: and .
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
.
Remark 1. One can also impose the horizontality condition , 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
:
An infinitesimal diffeomorphism is defined by means of a local parameter
:
. 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
, respectively, are the following:
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,
, and a
transformation, which transforms a scalar, vector, the metric and
in just the same way as
(these transformations are not rewritten here). In addition, we have the following cross-transformations:
It follows that the overall transformation
is nilpotent:
4.1. The Superfield Formalism
Our aim now is to reproduce the above transformations by means of the superfield formalism. The superspace coordinates are
, where
are the same anticommuting variables as above. A (super)diffeomorphism is represented by a superspace transformation
, where
3,
Here, is an ordinary diffeomorphism, are the generic anticommuting functions introduced before, and 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:
Now, we embed the scalar field
in a superfield
4The BRST interpretation is
. The horizontality condition, suggested by (
64), is the following:
Using (
63) with
, this becomes the following:
where · denotes index contraction. Then, (
66) implies the following:
where
.
Now, the BRST interpretation implies the following:
and
.
Inserting
and
into
C in (
68), we obtain the following:
This coincides with the expression of
C, (
68), if
Likewise,
which coincides with the expression of
, (
68), if the following holds:
Equating (
71) with (
73) we obtain the following:
which is possible if and only if the following CF condition is satisfied:
This condition is consistent, for applying and 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:
where
According to our prescription, horizontality means the following:
where
. Thus, we obtain the following:
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:
commute with
and
; and
anticommute with
. From (
A21), we obtain the following identifications:
and
One can see that
and
provided
which is possible if and only if the following CF condition is satisfied:
In particular,
can be rewritten as follows:
4.4. The Metric
The most important field for theories invariant under diffeomorphisms is the metric
. To represent its BRST transformation properties in the superfield formalism, we embed it in a supermetric
and form the symmetric 2-superdifferential as follows:
where ∨ denotes the symmetric tensor product and
while
, because the symmetric tensor product becomes antisymmetric for anticommuting variables:
.
The horizontality condition is obtained by requiring the following:
The explicit expression of the LHS of this equation can be found again in
Appendix B, from which the following identification follows:
Finally, we obtain the following:
This completes the verification of the horizontality condition. As expected, it leads to identifying the -and -superpartners of the metric as BRST and anti-BRST transforms.
4.5. Inverse of
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
, which is defined by first writing it as follows:
then in matrix terms as follows:
where
is the inverse of
g, i.e., the following:
and
is the ordinary metric inverse. Moreover,
This contains the correct BRST transformation properties. For instance, we have the following:
The simplest way to obtain (
121) is to proceed as follows:
4.6.
Now, we are ready to tackle the problem of the supermetric inverse. In ordinary Riemannian geometry, the inverse
of the metric is defined by the following:
. However,
can also be considered as a bi-vector such that
is invariant under diffeomorphisms.
We can try to define the analog of (
123) in the superspace, i.e.,
where
, and
This suggests immediately the horizontality condition
, i.e.,
The partial derivative
can be derived from
by inverting the relation as follows:
The matrix has the structure
, where
are commuting square matrices, while
are anticommuting rectangular ones (in this case
and
). Its inverse is
. Therefore, we have the following:
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
term, we obtain the following equation:
from which we deduce the following:
Similarly, from the
term we deduce the following:
and from
Therefore, only the components of
do not vanish. Equation (
A23) becomes the following:
This implies the following:
If we impose
to be the inverse of
, 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
; therefore, we must give up the idea of mimicking Riemannian geometry in the superspace. However, no obstacles exist if we limit ourselves to
. 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:
where
Similarly, we note the following:
and
in agreement with (
136).
The super-Riemann curvature is
where
This gives immediately the super-Ricci tensor
with
and
. Of course,
The super-Ricci scalar is the following:
It is easy to show the following:
7. HS-YM Models and Superfield Method
In this section, we apply the superfield method to higher-spin
5 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
, which are local in the phase space
, with
(
ℏ will be set to the value 1), where
are the operators whose classical symbols are
, according to the Weyl–Wignar quantization. The master field can be expanded in powers of
p as follows:
where
are ordinary tensor fields, symmetric in
. The indices
are upper (contravariant) Lorentz indices,
. 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
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
can undergo the following (HS) gauge transformations, whose infinitesimal parameter
is itself a master field:
where we have introduced the following covariant derivative:
The * product is the Moyal product, defined by the following:
between two regular phase-space functions
and
.
Like in ordinary gauge theories, we use the compact notation
and write (
245) as the following:
where it is understood that
.
Next, we introduce the curvature notation as follows:
with the transformation property
.
The action functionals we consider are integrated polynomials of
G or of its components
. 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:
From this, plus associativity, it follows that
where
is the Grassmann degree of
(this is usually referred to as
cyclicity property).
This property holds also when the is valued in a (finite dimensional) Lie algebra, provided that the symbol includes also the trace over the Lie algebra generators.
The HS Yang-Mills action. The curvature components, see (
249), are as follows:
with the following transformation rule:
If we consider the functional
, it follows from the above that
Therefore,
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:
which is covariant by construction under HS gauge transformations
We recall that is the covariant *-derivative and , 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
:
In
Appendix C, we show that the parameter
is the usual
gauge parameter for the field
, while
is the parameter for general coordinate transformations, and
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,
which, under a master gauge transformation (
245), transforms like
. The covariant derivative is
with the property
. With these properties, the kinetic action term
and the potential terms
are HS-gauge invariant.
In a quite similar manner, we can introduce master spinor fields,
where
is a Dirac field. The HS gauge transformations are
, and the covariant derivative is
with
.
With these properties, the following action integral is invariant.
BRST quantization of HS Yang-Mills. Fixing the Lorenz gauge with parameter
and applying the standard Faddeev–Popov approach, the quantum action becomes the following:
where
and
B are the ghost, antighost and Nakanishi–Lautrup master fields, respectively.
are anticommuting fields, while
B is commuting.
The action (
262) is invariant under the following BRST transformations:
which are nilpotent. In particular,
and
7.1. Anomalies in HS Theories
The effective action of HS-YM theories is functional of the master field
defined as follows:
where
are fermion master currents coupled to
. In a quantum theory, it may happen that the HS symmetry is not preserved (the Ward identity is violated):
but in this case, we have a consistency condition. Since the following holds,
we must have the following:
If we exclude the possibility that is trivial (i.e., , for an integrated local counterterm ), then we are faced with a true anomaly, which breaks the covariance of the effective action.
As illustrated above, in ordinary gauge theories, the form of chiral anomalies (and the CS action) is given by elegant formulas: the descent equations. It is natural to inquire whether in HS-YM theories there are similar formulas. We would be tempted to derive the relevant descent equations by mimicking the constructions of the previous sections. For instance, besides (
249), we can introduce the standard (ordinary gauge theory) definitions as follows:
The difference in the HS case is that, unlike in ordinary gauge theories, we cannot use graded commutativity. There is also another difficulty: the already signaled trace problem. We can use Equations (
250) and (
251), but we have to integrate over the full phase-space. Therefore, it is impossible to reproduce the unintegrated descent equations in the same way as in the ordinary gauge theories. The best we can do is try to reproduce each equation separately in integrated form. While it is rather simple, starting from the expression with
nG entries,
to derive the Chern–Simons action in
dimension
and prove that
, the remaining derivations are unfortunately lengthy and very cumbersome.
It is at this point that the superfield formulation of HS-YM theories comes to our rescue and gives us all these relations for free.
7.2. Superfield Formulation of HS YM
We introduce the master superfield 1-form as follows:
where all the component masterfields
are valued in the Lie algebra with generators
. The component master fields are as follows:
The supercurvature is as follows:
where
. Notice that the * commutator on the RHS persists also in the Abelian case. Setting the following,
the horizontality condition in this case is as follows:
The procedure is the same as in the ordinary YM case, except that the ordinary products are replaced by * products. From the vanishing of the
and
component of
, we obtain the following:
where
; see (
246). From the
,
and
components, we obtain the CF restriction as follows:
and
Let us recall again that the *-commutators are present also in the Abelian case we are considering. Expanding, for instance, the
c ghost master field in ordinary field components according to Equation (
258), we obtain the following:
where now all the component fields are anticommuting.
c is the gauge ghost field,
is the diffeomorphism ghost, and the subsequent are the HS ghost fields. For instance, the BRST transformations of
and
are the following:
For later use, we introduce the following notations:
and analogous formulas for
and
.
7.3. Derivation of HS-YM Anomalies
In order to limit the size and complication of the formulas, we limit ourselves to only one anticommuting variable
and consider the following superderivation (
):
where
. It follows that the supercurvature is the following:
From these formulas, it is immediately visible that the derivative with respect to
corresponds to the BRST transformations:
Let us start from the phase space integral with
n entries:
where
means integration over a phase space of dimension
. Let us introduce the following notations:
Then, consider the expression with
entries:
using the last of (
290). Then, using the first of (
290) together with (
251), this becomes the following:
Since they are integrated over a spacetime of dimension
, these expressions vanish. However, this is the way we identify the primitive functional action for HS CS (if the spacetime dimension is
). In other words, the HS CS action is the following:
where
means now integration over a phase space of dimension
.
Expressions relevant to anomalies appear if the spacetime dimension is
and the phase space one is
. In this case, the (unintegrated) expression
is a spacetime polyform of degree
:
i represents the ghost number and
is the spacetime form degree. Of course, its components of degree
and
vanish for dimensional reasons. Then, excluding vanishing factors and recalling that
corresponds to the BRST transform
, the integrand of Equation (
294) can be written as follows:
Now, this decomposition must be inserted inside the integration symbol
. This symbol needs a specification and must be interpreted as follows: any form
is understood to be multiplied by a trivial factor
, so that integration over spacetime makes sense. In conclusion, the following equation
means the following:
The anomaly is given by
, and the following:
where
. It follows that
Now, using
and
, one can easily find the more familiar expression of the anomaly as follows:
9. Conclusions and Comments
In this paper, we have reviewed (or proposed for the first time) several applications of the superfield method to represent the BRST and anti-BRST algebra in field theories with both gauge and diffeomorphism symmetries
8. We have shown, in many examples, that it correctly reproduces the transformations and, in more complicated cases, it helps finding them. Beyond that, we have shown that it is instrumental in practical applications, such as in the subject of consistent gauge anomalies and their integration (Wess–Zumino terms). In such instances, it can be of invaluable help as an effective algorithmic method, as opposed to laborious alternative trial and error methods.
In
Appendix A, we have reported a geometrical description of the BRST and anti-BRST algebras based on the geometry of principal fiber bundles and infinite dimensional groups of gauge transformations. Although elegant and with the appeal of classical geometry, this description can hardly be extended to the full quantum theory as defined by the perturbative expansion. The superfield description, which incorporates gauge, ghosts and auxiliary fields in a unique expression, seems instead to be more apt for this purpose, although a full attempt to exploit this possibility, to our best of knowledge, has never been made.
Elementary examples, in this sense, are the gauge-fixing action terms. Gauge fixing is a necessary step of quantization. There is freedom in choosing the gauge-fixing terms, except for a few obvious limitations: they must break completely the relevant gauge symmetry, have a zero ghost number, and then they must be real and of a canonical dimension not larger than 4 (in 4D) in order to guarantee unitarity and renormalizability. However, of course, they must be BRST (and, if possible, anti-BRST) invariant. Having at our disposal the superfield method, it is relatively easy to construct such terms. For instance, a well-known case is that of non-Abelian gauge theories. With reference to the notation in
Section 2.1, one such term is the trace of
, which, being the coefficient of
, is automatically BRST and anti-BRST invariant. It gives rise to the action term
. Another possibility is the trace of
and
, which give rise to the action terms
and
, respectively, and so on.
Against the backdrop of these well known examples, we would like to produce a few analogous terms in the case of gravity. The Einstein–Hilbert action for gravity in the superfield formalism can be written as follows [
54]:
On the same footing, we easily add matter fields and their interaction with gravity. Then, gauge-fixing terms invariant under BRST and anti-BRST transformations can be easily produced with the superfield formalism of
Section 4 and
Section 5, for instance, the coefficients of
in any local expression of the superfields of dimension 4. One such term is the following:
The dimensional counting is based on assigning to
dimension -1:
and to
dimension 0. Of course,
. This fixes the dimensions of all the component fields. For instance
Another possible term is the following:
which is, however, quadratic in
h.
Another way to obtain the BRST invariant term is to consider the following:
where
stands either for
or
. The only nonvanishing term is the following:
Using the supervierbein, one can construct the following:
and
It is remarkable that these gauge-fixing terms are generally more than quadratic in the fields, which implies, in particular, that not only linear gauge-fixing terms enjoy both BRST and anti-BRST symmetry. We see from this that aspects of BRST quantization of gauge and gravity field theories need further investigation and may reserve surprises. We plan to return to them.