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Remarks on Existence of Proper Action for Reducible Gauge Theories

In the field-antifield formalism, we review existence and uniqueness proofs for the proper action in the reducible case. We give two new existence proofs based on two resolution degrees called"reduced antifield number"and"shifted antifield number", respectively. In particular, we show that for every choice of gauge generators and their higher stage counterparts, there exists a proper action that implements them at the quadratic order in the auxiliary variables.


Introduction
This paper considers existence and uniqueness theorems for a proper action S of a general open reducible gauge theory in the field-antifield formalism [1,2]. We should stress that the term uniqueness theorem here should be understood in a generalized sense, i.e., as a theorem that specifies the natural arbitrariness/ambiguity of solutions (see Theorem 3.5). In our case, the uniqueness theorem states that all proper solutions to the classical master equation (S, S) = 0 can locally be reached from any other proper solution via a finite anticanonical transformation.
It is always possible to locally close or Abelianize the gauge algebra of a physical system by rotating and by adding off-shell contributions to the gauge generators [3,4]. Moreover, closed and Abelian theories have well-known proper actions, so why do we need another existence proof? The answer is that although closure and Abelianization are great theoretical tools, they have only limited practical use in field theory, where they often destroy space-time locality. Thus, ideally, one would like to establish the existence of the proper action S without tampering in any way with the original gauge algebra, and its higher-stage counterparts. This raises the question: How many terms in the action can one preserve while constructing the proper action? Specifically, we shall prove that the most general gauge generators R i α 0 and higher-stage counterparts Z α s−1 α s locally fit in unaltered form into a proper solution to the classical master equation, see Theorem 4.6. In other words, the Existence Theorem 4.6 states that any action S quad , which is quadratic in auxiliary variables, and which satisfies maximal rank conditions and Noether identities, can locally be completed into a proper action S = S quad +O (Φ * ) 2 , c 2 .
With this said, we must admit, that our current treatment will still make use of space-time non-local items, at least behind the screen. In particular, we will use the existence of a set of transversal and longitudinal fields ϕ i ≡ {ξ I ; θ A 0 }, which are often non-local, see Section 4.1. Also we should say that we will for simplicity use DeWitt's condensed index notation, where space-time locality is suppressed. So we work, strictly speaking, only with a finite number 2N of variables Γ A ≡ {Φ α ; Φ * α }.
There is of course an analogous story for the Hamiltonian/canonical formalism, which we omit for brevity. Also we do not discuss quantum corrections in this paper. Even today, there exists only a relatively limited number of general results in manifestly space-time local field-antifield formalism [5,6]. See Ref. [7] and Ref. [8] for a treatment of Yang-Mills type theories.
The standard existence proof is bases on two key elements. Firstly, the use of antifield number as resolution degree see Table 1. (The antifield number is sometimes called antighost number, because it is just the negative part of the ghost number.) Secondly, the use of a nilpotent acyclic Koszul-Tate operator s −1 of antifield number minus one, . . .
Historically, the existence and uniqueness theorems for an arbitrary irreducible gauge theory in an arbitrary basis were established in Ref. [9] and Ref. [10]. In the reducible case, an Sp(2) covariant proof (which at one point uses rotations of gauge generators R i α 0 and higher-stage counterparts Z α s−1 α s ) was given in Ref. [11]. A proof that does not change the gauge generators R i α 0 and higherstage counterparts Z α s−1 α s was given in Ref. [12]. The heart of the proof consists in showing the existence of a nilpotent and acyclic Koszul-Tate operator. This was done in Ref. [12] by referring to the analogous proof in the Hamiltonian setting [13,14]. We should warn, that it is easy to get the false impression after a first reading of Ref. [13], that the nilpotency of the Koszul-Tate operator is merely a consequence of its Grassmann-odd nature. This is of course wrong, and this is not what these authors are really saying, as becomes evident when reading Theorem 3 in Ref. [13]. (There are simple counterexamples of Grassmann-odd operators that satisfy a graded Leibniz rule, but are not nilpotent.) It should be stressed that the nilpotency of the Koszul-Tate operator is a non-trivial statement. It is equivalent to a L-stage tower of higher-stage Noether identities (2.1)-(2.3), and their consistency relations [15]. The first consistency relation appears at stage 2. The first few consistency relations read , (1.14) . . .
where "≈" means equality modulo equations of motion for ϕ i . The number of consistency relations grows with the reducibility stage. It is natural to wonder if these consistency relations can be satisfied without changing the original gauge generators R i α 0 and their higher-stage Z α s−1 α s counterparts [16,17]? This turns out to be possible, see Theorem 3 in Ref. [13] (or Theorem 10.2 in Ref. [14], or Lemma A.1 in this paper). The proof uses the acyclicity property of previous stages to prove the existence of a nilpotent extension of the Koszul-Tate operator up to a certain stage without changing R i α 0 and Z α s−1 α s .
Besides the gauge generators R i α 0 and Z α s−1 α s , s ∈ {1, . . . , L}, it turns out as an added bonus, that the antisymmetric first-stage structure function B ij α 1 in the first-stage Noether identity (2.2) can also be preserved. (This is because there is no consistency relations at first stage.) On the other hand, the proof does reveal that it will in general be necessary to change the given B iα s−2 α s structure functions in the higher-stage Noether identities (2.3) as The higher-stage Noether identities (2.3) are unaffected by such a change (1.15), due to the zero-stage Noether identity (2.1).
So what remains is to prove the acyclicity. Unfortunately, the treatments in Ref. [13] and Ref. [14] of acyclicity at higher stages are very brief. One of the main purposes to introduce reduced and shifted antifield number, is to properly spell out, in great detail and in a systematic way, an acyclicity proof for all stages.
It is very simple to motivate the construction of shifted antifield number "safn". Consider what happens if one raises the resolution degree of all the antifields Φ * α by 1 unit. Then the R i α 0 and the Z α s−1 α s terms, which are linear in the antifields, will have their resolution degree raised by 1, while the B iα s−2 α s terms and higher terms, which are at least quadratic in the antifields, will have their resolution degree raised by at least 2, and hence they become subleading, and can be dropped from the new "shifted" Koszul-Tate operator s (−1) . That is the good news! The bad news is that the very first term 16) in the Koszul-Tate operator now has resolution degree −2, which would be devastating. But there is a remedy. The term (1.16) is proportional to the original equations of motion. The remainder of our constructions is concerned with somehow assigning shifted antifield number ≥ 1 to the original equations of motion, so that the resolution degree of the term (1.16) becomes ≥ −1. In practice, we implement this by assigning shifted antifield number, safn(ξ I ) = 1, to a set of so-called transversal fields ξ I , see Section 4.1.
Another idea is to democratically assign resolution degree 1 to all antifields Φ * α . We call this degree for reduced antifield number. Then the terms will have resolution degree 0. Thus one would like the degree 0 sector to become the leading resolution sector. Again the B iα s−2 α s terms and higher terms, which are at least quadratic in the antifields, will become subleading in resolution degree, and can be dropped from the new "reduced" Koszul-Tate operator s −1 [0] . However, the very first term (1.16) now has resolution degree −1, which would be devastating. Again we cure this by assigning reduced antifield number, rafn(ξ I ) = 1, to a set of transversal fields ξ I , see Section 4.1.
The paper is organized as follows. In Section 2 we review the starting point of an arbitrary gauge theory, and introduce the field-antifield formalism. In Section 3 we consider an existence and uniqueness proof via the standard methods of antifield number and Koszul-Tate operator. And finally, in Section 4 we consider a complete existence proof via the new methods of reduced (shifted) antifield number, and reduced (shifted) Koszul-Tate operator, respectively. Note that the proofs in Section 3 are incomplete in the sense that one needs the new technology of reduced or shifted antifield number (which is developed in Section 4) to prove the acyclicity of the Koszul-Tate operator.
In an appendix B, we show for completeness how the B iα s−2 α s terms can be removed by (space-time non-local) change of the gauge generators R i α 0 and Z α s−1 α s .
General remarks about notation: An integer subindex without parenthesis refers to ordinary antifield number, while an integer subindex in round or square parenthesis refers to shifted or reduced antifield number, respectively. For instance, s −1 denotes the ordinary Koszul-Tate operator, which carries antifield number afn(s −1 ) = −1, while s (−1) is the shifted Koszul-Tate operator, which on the other hand carries shifted antifield number safn(s (−1) ) = −1. Strong equality "=" and weak equality "≈" refer to off-shell and on-shell equality with respect to the equations of motion for the original fields ϕ i , respectively.

Starting Point
In its purest form, the starting point for quantization is just an action S 0 = S 0 (ϕ), which depends on a set of fields ϕ i , i ∈ {1, . . . , n ≡ m −1 }. We shall hereafter refer to S 0 and ϕ i as the original action and the original fields, respectively.
Under some mild regularity assumptions, it is possible to recast the starting point into a form that we will use in this paper. (For a collection of starting points in the irreducible case, see Postulates 2-2 ′′′ in Ref. [10].) Explicitly, we will assume that all the gauge-symmetries, and in the reducible case, all the gauge(-for-gauge) s -symmetries, s ∈ {1, . . . , L}, have been properly identified. This means, in terms of formulas, that there should be given gauge-generators R i α 0 ≡ Z α −1 α 0 , and gauge(-for-gauge) sgenerators Z α s−1 α s , s ∈ {1, . . . , L}, such that where the indices runs over the following sets The letter L denotes the number of reducibility stages of the theory. A theory with reducibility stage L = 0 equal to zero is by definition an irreducible theory. For each stage s ∈ {0, . . . , L}, the number of gauge(-for-gauge) s -symmetries is denoted m s . It is convenient to define multiplicities m s := 0 for s > L. (Or, the other way around, L = min{s|∀r > s : m r = 0}.) The rank conditions read

6)
near the stationary ϕ-surface, i.e., the ϕ-surface of extremals for S 0 . Here we have defined The second equality in eq. (2.7) yields a recursion relation for M s . Note that M s = 0 for s > L. The word near means in some tubular ϕ-neighborhood of the stationary ϕ-surface. Note that it is not enough to impose rank conditions only at the stationary ϕ-surface, because the rank could jump once one leaves the stationary ϕ-surface. This is not allowed. Hence one must impose the rank conditions in an open ϕ-neighborhood of the stationary ϕ-surface.
A priori (outside the field-antifield formalism) it could in principle happen that the given B ji α 1 structure function in the first-stage Noether identity (2.2) does not have i ↔ j skewsymmetry (1.6). Nevertheless, one may show that there locally always exists a i ↔ j skewsymmetric B ji α 1 structure function satisfying the first-stage Noether identity (2.2), cf. Appendix C. On the other hand, within the field-antifield formalism, the B ji α 1 structure function is manifestly i ↔ j skewsymmetric, since it arises from an action term 1 2 (−1) ε j ϕ * j ϕ * i B ij α 1 c α 1 , cf. Section 2.2. We shall therefore for simplicity assume from now on, that the B ji α 1 structure function, that appears in the first-stage Noether identity (2.2), is a i ↔ j skewsymmetric tensor, that is defined in at least some tubular ϕ-neighborhood of the stationary ϕ-surface.
The choices of gauge(-for-gauge) s -generators Z α s−1 α s , s ∈ {0, . . . , L}, are not unique. An arbitrary other choice (with the same multiplicities) is locally given as To summarize, we shall assume that some gauge(-for-gauge) s -generators, Z α s−1 α s , s ∈ {0, . . . , L}, have been given to us (perhaps outside the field-antifield formalism), and they are hereafter considered as a part of the starting point.

Field-Antifield Formulation
Let us now reformulate the problem in the field-antifield language [1,2]. We shall for simplicity only consider the minimal sector. (The non-minimal sector, which is needed for gauge-fixing, can be treated by similar methods.) The minimal content of fields Φ α , α ∈ {1, . . . , N }, for a gauge theory of reducible stage L, is The c α 0 fields are the ghosts, or in a systematical terminology, the stage-zero ghosts. The c α s fields are the (ghost-for) s -ghosts, or stage-s ghost, s ∈ {1, . . . , L}. For Grassmann parity and ghost number assignments, see Table 1. To simplify notation, the stage s of a ghost c α s can only be identified though its index-variable α s . (Hopefully, this slight misuse of notation does not lead to confusion.) It is tempting to call the original fields ϕ i for stage-minus-one ghosts c α −1 ≡ ϕ i , but we shall not do so. From now on, the letter "c" without indices, and the word ghost, will always refer to a ghost with strictly positive ghost number. In particular, the symbol c * will not refer to original antifields ϕ * , but only the antifields for the ghost "c", i.e., gh(c * ) ≤ −2. We will collectively refer to ghosts "c" and antifields "Φ * " as auxiliary variables. Auxiliary variables are characterized by non-zero ghost number.
Remark: Mathematically, the various stucture functions, such as, e.g., R i α 0 = R i α 0 (ϕ) and B ji α 1 = B ji α 1 (ϕ), are tensors, or sections of appropriate vector bundles over the ϕ-basemanifold. They should be defined in at least some tubular ϕ-neighborhood of the stationary ϕ-surface. The auxiliary variables can be thought of as a local basis for the corresponding vector bundle. For instance, the antifield transforms as a co-vector under general coordinate transformations ϕ i → ϕ ′j , and can thus be identified with a local basis ∂ i for vector fields X = X i ∂ i . On the other hand, the indices α s , s ∈ {0, . . . , L}, do only transform under ridig ϕ-independent transformations, and the ghost c α s and their antifields c * α s can often be taken as global coordinates.
The total number N of fields is For each field Φ α , one introduces an antifield Φ * α of opposite Grassmann number ε(Φ * α ) = ε(Φ α )+ 1 and of ghost number gh(Φ * α ) = −gh(Φ α )−1. The antibracket is defined as The problem that we address in this paper is the existence of a proper solution S with ghost number zero gh(S) = 0 (2.14) to the classical master equation (S, S) = 0 , (2.15) such that S has the correct original limit 16) and satisfies the properness condition rank( at the stationary ϕ i -surface, when all the auxiliary variables are put to zero, Φ * α = 0 and c α s = 0, . . , 2N }, is a collective notation for both fields Φ α and antifields Φ * α . The half rank N is the maximal possible for a solution S to the classical master equation. The properness condition (2.17) is important, because it implies (after the non-minimal sector has been included) that the gauge-fixed proper action S(Φ, Φ * = ∂Ψ/∂Φ) is free of flat directions.
We will require one more condition besides eqs. (2.14)-(2.17). It will encode the gauge(-for-gauge) s symmetries, s ∈ {0, . . . , L}, into the proper action S. To explain it, lets us divide an arbitrary proper solution S into two parts, The first part S quad contains all terms that are at most quadratic in auxiliary variables, while the second part S non−quad = O (Φ * ) 2 , c 2 contains all terms that are at least cubic in auxiliary variables, which actually means all terms at least quadratic in ghosts or antifields, due to ghost number conservation (2.14). Because of ghost number conservation (2.14) and the original limit requirement (2.16), the action S quad must be of the form In particular, there are no action terms that are linear in the auxiliary variables. A priori the struc- iα s−2 α s structure functions. We will therefore demand that the structure functions R i α 0 and Z α s−1 α s in eq. (2.19) are the same as the given gauge generator R i α 0 and gauge(-for-gauge) s -generators Z α s−1 α s in the Noether identities (2.1)-(2.3), respectively.
With the above identification, the rank of the S quad part (2.19) of the Hessian is rank( at the stationary ϕ-surface as a result of the rank conditions (2.5) and (2.6). So the full action S = S quad + S non−quad is then guaranteed to be proper. The properness condition for the solution S at the stationary ϕ-surface implies, by continuity, properness of the S solution in some sufficiently small ϕ-neighborhood of the stationary ϕ-surface.

Classical BRST Operator
If a proper solution S to eqs. (2.14)-(2.17) exists, the corresponding classical BRST operator is defined as s := (S, · ) .
It is nilpotent as a consequence of the Jacobi identity for the antibracket and the classical master equation (2.15).
Theorem 2.1 (Acyclicity of the BRST operator) Let S be a proper solution defined in a tubular ϕ-neighborhood of the stationary ϕ-surface. Then the cohomology of the BRST operator s = (S, ·) is acyclic, i.e., in some tubular ϕ-neighborhood of the stationary ϕ-surface.
Proof of Theorem 2.1: Use Theorem 3.2, Theorem 3.3, Lemma D.1, and the fact that antifield number is (weakly) greater than minus ghost number, i.e., 3 Standard Methods

Introduction
In this Section 3 we try to attack the problem of existence of a field-antifield formulation for reducible theories by applying the standard method that is known to work in the irreducible case [9,10,12,14], i.e., generating a proper action S from a Koszul-Tate operator s −1 and keeping track of antifield number "afn". Unfortunately, it is cumbersome to directly verify the existence and acyclicity of the Koszul-Tate operator s −1 in the reducible case using this standard approach. The standard method will nevertheless serve as a simplified template, on which we will develop a complete existence proof in the next Section 4, using a reduced (shifted) Koszul-Tate operator s (−1) and a (shifted) antifield number "safn", respectively.
Elementary considerations reveal the following useful Lemma 3.1.
Lemma 3.1 Let f, g be two functions of definite antifield number. Then Assume that f also has definite ghost number. Then In particular, Proof of Lemma 3.1: There are only two possibilities. The first case is and the second case is with the rôles of f and g interchanged,

Koszul-Tate Operator s −1
If the proper solution S exists, the Koszul-Tate operator s −1 is then defined as the leading antifield number sector for the classical BRST operator, The Koszul-Tate operator s −1 is of the form where the functions M α = O((Φ * ) 2 ) are at least quadratic in the antifields. It follows from the requirement that the functions V α = V α (ϕ, Φ * ) cannot depend on the ghosts "c". In more detail, the V α functions read ) Phrased differently, the classical BRST operator "s" is a deformation of the Koszul-Tate operator, when one uses the antifield number "afn" as a resolution degree. The classical master eq. (2.15) implies that the classical BRST operator "s" is nilpotent s 2 = 0, and hence that the Koszul-Tate operator s −1 is nilpotent, The last rewriting shows that the proper action is nilpotent in some ϕ-neighborhood of a ϕ-point on the stationary ϕ-surface, where the structure functions V α are of the form (3.13)-(3.15). Then the cohomology of the Koszul-Tate operator s −1 is acyclic in some ϕ-neighborhood of the ϕ-point, i.e., ∀ functions f :   [13])) Assume that for every ϕ-point on the stationary ϕ-surface there exists some nilpotent and acyclic Koszul-Tate operator in a local ϕ-neighborhood around the ϕ-point. Then there exists a smooth, nilpotent and acyclic Koszul-Tate operator in a tubular ϕ-neighborhood of the whole stationary ϕ-surface.

Sketched Proof of Theorem 3.3:
Cover the stationary ϕ-surface with sufficiently small ϕneighborhoods U i such that the local Koszul-Tate operators (which may depend on the ϕ-neighborhoods U i ) are nilpotent and acyclic. Use a smooth partition of unity Here it is used that the Koszul-Tate operators do not act on the original ϕ-variables.

Existence of S
Now we would like to deduce a proper solution from the Koszul-Tate operator.
, that is defined in a ϕ-neighborhood U , with antifield number minus one, It follows from Lemma 3.1 that the rth classical master expression CME r can be written as where The second equality of eq. (3.21) follows because the antibracket itself has negative antifield number. The third equality of eq. (3.21) follows from the s −1 nilpotency (3.16). The proof of the main statement is an induction in the antifield number r. Assume that there exist such that 0 = CME 0 = CME 1 = . . . = CME r−1 . (3.23) It follows that the B r function (3.21) exists as well, because B r only depends on the previous S ≤r . We want to prove that there exists S r+1 such that CME r = 0. The Jacobi identity J = 0 gives because of the Noether identity (2.1). If r > 0, then the acyclicity condition (3.17) shows that there exists a function Here we used that Koszul-Tate operator s −1 does not depend on the ghosts c α s , s ∈ {0, . . . , L}.

Anticanonical Transformations
Consider finite anticanonical transformations e (Ψ, · ) , where Ψ is a Grassmann-odd generator with ghost number minus one, gh(Ψ) = −1. Such transformations form a group under composition is the Baker-Campbell-Hausdorff series expansion (with Lie brackets replaced by antibrackets).
Note that an anticanonical transformation e (Ψ, · ) S of a solution S to the classical master eq. (2.15) is again a solution to the classical master eq. (2.15). We shall now show that any two proper solutions are related via an anticanonical transformation modulo global obstructions. The induction assumption is true for m = 1, because one assumes that the correct original limit (2.16) is fulfilled, S 0 = S 0 . Now consider m = 2. One has that From nilpotency (3.16) of the two Koszul-Tate operators, one knows that Since the function V α = V α (ϕ, Φ * ) does not depend on the ghost variables c α s , s ∈ {0, . . . , L}, the antifield number is afn(V α r ) = −gh(V α r ) = gh(c α r ) = r + 1 .
Hence the difference s −1 (V α r − V α r ) = 0 is zero, so by acyclicity (3.17) of the Koszul-Tate operator s −1 , there exists One is allowed to change where with afn(W α r ) = −gh(W α r ) = r +3. The leading contribution proportional to c * α r−1 in eq. (3.37) and c * α r in eq. (3.38) read Alternatively, if one defines ∆ β r α r := (1− U ) β r α r , one has It follows from the rank conditions for Z α s−1 α s and Z α s−1 α s , that one may assume that ∆ β r α r is a regular invertible matrix, possibly after an allowed change (3.38). (This is the only place in the proof where one uses that the S solution is proper.) Therefore one may assume that the matrix U β r α r = (1 − ∆) β r α r has no eigenvalues equal to 1. Next apply the anticanonical transformation e −(Ψ, · ) to S, with Ψ = Ψ α r c α r , where and where the holomorphic function has a logarithmic singularity at z = 1. One may place the branch-cut of f away from the eigenvalue spectrum of the matrix U β r α r , so that f ( U ) is well-defined. The leading tilde piece of the Ψ function (3.43) is so that and hence where E(z) := e z −1 z = ∞ n=0 z n (n+1)! . One has (after renaming e −(Ψ, · ) S → S) which is the nested induction assumption (3.32) with r → r + 1. This finishes the proof that the induction assumption (3.30) is true for m = 2, and that the two Koszul-Tate operators s −1 = s −1 are equal.

Transversal and Longitudinal Fields
Because of the Noether identity (2.1), there exist locally M −1 independent on-shell gauge-invariants ξ I = ξ I (ϕ), which we will call the transversal fields. They satisfy in terms of formulas near the stationary ϕ-surface, where "≈" means equality modulo equations of motion and where K iI α 0 = K iI α 0 (ϕ) are some structure functions. Consider now an arbitrary ϕ-neighborhood U , where the transversal fields ξ I are defined, and where U is sufficiently close so that it intersects the stationary ϕ-surface. It follows from eq. (4.1) that the values of the gauge-invariants ξ I (ϕ) = ξ I cl do not depend on the point ϕ if one only varies the point ϕ within the M 0 -dimensional stationary subsurface T i (ϕ) = 0. By a redefinition ξ I → ξ I −ξ I cl it is possible to assume (and we will do so from now on), that ∀ϕ ∈ U : T i (ϕ) = 0 ⇒ ξ I (ϕ) = 0. Since transversal fields ξ I are independent, it follows that the equations of motion are equivalent to the vanishing of the transversal fields The transversal fields ξ I can locally be complemented with so-called longitudinal fields θ A 0 = θ A 0 (ϕ), such that the change of coordinates is a non-singular coordinate transformation. Here the indices runs over By definition The rectangular matrix must have maximal rank near the stationary ϕ-surface (If it didn't have maximal rank, it would signal the possibility to define at least one more transversal coordinate that satisfies eq. (4.1).) Therefore there exists a right inverse matrix N α 0 A 0 , such that The Noether identity (2.1) yields where The expression (4.11) is for later convenience. From the expression (4.10), one sees that By differentiating the Noether identity (2.1) with respect to ϕ i , one derives that and hence, Therefore the rank condition (2.5) implies that near the stationary ϕ-surface. The transversal and longitudinal fields are not uniquely defined.
If the K iI α 0 structure functions additionally satisfy the integrability condition then the second term of the expression (4.11) vanishes identically, so that instead of just eq. (4.13). Therefore, in case of the integrability condition (4.17), one has the following. In fact, the Gauge Principle 4.1 is precisely equivalent to the pair of eqs. (4.1) and (4.17). It was shown in Ref. [4] by integrating the generalized Lie equations that finite gauge transformations do exist, and in particular, that the gauge principle (4.1) holds, and hence that there exist ξ I such that both eqs. (4.1) and (4.17) are satisfied. This implies, among other things, that the set of stationary points for S 0 constitute a smooth submanifold. However, since ξ I generically will be space-time non-local, and since we will not actually need the integrability condition (4.17) in the following, we are reluctant to unnecessarily enforce eq. (4.17) on ξ I in what follows.

Reduced and Shifted Antifield Number
We would like to redefine the Koszul-Tate operator and the resolution degree, so that the Koszul-Tate operator is more directly related to the S quad part (2.19) of the action S in the reducible case. The new resolution degrees will be the so-called reduced and shifted antifield number. Transversal fields ξ I and antifields Φ * α are charged under reduced and shifted antifield number rafn(ξ I ) = 1 = safn(ξ I ) , All the other variables, i.e., the longitudinal fields θ A 0 and ghosts c α s , s ∈ {0, . . . , L}, carry no shifted or reduced antifield number, see Table 1. Reduced and shifted antifield number are not independent, because in general safn = rafn + afn .
where ghosts "c" are passive spectators, then the reduced antifield number "rafn" is the simplest resolution degree to work with, cf. Theorem 4.5. Reduced antifield number is also easy to transcribe into the Hamiltonian framework. On the other hand, when considering the whole BRST operator "s", where ghosts "c" are active, then one needs the shifted antifield number "safn", cf. Theorem 4.6. (For instance, some parts of the longitudinal derivative [14] would have leading resolution degree, if one only uses reduced antifield number as resolution degree, which would be devastating. On the other hand, shifted antifield number appropriately pushes the longitudinal derivative down the resolution hierarchy.) Reduced and shifted antifield number will depend on the local choice of transversal fields, so they are not globally defined.
Let us now decompose the action and the antibracket with respect to shifted antifield number. First rewrite the Jacobian matrix of the transformation (4.4), as functions Λ i j = Λ i j (ϕ) of transversal and longitudinal fields ϕ i ≡ {ξ I ; θ A 0 }. One can expand these functions in shifted antifield number. = const. , Here S 0(2) is of the form The antibracket (·, ·) expands as , safn(f, g) (r) = safn(f ) + safn(g) + r , (4.29) Assume that f also has definite ghost number. Then In particular, The equations of motion expand as Together with the Noether identity (2.1), this implies that We also have Eq. (4.40) implies that

Transversal and Longitudinal Ghosts and Antifields
The only purpose of this Section 4.3 is to device rotated auxiliary variables, in which the acyclicity of a shifted Koszul-Tate operator (4.51) becomes apparent, see Section 4.4. We stress that we do not use the rotated auxiliary variables thereafter. In particular, the rotated auxiliary variables do not enter the existence proofs in Sections 4.5-4.6.
We will first show by induction in the stage s ∈ {0, . . . , L}, that there exists a sequence of invertible rotation matrices Λ β s α s = Λ β s α s (ϕ), such that the first M s−1 rows in the rotated gauge(-for-gauge) sgenerator . From the rank condition (2.6), it follows that it is possible to find an invertible rotation matrix Λ β s α s = Λ β s α s (ϕ), such that the rotated gauge(-for-gauge) s -generator is (4.43) Due to eq. (4.43) and the Noether relation (2.3), the induction assumption (4.42) is then satisfied for the next stage, and so forth.
Remark: The operator k, when applied to an operator X, does not change the auxiliary variables in X. The operators δ and δ act trivially on ghosts c, but may annihilate or create a single original antifield ϕ * (or its derivative).

Existence of Koszul-Tate Operator s −1
Let there be given an original action S 0 , gauge(-for-gauge) s -generators Z α s−1 α s , s ∈ {0, . . . , L}, an antisymmetric first-stage stucture function B ij α 1 , and higher-stage stucture functions B iα s−2 α s , s ∈ {2, . . . , L}, that satisfy the Noether identities (2.1)-(2.3). In particular, there are given action parts that is defined in some ϕ-neighborhood of a ϕ-point on the stationary ϕ-surface, and with reduced antifield number zero. This guarantees the local existence (in some ϕ-neighborhood of the ϕ-point) of a nilpotent, acyclic Koszul-Tate operator

74)
with antifield number minus one, and that satisfies the boundary condition All such operators are of the form Proof of Theorem 4.5: Let us, for notational reasons, put a bar on top of the sought-for Koszul-Tate operator s −1 , and no bar on quantities associated with the given boundary conditions (4.66)-(4.72). We use ∆ to denote differences, e.g., ∆V Noether , and so forth. We shall below inductively define the bar solution s −1 to all orders in the reduced antifield number, but initially, we only fix the zeroth-order part s −1[0] := δ to be equal to the reduced Koszul-Tate operator δ. The rth nilpotency expression N [r] for a bar solution s −1 can then be written as (4.79) The proof of the main statement is an induction in the reduced antifield number r ≥ 1. Assume that there exists a bar solution [δ,       Table over the antibracket (f, g) of various functions f and g with ghost number zero, gh(f ) = 0 = gh(g). (4.102) The second and third term on the right-hand side of eq. (4.102) will change the structure functions B 106) which is induction assumption (4.83) for the next step p = r−1.

Existence of Proper Action S
Let there be given an original action S 0 , gauge(-for-gauge) s -generators Z α s−1 α s , s ∈ {0, . . . , L}, an antisymmetric first-stage stucture function B ij α 1 , and higher-stage stucture functions B that is defined in some ϕ-neighborhood of a ϕ-point on the stationary ϕ-surface, and with shifted antifield number minus one. This guarantees the local existence (in some ϕ-neighborhood of the ϕ-point) of a proper solution S to the classical master equation (2.15), that satisfies the boundary condition All such solutions are of the form First proof of Theorem 4.6 using reduced antifield number: Combine Theorem 3.4 and Theorem 4.5.
Second proof of Theorem 4.6 using shifted antifield number: Let us, for notational reasons, put a bar on top of the sought-for proper action S, and no bar on quantities associated with the given boundary conditions (4.66)-(4.68). We use ∆ to denote differences, e.g., ∆S Noether := S Noether − S Noether , ∆S fixed := S fixed −S fixed , and so forth. We shall below inductively define the bar solution S to all orders in the shifted antifield number, but initially, we only fix the zeroth-order, first-order and second-order part as where The proof of the main statement is an induction in the shifted antifield number r ≥ 2. Assume that there exist a bar solution is fulfilled, such that and such that the classical master equation holds up to a order r−1 in shifted antifield number 0 = CME (0) = CME (1) = . . . = CME (r−1) . (4.117) The action S (p) = S (p) (Φ α ; ϕ * i , c * α 0 , . . . , c * α p−3 ) cannot depend on antifields c * α q , for p − 3 < q ≤ L, because their shifted antifield number safn(c * α q ) = q+3 is too big. It follows from the induction assumption that the B (r) function (4.112) exists, and that it is a function B (r) = B (r) (Φ α ; ϕ * i , c * α 0 , . . . , c * α r−3 ). We want to prove that there exists S (r+1) = S (r+1) (Φ α ; ϕ * i , c * α 0 , . . . , c * α r−2 ), such that CME (r) = 0. The Jacobi identity J gives In the last equality of eq. (4.118) is used that the two shifted Koszul-Tate operators δ ≡ s (−1) and s (−1) agree on functions f = f (Φ α ; ϕ * i , c * α 0 , . . . , c * α r−2 ), due to the induction assumption (4.114). This choice S (r+1) may not meet the prescribed boundary condition (4.108). Let us probe the difference in terms of cohomology.
δ∆S 0(r+1) = 0 , (4.122) One starts at the top stage s = L and goes down successively to the zeroth stage s = 0 by induction. Assume that the strong Noether identities (B.2) holds for earlier (≡ higher) stages. Choose the s-stage gauge condition matrix χ α s α s−1 = χ α s α s−1 (ϕ), such that Define the s-stage Faddeev-Popov propagator D α s β s = D α s β s (ϕ) so that The feasibility to meet condition (B.4) for s < L can be seen from the induction assumption (B.2) (with substitution s → s+1). The Faddeev-Popov propagator D α s β s is typically space-time non-local. Define P α s−1 β s−1 := δ where the induction assumption (B.2) (with substitution s → s+1) was used in the last step.
Proposition C.1 If the structure function B ij α 1 locally satisfies the first-stage Noether identity (2.2), then there exists an antisymmetric structure function B ij α 1 = −(−1) ε i ε j B ji α 1 that does the same.
Proof of Proposition C.1: Recall that there exists transversal and longitudinal fields ϕ ı ≡ {ξ I ; θ A 0 } so that the original action S 0 only depends on ξ I , cf. the Gauge Principle 4.1. Define where we have used the Jacobian Λ ı i matrix (4.23). Then Now define antisymmetric tilde structure functions B ı α 1 = −(−1) ε ı ε  B ı α 1 as follows. It is easy to see that the antisymmetric structure functions B ı α 1 also satisfies the first-stage Noether identity (C.2).
Remark: If the structure functions B ij α 1 is a tensor under change of coordinates (as is normally assumed), the antisymmetric B ij α 1 in the proof of Proposition C.1 is not necessarily also a tensor.

D Deformation of Acyclicity
Acyclicity is stabile under deformations in the following sense.