# Unfree gauge symmetry in the BV formalism

## Abstract

The BV formalism is proposed for the theories where the gauge symmetry parameters are unfree, being constrained by differential equations.

## 1 Introduction

The Batalin–Vilkovisky (BV) formalism^{1} was initially proposed [1, 2, 3, 4] as a tool for quantizing classical gauge field theories. Later on, the scope of applications of the formalism has been extended to a large variety of problems in physics and mathematics ranging from consistent inclusion of interactions in gauge field theories to the characteristic classes of various manifolds.

Given the Lagrangian, the BV-BRST embedding of the theory is a well-known straightforward procedure [5] if certain regularity conditions are obeyed by the original field equations and their gauge symmetry. These regularity conditions are also generalized for not necessarily Lagrangian field equations to provide their BV-BRST embedding [6, 7]. We mention two conditions which are assumed to hold true for the original field theory to admit the usual BV-BRST embedding: (i) the gauge symmetry parameters are arbitrary functions of space-time coordinates, i.e. they are not constrained by any equations; (ii) any on-shell vanishing function of the fields and their derivatives can be spanned by the left hand sides of the field equations and their differential consequences. These two assumptions, being critical for the BV formalism construction, are violated in some field theory models of a current interest. Examples are given in Sect. 6. As one can see from the examples, both the assumptions are usually violated simultaneously. Once the gauge transformation parameters are constrained by equations, the gauge symmetry is named unfree.

In the recent article [8], the defining relations are found of the unfree gauge symmetry algebra. The algebra of gauge symmetry with unconstrained gauge parameters is constructed starting from two basic ingredients: the action functional and gauge symmetry generators. In the unfree case, two more basic ingredients are involved: the operators of gauge parameter constraints and the mass shell completion functions. The first extra ingredient defines the equations constraining gauge symmetry parameters. The completion functions constitute the generating set of the on-shell vanishing quantities such that do not reduce to the left hand sides of the Lagrangian equations and their differential consequences. Proceeding from the simplest case of the general unfree gauge algebra such that there is no off-shell disclosure, the Faddeev–Popov (FP) quantization recipe is deduced in the article [8]. Earlier, in the specific case of the unimodular gravity, the FP quantization recipe has been deduced in the paper [9, 10] making use of some nonlocal manipulations involving splitting the fields into longitudinal and transverse components and special gauge conditions. The set of ghosts involved in the general FP recipe in the case of unfree gauge parameters includes some extra variables comparing to the case of the unconstrained gauge symmetry. In the examples where the models with unfree gauge symmetry admit equivalent reformulations with unconstrained gauge parameters, the modified FP recipe can be explicitly reduced to the standard one [8].

In this paper, we propose the extension of the BV formalism to a general Lagrangian theory with unfree gauge symmetry. Our focus is at the algebraic aspects of the extension, while the subtleties are left aside concerning the functional aspects. In the next section, we describe the algebra of unfree gauge symmetry. Proceeding from the relaxed regularity assumptions such that admit irreducible unfree gauge symmetry, we deduce the basic structure relations of the gauge algebra including the most general off-shell terms. In Sect. 3, we propose the BV embedding for the Lagrangian theory with unfree gauge symmetry. This requires to introduce the set of ghosts and anti-fields adjusted for the unfree gauge algebra. The ghost and anti-field set is different from the theory with unconstrained gauge parameters. Given the set of fields and anti-fields, and the relaxed regularity conditions, we see that the classical master equation reproduces the structure relations of unfree gauge symmetry algebra. In Sect. 4, we consider the non-minimal sector and gauge fixing. As one can notice from the examples, the specifics of the unfree gauge symmetry is that the independent gauge fixing conditions break the relativistic symmetry, while the relativistic gauges are inevitably redundant even though the unfree gauge symmetry is irreducible. We give the non-minimal sector ghosts both for the independent and redundant gauges. For the theory without off-shell disclosure, we deduce the FP recipe by explicitly fixing the anti-fields by the gauge conditions. This is done both with independent and redundant gauges. In Sect. 5, we prove the existence theorem for the unfree gauge algebra. We use the homological perturbation theory (HPT) method. The key ingredient of the method is the Koszul–Tate differential. In the context of the existence theorem for the BV master equation, the Koszul–Tate differential has been first considered in the works [4, 11, 12]. The HPT method based on the Koszul–Tate complex was formulated in the article [13] as a tool for BV embedding of general Lagrangian gauge theories. This HPT procedure follows the pattern earlier suggested in the work [14] for the BFV^{2}-BRST embedding of the Hamiltonian systems with reducible first class constraints. For the basics of the HPT applications to the BV formalism, we refer to the book [5]. In the case of the unfree gauge symmetry, the Koszul–Tate resolution of the mass shell differs from that for the gauge theory with unconstrained gauge parameters. Once the resolution is identified, the HPT allows one to construct the BV master action. In Sect. 6, we review the specific models with unfree gauge symmetry. After that, we consider one simple model to exemplify all the stages of the BV construction for the theory with unfree gauge symmetry. Section 7 includes concluding remarks.

**Condensed notation** In this article except for Sect. 6, where the specific field theory models are discussed, we adopt the DeWitt condensed notation. In this notation, the space of all the field histories is mimicked by the finite dimensional manifold \({\mathcal {M}}\), while the fields \(\phi ^i\) are treated as the local coordinates on \({\mathcal {M}}\). The index *i* is “condensed”, i.e. it comprises the space-time argument *x* of the field, and all the discrete labels (like tensor, spinor, or flavor indices). The fields are supposed to obey certain boundary conditions when the spacial part of *x* tends to infinity, and the asymptotics are understood as a part of definition of \({\mathcal {M}}\). In this article we imply zero boundary conditions. All the other variables, like gauge transformation parameters, ghosts, anti-fields, are treated like if they were the coordinates on the fibers of appropriate bundles over \({\mathcal {M}}\). Summation over the condensed index includes integration over the space-time argument. The matrices with condensed indices represent differential operators. For example, \(\delta _{ij}\) includes \(\delta (x-y)\) and delta symbol of discrete labels, so the matrix \(M_{ij}\) can represent D’Alembertian with appropriate identification: \(i=x, j=y\), \(M_{ij}=\Box \delta (x-y)\). In this notation, the Klein–Gordon equation reads \((M_{ij}+m^2\delta _{ij})\phi ^i=0\). The local functionals, being integrals of the functions of the fields and their space-time derivatives \(F(\phi )=\int dx\, {\mathcal {F}}(\phi , \partial _x\phi , \partial ^2_x\phi , \ldots , \partial ^k_x\phi )\) are mimicked by functions on \({\mathcal {M}}\), so the linear space of local functionals is treated like it was just a special subspace of smooth functions on \({\mathcal {M}}\): \(F\in {\mathcal {C}}^\infty ({\mathcal {M}})\). We name the smooth functions of the fields and their derivatives \(O(\phi , \partial _x\phi , \partial ^2_x\phi , \ldots , \partial ^k_x\phi )\) as local functions. We denote the algebra of local functions as \(R({\mathcal {M}})\). Any local function can be viewed as a local functional, so \(R({\mathcal {M}})\subset {\mathcal {C}}^\infty ({\mathcal {M}})\). Derivatives with respect to \(\phi ^i\) are understood as functional derivatives of any local functional, including any local function. The condensed notation can be always unambiguously uncondensed. For further details of the condensed notation, we refer to the books [5, 15].

## 2 Algebra of unfree gauge symmetry

In this section, we at first address the issue of the Noether identities and their consequences, in the theories where the on shell vanishing local functions are not exhausted by the linear combinations of the left hand sides of the Lagrangian equations. In the second instance, we demonstrate that the modification of the Noether identities lead to the unfree gauge symmetry. Proceeding from the modified identities and unfree gauge symmetry, we deduce the higher structure relations of the unfree gauge symmetry algebra.

*I*is generated by \(\partial _i S\),

The BV formalism [1, 2, 3, 4, 5] relays on the assumption (5) in several crucial aspects. Let us mention two of them. First, the restriction (5) is included into the set of sufficient conditions that ensure the existence of solution to the BV master equation. Second, even if the solution exists for the master-equation, while (5) does not hold true, the ideal *I* of trivial local functions would not be isomorphic to the BRST-exact functions of zero ghost number. This would break the usual physical interpretation of the BRST cohomology groups.

The assumption (5) is violated in a number of field theories, see the examples in Sect. 6. Below, we elaborate on the general gauge symmetry algebra with a relaxed assumption (5).

*I*still admits a finite generating set. The ideal of the on-shell vanishing functions is supposed to be generated by \(\partial _i S(\phi )\) and by a finite set of the other trivial functions \(\tau _a (\phi )\),

*completion functions*of the Lagrangian system (1).

^{3}that all the completion functions (6) are essentially involved in the Noether identities (9), i.e.

In the gauge identities (9), the operators \(\Gamma ^i_\alpha \) and \(\Gamma ^a_\alpha \) are involved on an equal footing. However, they have different roles in the gauge symmetry transformations. The operator \(\Gamma ^i_\alpha \) generates the unfree gauge transformations (22), while \(\Gamma ^a_\alpha \) defines the equations (24) that restrict the gauge parameters. With this regard, we name \(\Gamma ^i_\alpha \) the generators of unfree gauge symmetry, while \(\Gamma ^a_\alpha \) are named the operators of gauge parameter constraints.

*E*are antisymmetric, \(E_{\alpha \beta }^{ij}=-E_{\alpha \beta }^{ji}, E_{\alpha \beta }^{ab}=-E_{\alpha \beta }^{ba}\). The off-shell relations (31) mean, in particular, that any two unfree gauge transformations (22), (24) commute on-shell to another unfree gauge transformation. The off-shell relations (32) ensure that the equations (24) constraining the gauge parameters are on-shell gauge invariant themselves. This allows one to conclude that the unfree gauge symmetry transformations define on-shell integrable distribution. It foliates the mass shell into the gauge orbits, much like the gauge transformations would do if the gauge parameters were not constrained. This allows one to define physical observables in the usual way, as the equivalence classes (4) of the on-shell gauge invariant local function(al)s (26), (27). Any two observables are considered equivalent if they coincide on shell (4). Let us denote the subalgebra of on-shell gauge invariant local functions as \(G({\mathcal {M}})\), i.e.

*G*/

*I*.

Let us summarize the most important specifics of the unfree gauge symmetry algebra. First, the generating set for the ideal of trivial local functions is not exhausted by the left hand sides of Lagrangian equations, it also includes the completion functions (6). The Noether identities are modified (9) also involving completion functions. Second, the gauge transformation parameters (22) are unfree, being constrained by the equations (24). Third, the regularity/completeness assumptions involve the equations of motion, completion functions, gauge generators and gauge parameter constraint operators (7), (13), (14). This specifics has to be accounted by an appropriate modification of the BV formalism such that can cover the systems with unfree gauge symmetry.

### Remark

^{4}We denote these arbitrary functions \(\omega ^A\). The condensed index

*A*includes the space-time argument

*x*, so \(\omega ^A\) are the arbitrary functions of

*x*indeed. They can be considered as the unconstrained gauge symmetry transformation parameters. In this setting, the solution to the equations (24) read

## 3 Master equation

Construction of the BV-BRST enbedding for the gauge system begins with the definition of the ghost and anti-field extension of the original set of the fields \(\phi ^i\). Below we provide some reasons for certain ghost/anti-field extension of \({\mathcal {M}}\) and formulate the master equation for the action. After that, we shall see that the master equation indeed reproduces the structure relations of the unfree gauge symmetry algebra deduced in the previous section. Then we shall see that the algebra of the gauge invariants of the original theory is mapped to the BRST cohohomology of the BV formalism. In the end of the section, we provide a reinterpretation of the constructed BV formalism in terms of “compensator fields”. In the next section we consider the gauge fixing in the BV formalism. A formal justification of the specific ghost and anti-field set is provided in Sect. 5, where we identify the Koszul–Tate differential for the mass shell of the theory with the unfree gauge symmetry and prove the existence theorem for the master equation.

*n*is the ghost number of the equation. Every generator of gauge identity is also assigned with the anti-field whose ghost number is \(-n-2\), where

*n*is the ghost number of the equations involved in the identity. Every generator of the gauge symmetry is assigned with the ghost whose ghost number is \(k+1\), where

*k*is the ghost number of the gauge parameter. Notice that for the non-Lagrangian systems, the gauge symmetries are not necessarily paired with the gauge identities, so the corresponding generators can be different, while in the Lagrangian case the same operator generates both gauge symmetry and gauge identity. In the Lagrangian systems with unfree gauge symmetry, the non-Lagrangian pattern works well for introducing ghosts and anti-fields after two adjustments. The first is that the anti-fields are to be assigned to every element of the generating set for the ideal of on-shell vanishing local functions. This means, the anti-fields are introduced both for Lagrangian equations (1) and completion functions (6). We denote these anti-fields \(\phi ^*_i\) and \(\xi ^*_a\), respectively. The second is that the ghost \(C^\alpha \) is assigned to every gauge symmetry generator \(\Gamma _\alpha ^i\) (22) even though the gauge parameters are unfree (24). The constraints on the gauge parameters are accounted for by imposing the same constraints on the ghosts:

*I*. This means, the anti-field has to be introduced for every ghost constraint (40). As the equations (40) are of the ghost number 1, the anti-fields should be assigned with the ghost number zero. We denote these anti-fields \(\xi ^a\). It does not mean the mere extension to the set of original fields \(\phi ^i\), because the BV formalism also implies one more grading: the resolution degree (also referred to as the anti-ghost number in some literature, e.g. in [5]). The anti-field \(\xi ^a\) has the resolution degree 1, unlike \(\phi ^i\). And finally, the anti-fields \(C_\alpha ^*\) with ghost number \(-2\) are assigned to the modified Noether identities (9) alike the Lagrangian theory with unconstrained gauge parameters, even though the identities involve completion functions, unlike the usual case.

Gradings of fields and anti-fields in a theory with unfree gauge symmetry

Grading\(\backslash \)variable | \(\phi ^i\) | \(\xi ^a\) | \(C^\alpha \) | \(\phi ^*_i\) | \(\xi ^*_a\) | \(C^*_\alpha \) |
---|---|---|---|---|---|---|

\(\varepsilon \) | 0 | 0 | 1 | 1 | 1 | 0 |

\(\text {gh}\) | 0 | 0 | 1 | \(-\) 1 | \(-\) 1 | \(-\) 2 |

\(\text {deg}\) | 0 | \(\mathbf 1 \) | 0 | 1 | 1 | 2 |

*I*are assigned with the resolution degree 1. It is the same principle as in the gauge theory with unconstrained gauge parameters, with one adjustment: the anti-fields are introduced not only for the original Lagrangian equations (1) – \(\phi ^*_i\), but also to the completion functions (6) – \(\xi ^*_a\), and to the equations constraining the ghosts (40) – \(\xi ^a\). The anti-fields are also introduced being paired with the Noether identities (9) – \(C^*_\alpha \). These are assigned with the resolution degree 2, much alike the theory with unconstrained gauge parameters.

*Z*-graded manifold \(\bar{{\mathcal {M}}}\). The fields \(\varphi ,\varphi ^*\) are considered as coordinates on \(\bar{{\mathcal {M}}}\). The original equations of motion (1) and the constraints imposed on the ghosts (40) define the extended mass shell \(\bar{\Sigma }\subset \bar{{\mathcal {M}}}\),

*s*squares to zero because of the master equation (50) and Jacobi identity for the anti-bracket. It is Grassmann odd vector field of the ghost number 1,

*s*, squares to zero, so it is a differential in itself. Explicitly, \(\delta \) reads

*G*/

*I*and BRST cohomology \(H^0(s)\). Cohomological proof of the isomorphism is provided in Sect. 5. Here, we just demonstrate the explicit iterative construction of the BRST invariant corresponding to gauge invariant up to the first order with respect to the resolution degree. Consider the expansion

So, any gauge invariant local function(al) \(O(\phi )\) can be extended to the BRST invariant function(al) at least up to the first order with respect to the resolution degree. The extension has a natural ambiguity of the \(\delta \)-exact terms. This precisely corresponds to the equivalence relations (4) for the local function (4). So, one can see that the algebra of physical observables of original theory *G* / *I* (33) is indeed isomorphic to the BRST cohomology \(H^0(s)\) of zero ghost number to the first order in resolution degree. The higher orders exist, and they all can be iteratively constructed by the HPT method. This is explained in Sect. 5. Let us make one more remark concerning the solution to master equation. The variable \(\xi ^a\) has zero ghost number, and it is Grassmann even if the gauge parameters are even while it has the resolution degree one. This means, the expansion can be infinite in \(\xi ^a\). It might seem not a plausible iterative procedure for solving the master equation once it does not terminate in the final number of iterations. In fact, one can avoid explicitly finding all the orders in \(\xi ^a\). As we shall see in the next section, \(\xi ^a\) can be always fixed by an appropriate choice of gauge conditions, so only the first orders in \(\xi ^a\) can matter.

### Remark

The conversion of the theory with the unfree gauge symmetry parameters into an equivalent theory with extended set of fields, and without constraints on the gauge parameters is analogous to some extent to the conversion of the second class constraints into the first class ones [20, 21], or to inclusion of the Stückelberg fields. The difference is that the second class constraints are absorbed by the first class ones, while in the theories with unfree gauge parameters, the operators of gauge parameter constraint operators are absorbed by the gauge generators, and the completion functions are absorbed by the action. One more common feature is that the HPT in both cases involves the conjugate variables such that have non-zero resolution degree, though in the conversion of Hamiltonian constraints the bracket is even, while in the unfree gauge theory it is odd.

From the viewpoint of the re-interpretation of the anti-fields \(\xi ^a\) as compensators, it is clear that the gauge conditions can be imposed such that fix \(\xi ^a\), and even force them to vanish. With these gauge imposed, it becomes unnecessary to explicitly find all the orders in \(\xi \). It becomes sufficient to explicitly know the first order in \(\xi \), once the higher orders can be excluded from the gauge fixed BRST invariant action by an appropriate gauge condition.

## 4 Gauge fixing

In this section, we briefly consider the gauge fixing procedure in the BV formalism of the theories with unfree gauge symmetry algebra.

Let us first discuss the options of imposing the gauge fixing conditions on the original fields, and then turn to introducing the ghosts of non-minimal sector.

## 5 The unique existence of solution to the master equation

In Sect. 3, the master equation has been formulated for the theories with unfree gauge symmetry algebra. We have explicitly found the solution in the second order approximation with respect to the resolution degree (48), (49). Now, we consider the existence problem in any order. At first, we shall see that the problem can be brought to the usual HPT setup with respect to the Koszul–Tate differential (57). Then, we consider the cohomology of \(\delta \). The differential \(\delta \) can be understood a resolution for the ideal \(\bar{I}\) (43) of the on-shell vanishing local function(al)s. The ideal \(\bar{I}\) is generated not only by the left hand sides of the Lagrangian equations, but also by the completion functions (6), (7), and by the constraints imposed on the ghosts (40). Once the generating set is different for \(\bar{I}\) comparing to the corresponding ideal in the case of the Lagrangian theory with unconstrained gauge symmetry parameters, the cohomology of the differential has to be examined in this case. We shall demonstrate that \(\delta \) is acyclic in the strictly positive resolution degrees. Given the acyclicity of \(\delta \), and relation (56), we shall see that the master equation can be iteratively solved by the usual HPT tools. And finally, we shall address the issue of physical observables.

^{5}existence of all the higher order items in the expansion (89).

*R*/

*I*of the nontrivial gauge invariant function(al)s is isomorphic to the BRST cohomology group of zero ghost number.

## 6 Examples

*T*-diff.

*T*-diff is the gauge symmetry of unimodular gravity and various modifications, see [22, 23, 24, 25, 26, 27, 28, 29, 30, 31] and references therein. Once the unimodularity condition \(\det g_{\mu \nu }=-1\) is imposed, the gauge variation \(\delta _\epsilon g_{\mu \nu }=\nabla _\mu \epsilon _\nu +\nabla _\nu \epsilon _\mu \,\) is unfree of the metric, as the transformation have to be consistent with the fixed volume. This means, the transformation parameter \(\epsilon ^\mu (x)\) is constrained by the transversality condition,

*s*[34]. The gauge parameters, being the symmetric traceless tensors of rank \(s-1\) turn out unfree again. The Maxwell-like models [35, 36] of higher spins involve tracefull tensors, whose gauge symmetry is parameterized by the tracefull tensors of a lower rank, while the differential equations are still imposed on the gauge parameters. So, the phenomenon of the unfree gauge symmetry is not necessarily related to any constraint (like trace condition) imposed on the fields.

^{6}With zero boundary conditions, the constant should vanish, so we have

*R*is the Ricci curvature of the unimodular metric. This means,

*T*-diffs,

*W*(29), (30) does not vanish. In the case at hands

*x*is the space-time argument. The gradings of the fields and anti-fields are arranged in Table 2.

Gradings of fields and anti-fields in the linearized unimodular gravity

Grading\(\backslash \)variable | \(h^{\mu \nu }\) | \(\xi \) | \(C^\mu \) | \(h^*_{\mu \nu }\) | \(\xi ^*\) | \(C^*_\mu \) |
---|---|---|---|---|---|---|

\(\varepsilon \) | 0 | 0 | 1 | 1 | 1 | 0 |

\(\text {gh}\) | 0 | 0 | 1 | \(-\)1 | \(-\)1 | \(-\)2 |

\(\text {deg}\) | 0 | 1 | 0 | 1 | 1 | 2 |

*M*(82) is the D’Alembert operator,

*Z*. To make explicit comparison of results with independent and redundant gauge, we bring (135) to the form (124). We proceed with making the change of ghost variables,

## 7 Concluding remarks

Let us summarize and discuss the results.

Proceeding from the observation that the reasonable gauge field theories can admit the local quantities termed the completion functions such that vanish on shell and do not reduce to the differential consequences of equations of motion (6), we deduce the most general gauge symmetry algebra for this case. It turns out that the existence of the completion functions in the theory results in the unfree gauge symmetry, with gauge parameters constrained by the equations (24). And vice versa, the unfree gauge symmetry implies the existence of completion functions. This is a consequence of the modified Noether identities (9) which involve both the Lagrangian equations and completion functions. The modified identities result in the unfree gauge algebra. Given the unfree gauge symmetry algebra, we work out a systematic procedure for the BV-BRST embedding of the theory. The extension of the BV formalism to the systems with unfree gauge symmetry algebra has some distinctions from the case where the gauge parameters are unconstrained. The source of distinctions is two-fold. First, the Koszul–Tate resolution for the ideal of on-shell vanishing local function(al)s *I* should involve the completion functions as the l.h.s. of the Lagrangian equations do not generate *I*. Second, the ghost are constrained by the equations (40) as the corresponding gauge parameters are unfree (24). The equations constraining the ghosts (40) have to be also accounted for by the Koszul–Tate resolution. These reasons define the minimal set of the fields and anti-fields needed for the proper BV embedding, see Table 1 in Sect. 3. The set involves the anti-fields \(\xi ^a\) to the constraints imposed on the ghosts (40). What is unusual, these anti-fields have the ghost number zero, while their resolution degree is 1. The completion functions are also assigned with the anti-fields \(\xi ^*_a\). The usual BV formalism with unconstrained gauge symmetry parameters does not involve \(\xi ,\xi ^*\). These new variables are naturally conjugate with respect to the anti-bracket. The boundary conditions for the BV master-action are defined by the original action, gauge generators, completion functions, and operators of gauge parameter constraints (47), (48). In the case of the gauge symmetry with unconstrained gauge parameters, only first two constituents are involved. Given the regularity conditions imposed on the boundary, the master equations admits a solution which is unique modulo the natural ambiguity. The BV formalism for the unfree gauge symmetry admits the re-interpretation in terms of the usual BV formalism for the theory with the “compensator fields”, see the remark in the end of Sect. 3. Given the master-action, \(\xi \) can be considered on an equal footing with the original gauge fields \(\phi \). Then, the theory would correspond to the theory of the fields \(\phi ,\xi \) with the action \(S'(\phi ,\xi )\) (70) and unconstrained gauge symmetry of the extended set of fields. From the viewpoint of this re-interpretation, the existence theorem of Sect. 5 provides a systematic procedure for inclusion of the compensator fields such that the gauge symmetry involves unconstrained parameters of the extended theory. The equivalence is obvious between the original theory and theory with compensators as they correspond to the same BRST complex. In all the examples of specific models with unfree gauge symmetry reviewed in Sect. 6, the compensator fields are known. From the viewpoint of Sect. 5, it looks as an expected fact rather than a coincidence. The reinterpretation of the anti-fields \(\xi \) to the equations constraining the ghosts (40) as a compensators can be further extended, in principle, in another direction. The Lagrangian equations can have the lower order *differential* consequences. This is a typical case for the Lagrangian theories having the second class constraints in the Hamiltonian formalism, for example. Then, Lagrangaian equations do not constitute the involutive PDE system. Concerning the specifics of (non-)involutive equations, see [37]. In particular, the theory admits the implicit Noether identities which involve the original Lagrangian equations and lower order consequences. These consequences and identities control the degree of freedom number on equal footing with the original equations and their gauge symmetries. So, the consistent deformation of the non-involutive Lagrangian theory is not controlled by the naive BV master equation once it does not respect the consequences and implicit identities [37]. From the algebraic standpoint, the implicit identities are similar to the modified Noether identities (9) if the completion functions \(\tau \) are replaced by the lower order differential consequences of the Lagrangian equations. If the BV embedding procedure of Sect. 3 is applied to the non-involutive Lagrangian system, with \(\tau _a\) being the lower order consequences of the Lagrangian equations, the variables \(\xi ^a\) would play the role of the Stückelberg fields. So, this BV embedding algorithm would work as a systematic procedure of consistent inclusion of Stückelberg fields. In terms of Hamiltonian formalism, the general methods are known of conversion the second class constraints into the first class ones, [20, 21], while at Lagrangian level the “Stückelbergization” is rather a series of ad hoc tricks adjusted to specific models than a general systematic method. We expect that the BV embedding procedure of Sect. 3 can be reshaped into the general method of “Stückelbergization” in the Lagrangian formalism. One more aspect of the general connection between the theory with unfree gauge symmetry (hence, with nontrivial completion functions) and its equivalent with the compensator fields and without constraints on the gauge parameters is related to the “global modes”. As one can see from the examples, the completion functions usually reduce to the arbitrary constants on shell (see in Sect. 6). With the fixed boundary conditions imposed on the fields, these constants take fixed values defined by the boundary. If the setup is adopted with unfixed boundary conditions for the fields, these constants would play the role of the global conserved quantities. The corresponding global degrees of freedom can be understood as modular parameters. These degrees of freedom have to be accounted for in the BV formalism once the fields are not fixed at the boundary. This issue is not addressed in the present work, though the formalism can accommodate the modular parameters, in principle.

## Footnotes

- 1.
Also known as BRST (Becci-Ruet-Stora-Tutin) field-anti-field formalism.

- 2.
Batalin–Fradkin–Vilkovisky.

- 3.
This assumption is obeyed by all the presently known theories with unfree gauge symmetry. The examples are provided in Sect. 6. It could be be relaxed, however this would lead to a more involved set of ghosts and anti-fields needed for the proper BV-BRST embedding of the theory. At the moment, this option seems having only the academic interest.

- 4.
The unfree gauge transformations should not be confused with so-called semi-local symmetries, see [16, 17, 18] and references therein. In both the cases, the transformation parameters have to obey the differential equations. The difference is that the solutions to the equations on the unfree parameters involve the arbitrary functions of

*d*coordinates in*d*dimensional space, while in the semi-local case the arbitrary functions depend on \(d-1\) coordinates or less. - 5.
Up to the \(\delta \)-exact terms. This is a natural ambiguity related to the fact, that the structure functions of the gauge algebra are defined modulo the on shell vanishing contributions.

- 6.
The boundary conditions should not be confused with Cauchy data which define the initial values of the fields and their time derivatives on some space-like hyper-surface, not at spacial infinity. Zero boundary conditions do not contradict to nontrivial Cauchy data.

## Notes

### Acknowledgements

We thank A. Sharapov for discussions. The work is partially supported by Tomsk State University Competitiveness Improvement Program. The work of SLL is supported by the project 3.5204.2017/6.7 of Russian Ministry of Science and Education.

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