# Unimodular Gravity in Restricted Gauge Theory Setup

A.O. Barvinsky†, D.V. Nesterov⋆

###### Abstract

We develop Lagrangian quantization formalism for a class of theories obtained
by the restriction of the configuration space of gauge fields from a wider
(parental) gauge theory. This formalism is based on application of Batalin-
Vilkovisky technique for quantization of theories with linearly dependent
generators, their linear dependence originating from a special type of
projection from the originally irreducible gauge generators of the parental
theory. The algebra of these projected generators is shown to be closed for
parental gauge algebras closed off shell. We demonstrate that new physics of
the restricted theory as compared to its parental theory is associated with
the rank deficiency of a special gauge restriction operator reflecting the
gauge transformation properties of the restriction constraints functions –
this distinguishes restriction of the theory from its partial gauge fixing. As
a byproduct of this technique a workable algorithm for one-loop effective
action in generic first-stage reducible theory was constructed, along with the
explicit set of tree-level Ward identities for gauge field, ghost and ghosts-
for-ghosts propagators, which guarantee on-shell gauge independence of this
effective action. The restricted theory formalism with reducible set of
projected gauge generators is applied to unimodular gravity theory by using
background-covariant gauge-fixing procedure. Its one-loop effective action is
obtained in terms of functional determinants of minimal second order
differential operators calculable on generic backgrounds by Schwinger-DeWitt
technique of local curvature expansion. The result is shown to be equivalent
to Einstein gravity theory with a cosmological term up to a special
contribution of the global degree of freedom associated with the variable
value of the cosmological constant. The role of this degree of freedom is
briefly discussed.

Theory Department, Lebedev Physics Institute, Leninsky Prospect 53,
Moscow 119991, Russia

†e-mail<EMAIL_ADDRESS>⋆e-mail<EMAIL_ADDRESS>

###### Contents

  1. 1 Introduction
  2. 2 Effective action for theories with first-stage reducible gauge generators
    1. 2.1 Batalin-Vilkovisky formalism for reducible gauge theories
    2. 2.2 Stationary point of the gauge-fixed master action
    3. 2.3 One-loop contribution to the generating functional
    4. 2.4 Ward identities and gauge independence of the effective action
  3. 3 Reducible gauge structure of restricted gauge theories
    1. 3.1 Two representations of a restricted theory
    2. 3.2 Gauge symmetry and reducibility
    3. 3.3 Gauge algebra of a restricted theory
  4. 4 One-loop effective action of restricted gauge theory
    1. 4.1 Two representations of one-loop effective action
    2. 4.2 Independence of projector parameter
    3. 4.3 Canonical normalization of generators
    4. 4.4 Difference between parental and restricted theories in one loop
  5. 5 Unimodular gravity theory
    1. 5.1 Gauge fixing and propagators
    2. 5.2 Reduction of functional determinants
  6. 6 Conclusions
  7. A Appendix: Moore-Penrose inverse and variation of projectors
  8. B Appendix: Gauge algebra of the restricted theory from a parental theory
  9. C Appendix: Determinant relation

##  1 Introduction

The purpose of this paper is to discuss interesting property that arises when
one restricts a configuration space of the gauge theory. This is the origin of
a reducible gauge structure characterized by linear dependence of the
generators of gauge transformations. Both mechanisms – reduction of the
configuration space and reducibility of gauge symmetry – are widely known and
applied in various areas of field theory. In gravity theory context one of the
most important examples of the field space restriction is unimodular gravity
theory (UMG) that was suggested by Einstein soon after the invention of
general relativity [1], much later considered in context of particle physics
[2], in context of a spacetime covariant formulation [3], problem of time and
the cosmological constant problem [4] and then applied within the dark energy
paradigm [5] with the emphasis on purely technical issues of perturbation
theory, etc. Extension beyond the unimodular constraint on the metric of
spacetime – the so-called generalized unimodular gravity (GUMG) [6] also
turned out to be rather fruitful from the viewpoint of generation of viable
dark energy and inflation scenarios [7, 8]. From field-theoretical point of
view, unimodular gravity is an interesting object of quantization [9, 10, 11,
12, 13] especially regarding its relation to quantization of Einstein theory
[14] as a sort of a parental theory whose field space reduction leads to
unimodular gravity model.

On the other hand, it was sufficiently long ago conjectured [6] that the
formalism of quantized unimodular and generalized unimodular gravity theories
can be developed along the lines of gauge theory quantization in models with
linearly dependent (or reducible) generators [15, 16]. This observation
follows from a simple fact that the restriction of the configuration space of
metrics to the subspace of metrics with a unit determinant results in the
relevant reduction in the space of diffeomorphism invariance parameters, which
can be attained by a projection procedure. Then, if one wants for the sake of
retaining manifest covariance to describe the restricted theory in terms of
the parental one, this projection applied to the original gauge generators
makes them linearly dependent.

To the best of our knowledge this relation between the reducible nature of
gauge generators and the reduction of the field space has not yet been fully
exploited even in unimodular gravity context, not to say about its extension
to more or less general gauge theories. The studies of theories with
restrictions on gauge transformations, the theories with the so-called unfree
gauge parameters, were recently intensively conducted in a series of
publications [17, 18] with the purpose of constructing the BRST-BFV formalism
transcending these restrictions to the ghost sector of the model and starting
from first principles of Hamiltonian formalism and canonical quantization.
Here we choose a somewhat different approach closer to the Lagrangian
formalism and try to develop the idea of [6] for a generic parental gauge
field theory with the closed algebra of its irreducible generators becoming
linearly dependent in the course of projection induced by the restriction of
the original configuration space.

Our method is based on the explicit construction of projector operator with
which the generators of the original parental theory are projected onto the
generators of the residual symmetries of the restricted theory. Linear
dependence of the latter is then treated by the quantization method of Batalin
and Vilkovisky (BV) for reducible gauge theories [15, 16] – a brief review of
this formalism with all necessary notations and definitions is presented in
Sect.2.1. As a by-product of using BV formalism we develop in much detail the
gauge-fixing procedure for gauge theory of the first-stage reducibility. Quite
interestingly, despite numerous applications of this method, current
literature does not present a workable algorithm even for the one-loop
approximation in generic reducible gauge theories. Implicitly this algorithm
is, of course, contained in the pioneering works [15, 16], but its concrete
realization with all the details of Ward identities providing gauge
independence of the on-shell effective action is still missing. We seem to
close this omission and suggest the recipe for all the elements of Feynman
diagrammatic technique and the relevant algorithm of the one-loop effective
action in the above class of theories at the level of presentation
characteristic of a folklore use of the well-known Faddeev-Popov prescription.

The gauge-fixing algorithm here turns out to be more involved than in a
conventional Faddeev-Popov formalism for irreducible theories, because it
includes gauge fixing for original gauge fields, gauge fixing for ghosts and
the corresponding ghosts for ghosts. Even for first-stage reducible theories,
for which the hierarchy of structure functions of the gauge algebra terminates
at the first step (when there is no higher order zero vectors for reducibility
generators – zero vectors of original gauge generators), this gauge-fixing
procedure involves, apart the usual gauge conditions, at least three extra
gauge-fixing elements and the construction of a special projector in the
gauge-breaking term of the theory. Main result for the one-loop effective
action of gauge theories of the first-stage reducibility is assembled in the
sequence of equations (2.66)-(2.72) of Sect.2.3.

Then in Sect.3 we explain how reducible gauge structure with linearly
dependent generators arises in theories with restricted configuration space of
gauge fields. In particular, we raise the question why and when constraining
this space can be regarded merely as a partial gauge fixing, or on the
contrary forming a new physical theory inequivalent to the parental one. It
turns out that the answer is based on the rank of a special gauge restriction
operator. The rank deficiency of this operator, or the presence of its
nontrivial zero modes, signifies new physics of a restricted theory as
compared to the parental model whose field space is subject to restriction.
This gauge restriction operator allows one to construct the projector on the
space of residual gauge symmetries. Projected gauge generators form the
algebra which is closed as long as the original parental theory has a closed
algebra both on and off shell.

Zero vectors of this projector become first-stage reducibility generators of
the BV method of treatment of the restricted theory, and they represent a free
element of the formalism, whose choice is limited only by a certain rank
restriction condition. It turns out, however, that their special normalization
fully provides on-shell independence of the one-loop effective action of this
choice, whereas the normalization of the gauge generators of the parental
theory can be fixed by matching it with the canonical formalism and canonical
quantization of parental theory.

One-loop effective action for generic restricted theory is discussed in 4.1.
Discussion of correct gauge generator normalization in covariant formalism is
the subject of Sects.4.2 ,4.3. In Sect.4.4 we discuss the factor in one-loop
effective action which distinguishes restricted and parental theories at the
quantum level.

In the last section (Sect.5) we apply the developed formalism to unimodular
gravity theory treated as descending from its parental version – Einstein
general relativity. We construct all the elements of its gauge-fixing
procedure in the background covariant gauge [19] with the background
covariance property extended to all the objects of the Feynman diagrammatic
technique. Then, according to the constructed general algorithm, we build on
shell the one-loop effective action of unimodular gravity. It coincides with
the one-loop effective action of the Einstein theory with a cosmological term
for a generic value of its cosmological constant modulo a special contribution
of the global degree of freedom associated with the variable value of this
constant. This confirms conclusions of [11] which were attained, however, with
certain assumptions on the derivation of the group of diffeomorphisms volume
to be factored out off the partition function. No such assumptions are needed
in our approach which is fully fixed by the requirement of gauge independence
of on-shell physical results. Moreover, the final expression for the effective
action, which is usually presented in terms of functional determinants on
irreducible (transverse and transverse-traceless) subspaces of the full tensor
and vector field space, is built in our case in terms of the functional
determinants of the minimal operators which admit for their calculation
application of the heat kernel method and Schwinger-DeWitt local curvature
expansion [20]. This property is important for applications on generic
inhomogeneous backgrounds not exhausted by Einstein spaces.

In Conclusions we summarize the results and briefly discuss the role of global
extra degree of freedom as a cosmological constant and its contribution to the
one-loop quantum corrections. This opens further prospects of BV formalism in
application to restricted gauge theories, probably involving extension to
second-stage reducibility. In appendices we discuss Moore-Penrose construction
of inverse operators for degenerate operators [21], the derivation of gauge
algebra of the projected generators and some technical details of determinant
relations needed for the construction of the generator basis.

##  2 Effective action for theories with first-stage reducible gauge
generators

In this section we briefly review Batalin-Vilkovisky (BV) formalism for the
first-stage reducible gauge theories. We mainly regard this as the gauge
extension of the theory which prescribes suitable way to construct quantum
effective action.

Throughout this section (and in most other sections unless otherwise
indicated) we use DeWitt condensed notations. All indices are of condensed
nature and combine discrete bundle and continuous spacetime point components.
Two-index quantities are two-point kernels and summation over contracted
indices implies the corresponding spacetime integration. Ranges of condensed
indices to be summed and integrated over formally reflect the continuum of
spacetime points and discrete spin-tensor labels of fields. Ranks of two-point
kernels of linear operators and forms thus refer to their functional space
dimensionality.

###  2.1 Batalin-Vilkovisky formalism for reducible gauge theories

Consider a generic gauge theory of fields $\phi^{i}$ (with the range of field
indices ${i}$ formally denoted in what follows by $n$). Let it be described by
the action $S[\,\phi\,]$ which is invariant with respect to local gauge
transformations with gauge generators
$\mathcal{R}^{i}_{\alpha}=\mathcal{R}^{i}_{\alpha}(\phi)$ and infinitesimal
gauge parameters $\xi^{\alpha}$ (the range of gauge indices $\alpha$ being
denoted by $m_{0}<n$)

$\displaystyle\delta_{\xi}\phi^{i}=\mathcal{R}^{i}_{\alpha}\xi^{\alpha},$
(2.1) $\displaystyle S_{,{i}}\mathcal{R}^{i}_{\alpha}=0,\quad
S_{,{i}}\equiv\frac{\delta S}{\delta\phi^{i}}.$ (2.2)

Here and in what follows comma denotes a functional derivative with respect to
the relevant field variable.

The algebra of gauge generators begins with the relations – corollaries of the
Jacobi identity for a double commutator of gauge transformations

$\mathcal{R}^{i}_{\alpha,{j}}\mathcal{R}^{j}_{\beta}-\mathcal{R}^{i}_{\beta,{j}}\mathcal{R}^{j}_{\alpha}=\mathcal{R}^{i}_{\gamma}C^{\gamma}_{\alpha\beta}+E^{{i}{j}}_{\alpha\beta}S_{,{j}}$
(2.3)

with the structure functions
$C^{\gamma}_{\alpha\beta}(\phi)=-C^{\gamma}_{\beta\alpha}(\phi)$,
$E^{{i}{j}}_{\alpha\beta}(\phi)=-E^{{i}{j}}_{\beta\alpha}(\phi)=-E^{{j}{i}}_{\alpha\beta}(\phi)$.
For closed algebras (in contrast to so-called open algebras)
$E^{{i}{j}}_{\alpha\beta}(\phi)$ and higher structures functions of the gauge
algebra vanish [24], [25].

The theory is reducible when its gauge generators are linearly dependent on
shell, that is on solutions of classical equations $S_{,{i}}=0$, so that there
exist $m_{1}$ reducibility generators
$\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}=\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}(\phi)$
– right zero eigenvalue eigenvectors of the original generators
$\mathcal{R}^{i}_{\alpha}$ parameterized by indices
${{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ of range $m_{1}$. This
implies relations

$\mathcal{R}^{i}_{\alpha}\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}=2S_{,{j}}B^{{j}{i}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
(2.4)

with some coefficient functions
$B^{{j}{i}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}(\phi)$. For
the first-stage reducible theories, which we consider in this section, the
generators
$\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
form on shell a _complete_ independent set, whereas for higher order stages of
reducibility they also become linearly dependent with higher order
reducibility generators and so on.

#### Master action

Batalin-Vilkovisky quantization of gauge theories starts with the construction
of _master action_ $S^{\scriptscriptstyle{BV}}[\,\varPhi,\varPhi^{*}\,]$ which
is a certain extension of the original action classical $S[\,\phi\,]$ onto the
configuration space of fields and antifields
$(\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}},\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}})$.
This even-dimensional space is $\mathbb{Z}$-graded with respect to the so-
called ghost number
$\mathop{gh}{(\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}})}=-\mathop{gh}{(\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}})}-1$
and $\mathbb{Z}_{2}$-graded with respect to their Grassmann parity $\epsilon$,
$\epsilon(\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}})=\epsilon(\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}})+1$.
Original fields $\phi^{i}$ become a part of extended fields set
$\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}$
and carry zero ghost number. To avoid messy sign factors we assume that all
$\phi^{i}$ are bosonic (even) fields, $\epsilon(\phi^{i})=0$, and Grassmann
parity of ghost fields and antifields coincides with $\mathbb{Z}_{2}$ parity
of their ghost number (those with even ghost numbers are commuting fields,
those with odd ghosts number – anticommuting).

Extension from $S[\,\phi\,]$ to
$S^{\scriptscriptstyle{BV}}[\,\varPhi,\varPhi^{*}\,]$ is governed by the
_master equation_ on $S^{\scriptscriptstyle{BV}}[\,\varPhi,\varPhi^{*}\,]$

${\mathchoice{\big{(}S^{\scriptscriptstyle{BV}}{\hskip 0.4pt,\hskip
0.4pt}S^{\scriptscriptstyle{BV}}\big{)}}{\big{(}S^{\scriptscriptstyle{BV}}{\hskip
0.4pt,\hskip
0.4pt}S^{\scriptscriptstyle{BV}}\big{)}}{(S^{\scriptscriptstyle{BV}}\,,\,S^{\scriptscriptstyle{BV}})}{(S^{\scriptscriptstyle{BV}},S^{\scriptscriptstyle{BV}})}}\equiv\frac{{\delta^{(r)}}S^{\scriptscriptstyle{BV}}}{\delta{\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}}}\frac{{\delta^{(l)}}S^{\scriptscriptstyle{BV}}}{\delta{\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}}}-\frac{{\delta^{(r)}}S^{\scriptscriptstyle{BV}}}{\delta{\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}}}\frac{{\delta^{(l)}}S^{\scriptscriptstyle{BV}}}{\delta{\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}}}=0$
(2.5)

in terms of the antibracket $(G,H)$ which is defined for any two functionals
$G$ and $H$ of fields and antifields. Here $(r)$ and $(l)$ label respectively
right and left derivatives.

The solution $S^{\scriptscriptstyle{BV}}[\,\varPhi,\varPhi^{*}\,]$ is obtained
as a perturbative expansion in powers of ghost fields and antifields. To make
this solution proper and allow one to perform a further gauge-fixing procedure
one should specify BV-extended configuration space and impose certain initial
conditions to (2.5) which should explicitly encode gauge structure of the
original action $S[\,\phi\,]$.

Any proper solution of master equation is inevitably gauge invariant
[22],[23],[24],[25]. In terms of the total set of BV fields and antifields
$\varPhi\vphantom{\big{|}}_{\scriptscriptstyle{\\!BV}}^{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}=(\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}},\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}})$,
${\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}=1,..,2N$,
${\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}=1,..,N$,
the master equation reads

$\frac{{\delta^{r}}S^{\scriptscriptstyle{BV}}}{\delta\varPhi\vphantom{\big{|}}_{\scriptscriptstyle{\\!BV}}^{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}}\zeta^{{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}{\mathchoice{\mathcal{B}}{\mathcal{B}}{\scriptstyle{\mathcal{B}}}{\mathcal{B}}}}\frac{{\delta^{l}}S^{\scriptscriptstyle{BV}}}{\delta\varPhi\vphantom{\big{|}}_{\scriptscriptstyle{\\!BV}}^{\mathchoice{\mathcal{B}}{\mathcal{B}}{\scriptstyle{\mathcal{B}}}{\mathcal{B}}}}=0,\;\quad\;\zeta^{{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}{\mathchoice{\mathcal{B}}{\mathcal{B}}{\scriptstyle{\mathcal{B}}}{\mathcal{B}}}}=\left[\begin{array}[]{cc}0&\mathbb{I}\vphantom{\big{|}^{i}}\\\
\\!-\mathbb{I}\\!&0\vphantom{\big{|}}\\\ \end{array}\right].$ (2.6)

By differentiating (2.6) one gets Noether identities

$\frac{{\delta^{r}}S^{\scriptscriptstyle{BV}}}{\delta\varPhi\vphantom{\big{|}}_{\scriptscriptstyle{\\!BV}}^{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}}\mathcal{R}^{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}_{\,{\mathchoice{\mathcal{C}}{\mathcal{C}}{\scriptstyle{\mathcal{C}}}{\mathcal{C}}}}=0,\;\quad\;\mathcal{R}^{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}_{\,{\mathchoice{\mathcal{C}}{\mathcal{C}}{\scriptstyle{\mathcal{C}}}{\mathcal{C}}}}=\zeta^{{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}{\mathchoice{\mathcal{B}}{\mathcal{B}}{\scriptstyle{\mathcal{B}}}{\mathcal{B}}}}\frac{\delta^{l}\delta^{r}S^{\scriptscriptstyle{BV}}}{\delta\varPhi\vphantom{\big{|}}_{\scriptscriptstyle{\\!BV}}^{\mathchoice{\mathcal{B}}{\mathcal{B}}{\scriptstyle{\mathcal{B}}}{\mathcal{B}}}\delta\varPhi\vphantom{\big{|}}_{\scriptscriptstyle{\\!BV}}^{\mathchoice{\mathcal{C}}{\mathcal{C}}{\scriptstyle{\mathcal{C}}}{\mathcal{C}}}}.$
(2.7)

featuring $2N$ gauge generators
$\mathcal{R}^{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}_{\,{\mathchoice{\mathcal{C}}{\mathcal{C}}{\scriptstyle{\mathcal{C}}}{\mathcal{C}}}}$
of master action gauge symmetry.

These generators however form a reducible set because the second variational
derivative of (2.6) shows their on-shell $N$-reducibility,
$\zeta^{-1}_{{\mathchoice{\mathcal{D}}{\mathcal{D}}{\scriptstyle{\mathcal{D}}}{\mathcal{D}}}{\mathchoice{\mathcal{B}}{\mathcal{B}}{\scriptstyle{\mathcal{B}}}{\mathcal{B}}}}\mathcal{R}^{\mathchoice{\mathcal{B}}{\mathcal{B}}{\scriptstyle{\mathcal{B}}}{\mathcal{B}}}_{\,{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}}\mathcal{R}^{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}_{\,{\mathchoice{\mathcal{C}}{\mathcal{C}}{\scriptstyle{\mathcal{C}}}{\mathcal{C}}}}\propto\delta
S^{\scriptscriptstyle{BV}}/\delta\varPhi\vphantom{\big{|}}_{\scriptscriptstyle{\\!BV}}^{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}=0$.
Thus master action $S^{\scriptscriptstyle{BV}}$ possess $N$ on-shell
independent symmetries – half the dimension of the BV configuration space
$(\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}},\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}})$.
111 Nilpotence of $2N\times 2N$ matrix implies that its rank is at most $N$.
Considering _proper_ solution implies that rank of the hessian
$\delta^{l}\delta^{r}S^{\scriptscriptstyle{BV}}/\delta\varPhi\vphantom{\big{|}}_{\scriptscriptstyle{\\!BV}}^{\mathchoice{\mathcal{A}}{\mathcal{A}}{\scriptstyle{\mathcal{A}}}{\mathcal{A}}}\delta\varPhi\vphantom{\big{|}}_{\scriptscriptstyle{\\!BV}}^{\mathchoice{\mathcal{B}}{\mathcal{B}}{\scriptstyle{\mathcal{B}}}{\mathcal{B}}}$
equals $N$. Otherwise the set of Noether identities (2.7) is incomplete and
there are more then $N$ gauge generators since there are more then $N$ zero
modes of the hessian.

Convenient gauge fixing of master action gauge symmetry (2.7) consists in
expressing antifields in terms of fields via $N$ gauge conditions
[22],[23],[24],[25]

$\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}-\frac{\delta\varPsi[\,\varPhi\,]}{\delta\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}}=0.$
(2.8)

It is parameterized by a single functional of fields – gauge fermion
$\varPsi[\,\varPhi\,]$ of ghost number $-1$ and leads to gauge-fixed action

$S_{\varPsi}[\,\varPhi\,]=S^{\scriptscriptstyle{BV}}\big{[}\,\varPhi,\delta\varPsi[\,\varPhi\,]/\delta\varPhi\,\big{]}$
(2.9)

which is functional of only BV fields
$\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}$.
For correctly chosen gauge fermion the gauge-fixed action (2.9) appears to be
nondegenerate
$\mathrm{sdet}\,[\,\delta^{(l)}\delta^{(r)}S_{\varPsi}/\delta\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}\,\delta\varPhi^{\mathchoice{\mathcal{J}}{\mathcal{J}}{\scriptstyle{\mathcal{J}}}{\mathcal{J}}}\,]_{\,\delta
S_{\varPsi}/\delta\varPhi=0}\neq 0$ and yields perturbatively consistent path
integral for the generating functional which modulo the contribution of local
measure reads

$Z=\int D\varPhi\;e^{\,i\,S_{\varPsi}[\,\varPhi\,]}.$ (2.10)

Master equation for $S^{\scriptscriptstyle{BV}}[\,\varPhi,\varPhi^{*}\,]$
(2.5) provides the corner stone of gauge theory quantization – independence of
the on-shell generating functional of the choice of gauge fermion. Under the
gauge fermion change $\varPsi\to\varPsi+\Delta\varPsi$ in view of integration
by parts the change in $Z$ reads

$\displaystyle\Delta Z=-\int
D\varPhi\,\Delta\varPsi\,\left(\,i\,\frac{\delta^{(r)}S^{\scriptscriptstyle{BV}}}{\delta\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}}\frac{\delta^{(l)}S^{\scriptscriptstyle{BV}}}{\delta\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}}+\frac{\delta^{(r)}}{\delta\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}}\frac{\delta^{(l)}S^{\scriptscriptstyle{BV}}}{\delta\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}}\right)\Big{|}_{\,\varPhi^{*}=\frac{\delta\varPsi}{\delta\varPhi}}\,e^{\,i\,S_{\varPsi}},$
(2.11)

where extra terms vanish in view of Grassmann symmetry properties. This
expression equals zero, because the first term vanishes in view of the master
equation
$\delta^{(r)}S^{\scriptscriptstyle{BV}}/\delta\varPhi^{*}_{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}\,\delta^{(l)}S^{\scriptscriptstyle{BV}}/\delta\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}=\tfrac{1}{2}(S^{\scriptscriptstyle{BV}},S^{\scriptscriptstyle{BV}})$,
while the second term in local theories is proportional to
$\partial...\partial\,\delta(0)$ and is supposed to be either cancelled by a
local measure or killed by dimensional regularization. The local measure is
not rigorously available within BV formalism whose incompleteness can be
disregarded for theories with local gauge algebra and within the class of
local gauge fermions. This measure can be attained within canonical BFV
quantization formalism, which will be used below to justify the application of
BV method for restricted gauge theories in which locality of their generators
will be spoiled by generically nonlocal projectors.

#### Minimal and nonminimal proper solutions

Construction of BV master action begins with finding proper solution of master
equation (2.5) on the minimal sector of fields and antifields. For the first-
stage reducible theories (2.2),(2.4) this requires introduction of ghosts
$\mathcal{C}^{\alpha}$, $\mathop{gh}{(\mathcal{C}^{\alpha})}=1$, the ghosts
for ghosts $\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$,
$\mathop{gh}{(\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}})}=2$,
and antifields to all fields and ghosts [15],[24],[25],

$\varPhi_{\rm
min}=(\phi^{i},\mathcal{C}^{\alpha},\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}),\;\;\varPhi^{*}_{\rm
min}=(\phi^{*}_{i},\mathcal{C}^{*}_{\alpha},\mathcal{C}^{*}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}})$
(2.12)

The _minimal_ proper solution $S^{\rm min}[\,\varPhi_{\rm
min},\varPhi^{*}_{\rm min}\,]$ of master equation can be represented as the
series in powers of antifields

$S^{\rm
min}=S+\phi^{*}_{i}\mathcal{R}^{i}_{\alpha}\mathcal{C}^{\alpha}+\mathcal{C}^{*}_{\alpha}\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}+...$
(2.13)

where $S=S[\,\phi\,]$ is the action of the initial gauge theory, two terms
bilinear in ghost fields and antifields serve as initial conditions for master
equation which guaranty that such solution of master equation is proper,
$\mathcal{R}^{i}_{\alpha}$ and
$\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
are gauge and reducibility generators (2.2),(2.4), dots stand for higher order
terms in powers of ghost fields and antifields. All higher order terms can be
found within the iterative procedure of solving the master equation.222
Minimal master action with numerous higher-order terms reads [24],[25]
$\displaystyle S^{min}$ $\displaystyle\;=$ $\displaystyle
S[\,\phi\,]+\phi^{*}_{i}\mathcal{R}^{i}_{\alpha}\mathcal{C}^{\alpha}+\mathcal{C}^{*}_{\alpha}\left(\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}+\tfrac{1}{2}C_{\beta\gamma}^{\alpha}\mathcal{C}^{\gamma}\mathcal{C}^{\beta}\right)+\mathcal{C}^{*}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\left(A_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}\alpha}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}\mathcal{C}^{\alpha}+\tfrac{1}{2}F_{\alpha\beta\gamma}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\mathcal{C}^{\gamma}\mathcal{C}^{\beta}\mathcal{C}^{\alpha}\right)$
$\displaystyle+\phi^{*}_{i}\phi^{*}_{j}\left({\tfrac{1}{2}B_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}^{{j}{i}}\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}+\tfrac{1}{4}E_{\alpha\beta}^{{j}{i}}\mathcal{C}^{\beta}\mathcal{C}^{\alpha}}\right)+\ldots\quad$
where the coefficient functions of higher-order terms originate from higher
order structure functions of gauge algebra and the dots denote terms with
antifields of the total ghost number $-3$ and lower.

Meaningful gauge fixing of the BV master action requires extension of the
proper solution to nonminimal configuration space. Motivation for this is that
gauge fixing (2.8) is performed through properly constructed _gauge fermion_
$\varPsi[\,\varPhi\,]$ – the functional of only the BV fields
$\varPhi^{\mathchoice{\mathcal{I}}{\mathcal{I}}{\scriptstyle{\mathcal{I}}}{\mathcal{I}}}$.
But the gauge fermion has a negative ghost number $-1$ and thus can not be
constructed as a functional of the minimal fields
$\phi^{i},\mathcal{C}^{\alpha},\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$
with nonnegative ghost numbers. Therefore the set of fields should be further
extended to contain auxiliary ghosts with negative ghost numbers.

Standard scheme for first-stage reducible theories presumes introducing
auxiliary ghosts
$\bar{\mathcal{C}}_{\alpha},\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$,
their attendant companions
$\pi_{\alpha},\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
with ghost number increased by one and the corresponding set of
antifields333Fields
$\bar{\mathcal{C}}_{\alpha},\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$
are often referred as antighosts,
$\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$ –
extraghosts,
$\pi_{\alpha},\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
– Lagrange multiplier or Nakanishi-Lautrup fields. [24],[25] so that
nonminimal BV configuration space reads

$\varPhi=(\underbrace{\phi^{i},\mathcal{C}^{\alpha},\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\varPhi_{\rm
min}},\bar{\mathcal{C}}_{\alpha},\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}},\pi_{\alpha},\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}),\;\;\varPhi^{*}=(\underbrace{\phi^{*}_{i},\mathcal{C}^{*}_{\alpha},\mathcal{C}^{*}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\varPhi^{*}_{\rm
min}},{\bar{\mathcal{C}}^{*}}^{\alpha},{\bar{\mathcal{C}}^{*}}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\mathcal{C}^{\prime*}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},{\pi^{*}}^{\alpha},{\pi^{*}}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},{\pi^{\prime*}}_{\\!\\!\\!{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}})$
(2.14)

The _nonminimal_ proper solution
$S^{\scriptscriptstyle{BV}}[\varPhi,\varPhi^{*}]$ of the master equation on
such space is just a sum of the minimal master action $S^{\rm
min}[\varPhi_{\rm min},\varPhi^{*}_{\rm min}]$ (2.13) and the contribution of
auxiliary fields $S^{\rm
aux}[\pi_{\alpha},\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}},{\bar{\mathcal{C}}^{*}}^{\alpha},{\bar{\mathcal{C}}^{*}}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\mathcal{C}^{\prime*}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}]$

$\displaystyle S^{\scriptscriptstyle{BV}}=S^{\rm min}+S^{\rm aux},$ (2.15)
$\displaystyle S^{\rm
aux}=\,\pi_{\alpha}{\bar{\mathcal{C}}^{*}}^{\alpha}+\,\pi_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}{\bar{\mathcal{C}}^{*}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}+\,\mathcal{C}^{\prime*}_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}.$
(2.16)

Master action (2.15) obviously satisfies master equation (2.5) on nonminimal
configuration space since auxiliary and minimal sectors are so far decoupled
(these sectors will be mixed after gauge-fixing) so that $(S^{\rm min}+S^{\rm
aux},S^{\rm min}+S^{\rm aux})=(S^{\rm min},S^{\rm min})+(S^{\rm aux},S^{\rm
aux})$, and by construction $S^{\rm min}$ and $S^{\rm aux}$ separately
nilpotent in antibracket.

Nonminimal set of fields and antifields for the first-stage reducible gauge
theory is listed in Table 1 where companion antifields
${\pi^{*}}^{\alpha},{\pi^{*}}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},{\pi^{\prime*}}_{\\!\\!\\!{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
are omitted since they do not appear in BV procedure.

|  $\varPhi$ — | $\mathop{gh}{(\varPhi)}$ |  |  $\varPhi^{*}$ — | $\mathop{gh}{(\varPhi^{*})}$ |
---|---|---|---|---|---|---
$\,\phi^{i}$$(\bar{\mathcal{C}}_{\alpha},\pi_{\alpha})$$\mathcal{C}^{\alpha}$$(\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}})$$(\bar{\mathcal{C}}_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}},\pi_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}})$$\mathcal{C}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$The triangular field graph | minimal sector
$\phi^{i}$ | $0$ |  | $\phi^{*}_{i}$ | $-1$ |
$\mathcal{C}^{\alpha}$ | $+1$ |  | $\mathcal{C}^{*}_{\alpha}$ | $-2$ |
$\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ | $+2$ |  | $\mathcal{C}^{*}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ | $-3$ |
auxiliary sector
$\bar{\mathcal{C}}_{\alpha}$ | $-1$ |  | ${\bar{\mathcal{C}}^{*}}^{\alpha}$ | $0$ |
$\qquad\pi_{\alpha}$ | $\qquad 0$ |  |  |  |
$\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ | $-2$ |  | ${\bar{\mathcal{C}}^{*}}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ | $+1$ |
$\qquad\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ | $\qquad-1$ |  |  |  |
$\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$ | $0$ |  | $\mathcal{C}^{\prime*}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ | $-1$ |
| $\qquad\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$ | $\qquad+1$ |  |  |  |
Table 1: BV field content of first-stage reducible gauge theory

#### Choice of a gauge fermion for the Gaussian gauge fixing

Grading restriction $\mathop{gh}{(\varPsi)}=-1$ on gauge fermion
$\varPsi[\,\varPhi\,]$ admits the following form at most quadratic in ghosts
and auxiliary fields

$\displaystyle\varPsi[\,\varPhi\,]\\!$ $\displaystyle=$
$\displaystyle\\!\bar{\mathcal{C}}_{\alpha}\big{(}\chi^{\alpha}(\phi)+\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}(\phi)\,\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\big{)}+\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}(\phi)\,\mathcal{C}^{\alpha}+\tfrac{1}{2}\,\big{(}\bar{\mathcal{C}}_{\alpha}\varkappa^{\alpha\beta}\pi_{\beta}+\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}{+\,}\pi_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\big{)}.\;\;\;\;$
(2.17)

The latter group of terms linear in
$\pi_{\alpha},\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
allows to implement _Gaussian_ (Faddeev-Popov) type of gauge fixing when
$\varkappa^{\alpha\beta},\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
are _nondegenerate_

$\det{\varkappa^{\alpha\beta}}\neq
0,\qquad\det{\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}\neq
0.$ (2.18)

In generic theories it is convenient to choose $\phi$-independent coefficient
functions
$\varkappa^{\alpha\beta},\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$.
Such choice guarantees possibility to integrate out companion fields
$\pi_{\alpha},\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
to obtain Faddeev-Popov representation for generating functional.444 For
$\phi$-dependent functions
$\varkappa^{\alpha\beta}(\phi),\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}(\phi)$
terms linear in companion fields
$(\pi_{\alpha},\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}})$
will arise in gauge-fixed minimal master action $S_{\varPsi}^{\rm min}$
instead of each antifield:
$\phi^{*}_{i}=\delta\varPsi/\delta\phi^{i}=\tfrac{1}{2}{\mathcal{C}}_{\alpha}(\delta\varkappa^{\alpha\beta}/\delta\phi^{i})\,\pi_{\beta}+...\,$.
Therefore for open algebras and algebras with higher structure functions,
whose $S^{\rm min}$ contains second and higher powers of antifields,
$(\pi_{\alpha},\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}})$-dependence
of $S_{\varPsi}^{\rm min}$ becomes more than quadratic, which does not allow
to integrate them out via Gaussian integration. Also one can assume
$\varkappa^{\alpha\beta}$ to be _symmetric_ , because after gauge fixing the
nonminimal action antisymmetric part of $\varkappa^{\alpha\beta}$ is lost.

After the gauge fixing (2.9), $\varPhi^{*}\to\delta\varPsi/\delta\varPhi$ with
the gauge fermion (2.17) one gets the nonminimal gauge-fixed action (2.9)
which according to (2.15) is $S_{\varPsi}=S_{\varPsi}^{\rm
min}+S_{\varPsi}^{\rm aux}$ with

$\displaystyle S^{\rm min}_{\varPsi}$ $\displaystyle=$ $\displaystyle
S\,+\,\bar{\mathcal{C}}_{\alpha}\big{(}X^{\alpha}_{,{i}}\mathcal{R}^{i}_{\beta}\big{)}\mathcal{C}^{\beta}+\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\big{(}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\big{)}\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}+...,$
(2.19) $\displaystyle S^{\rm aux}_{\varPsi}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\pi_{\alpha}\varkappa^{\alpha\beta}\pi_{\beta}+\pi_{\alpha}X^{\alpha}\,+\,\pi_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}+\pi_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}\mathcal{C}^{\alpha}+\bar{\mathcal{C}}_{\alpha}\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}},$
(2.20)

where $S=S[\,\phi\,]$ is the action of the initial gauge theory and dots hide
terms more then quadratic in ghost fields with nonzero ghost numbers.555 Terms
explicated in (2.19) and (2.20) are sufficient in the one-loop approximation.
Gauge fixed action with several higher order terms reads  $\displaystyle
S^{\rm min}_{\varPsi}[\varPhi]$ $\displaystyle=$ $\displaystyle
S[\phi]\,+\,\bar{\mathcal{C}}_{\alpha}\big{(}X^{\alpha}_{,{i}}\mathcal{R}^{i}_{\beta}\big{)}\mathcal{C}^{\beta}+\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\big{(}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\big{)}\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}+\,\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\big{(}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\alpha,{i}}\mathcal{R}^{i}_{\beta}+\tfrac{1}{2}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}C_{\beta\gamma}^{\alpha}\big{)}\mathcal{C}^{\gamma}\mathcal{C}^{\beta}+\bar{\mathcal{C}}_{\alpha}\bar{\mathcal{C}}_{\beta}\big{(}\tfrac{1}{2}X^{\alpha}_{,{i}}X^{\alpha}_{,{j}}B_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}^{{j}{i}}\big{)}\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}+\ldots\,\,.$
We have introduced combination

$X^{\alpha}\equiv\chi^{\alpha}(\phi)+\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}(\phi)\,\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}$
(2.21)

which depends on fields
$\phi^{i},\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
with zero ghost number. All the dependence on $\mathcal{C}^{\prime}$ in the
gauge-fixed action $S_{\varPsi}$ is now hidden within $X^{\alpha}$ and its
variational derivative

$\displaystyle X^{\alpha}_{,{i}}\equiv\delta
X^{\alpha}/\delta\phi^{i}=\chi^{\alpha}_{\,,{i}}+\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}},{i}}\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}.$
(2.22)

#### Reduction of companion auxiliary fields

For $\phi$-independent matrices $\varkappa^{\alpha\beta}$ and
$\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
– the case which we consider in what follows – Gaussian integration over the
fields
$\pi_{\alpha},\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
in the generating functional (2.10) gives

$Z=\int D\varPhi_{\rm
red}\;\big{(}\\!\det\varkappa^{\alpha\beta}\big{)}^{-1/2}\;{\det}\,\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\;e^{\,i\,S^{\scriptscriptstyle{FP}}[\,\varPhi_{\rm
red}\,]},$ (2.23)

where $\varPhi_{\rm red}$ is the reduced nonminimal set of fields
$\varPhi_{\rm
red}=(\phi^{i},\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}},\mathcal{C}^{\alpha},\bar{\mathcal{C}}_{\alpha},\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}})$,
cf. Eq.(2.14), and $S^{\scriptscriptstyle{FP}}[\,\varPhi_{\rm red}\,]$ is the
corresponding Faddeev-Popov reduced gauge-fixed action

$\displaystyle S^{\scriptscriptstyle{FP}}[\,\varPhi_{\rm red}\,]\\!$
$\displaystyle=$
$\displaystyle\\!S[\,\phi\,]-\tfrac{1}{2}X^{\alpha}{\varkappa}_{\alpha\beta}X^{\beta}\,+\,\bar{\mathcal{C}}_{\alpha}\big{(}X^{\alpha}_{,{i}}\mathcal{R}^{i}_{\beta}-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\,\rho^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\,\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}\big{)}\mathcal{C}^{\beta}+\,\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\big{(}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\big{)}\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}+...\,.\;\;\;\;\;$
(2.24)

Here we have introduced the matrices ${\varkappa}_{\alpha\beta}$,
${\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
inverse respectively to $\varkappa^{\beta\alpha}$,
$\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$.
Dots denote terms which are more then quadratic in ghost fields
$\mathcal{C}^{\alpha},\bar{\mathcal{C}}_{\alpha},\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$.
Such terms are irrelevant in one-loop approximation.

The obtained reduced action is non-degenerate in the sense that its Hessian at
the stationary points of the classical BV action represents the nondegenerate
operator

$\mathrm{sdet}\,{\frac{\delta^{(l)}\delta^{(r)}S^{\scriptscriptstyle{FP}}}{\delta\varPhi_{\rm
red}\,\delta\varPhi_{\rm red}}}\Bigg{|}_{\,\delta
S^{\scriptscriptstyle{FP}}/\delta\varPhi_{\rm
red}=0}\\!\\!\\!\\!\\!\\!\\!\\!\\!\neq 0$ (2.25)

for the relevant set of boundary conditions. This property is inherited from
the nondegeneracy of unreduced gauge-fixed action $S_{\varPsi}[\,\varPhi\,]$
with appropriately chosen gauge fermion, because integrated out were the
auxiliary nondynamical variables, which were expressed from their own
equations of motion. For nondegenerate gauge-fixing matrices
$\varkappa^{\beta\alpha}$,
$\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
companion fields
$\pi_{\alpha},\pi_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\pi^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
form the set of such auxiliary variables.

Now we will analyze nondegeneracy in various sectors of the reduced
configuration space.

###  2.2 Stationary point of the gauge-fixed master action

The simplest is the ghost sector consisting of the fields
$\mathcal{C}^{\alpha},\bar{\mathcal{C}}_{\alpha},\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$
(the field
$\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$ with
zero ghost number does not belong to this sector even though it is usually
called extra ghost [15] – rather it belongs to zero ghost number sector where
it plays a special role in gauge-fixing procedure for $\phi^{i}$). The
variational equations for the fields
$\mathcal{C}^{\alpha},\bar{\mathcal{C}}_{\alpha},\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$
obviously make them vanishing under zero boundary conditions

$\displaystyle\mathcal{C}^{\alpha}=\bar{\mathcal{C}}_{\alpha}=\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=0,$
(2.26)

provided the kernels of their bilinear forms in gauge-fixed action (2.24)
represent invertible operators having well defined Green’s functions

$\displaystyle
F^{\alpha}_{\;\beta}=X^{\alpha}_{,{i}}\mathcal{R}^{i}_{\beta}-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}{\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\,\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta},\quad\det
F^{\alpha}_{\;\beta}\neq 0,$ (2.27) $\displaystyle
F^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}},\qquad\qquad\qquad\quad\det
F^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\neq
0.$ (2.28)

The interpretation of these operators and their properties is obvious. The
first term of the ghost operator $F^{\alpha}_{\;\beta}$ is degenerate because
of the reducibility of gauge generators, so the conventional Faddeev-Popov
ghosts $\mathcal{C}^{\alpha},\bar{\mathcal{C}}_{\alpha}$ themselves become
gauge fields with the local symmetry induced by the reducibility generators
$\mathcal{Z}^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$.
The second term of $F^{\alpha}_{\;\beta}$ in (2.27) plays the role of gauge
breaking term for this symmetry, and its effect is the invertibility of
$F^{\alpha}_{\;\beta}$. Correspondingly,
$F^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
is the Faddeev-Popov operator of new ghosts
$\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$
for original ghosts $\mathcal{C}^{\alpha},\bar{\mathcal{C}}_{\alpha}$ treated
as gauge fields, that is why the fields
$\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$
bear the name of ghosts for ghosts.

The situation is trickier in the zero ghost number sector of fields
$\phi^{i},\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$.
Their equations of motion read

$\displaystyle\frac{\delta S^{\scriptscriptstyle{FP}}}{\delta\phi^{i}}$
$\displaystyle=$ $\displaystyle
S_{,{i}}-X^{\alpha}_{,{i}}{\varkappa}_{\alpha\beta}X^{\beta}=0,$ (2.29)
$\displaystyle\frac{\delta
S^{\scriptscriptstyle{FP}}}{\delta\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}}$
$\displaystyle=$
$\displaystyle-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}{\varkappa}_{\alpha\beta}X^{\beta}=0.$
(2.30)

Contracting them respectively with $\mathcal{R}^{i}_{\gamma}$ and
${\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\gamma}$
and subtracting from one another we obtain on account of the Noether identity
$S_{,{i}}\mathcal{R}^{i}_{\gamma}=0$ the following on-shell relation

$\Big{(}X^{\alpha}_{,{i}}\mathcal{R}^{i}_{\gamma}-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}{\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\gamma}\Big{)}{\varkappa}_{\alpha\beta}X^{\beta}\;=\;F^{\alpha}_{\;\gamma}{\varkappa}_{\alpha\beta}X^{\beta}=0,$
(2.31)

which allows one in view of invertibility of $F^{\alpha}_{\;\gamma}$ and
${\varkappa}_{\alpha\beta}$ to rewrite the equations of motion in the zero
ghost number sector as

$\displaystyle S_{,{i}}=0,$ (2.32) $\displaystyle
X^{\alpha}\equiv\chi^{\alpha}\\!+\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}=0.$
(2.33)

This can be interpreted as equations of motion for $\phi^{i}$ supplied by the
set of gauge conditions $X^{\alpha}=0$. The latter look overcomplete because
the number of independent equations of motion $S_{,{i}}=0$ on $n$ variables
$\phi^{i}$ is $n-m_{0}+m_{1}$ whereas the number of gauges is $m_{0}$. The
$m_{1}$ mismatch is, however, corrected by the $m_{1}$ extra-ghost fields
$\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$. This
goes as follows.

Contracting the nondegenerate rank $m_{0}$ operator $F^{\alpha}_{\;\beta}$
with the full the rank $m_{1}$ generator
$\mathcal{Z}^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}$ one
gets the full rank condition,
$\operatorname{\,\mathrm{rank}\,}{(\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}{\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\,\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}\mathcal{Z}^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}})}=m_{1}$,
which due to nondegeneracy of
$F^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}=\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}\mathcal{Z}^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}$
and
${\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
implies the full rank conditions
$\operatorname{\,\mathrm{rank}\,}{\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}}=m_{1}$,
$\operatorname{\,\mathrm{rank}\,}{\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}}=m_{1}$.
This property in particular guarantees that all
$\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$ are
expressible in terms of $\phi^{i}$ from the equations $X^{\alpha}=0$.

By introducing an arbitrary matrix
${s}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}(\phi^{i})$
such that
$\det{({s}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}})}\neq
0$ one can express the solution for
$\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$ in
terms of $\phi^{i}$ from Eq. (2.33)

$\mathcal{C}^{\prime\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}=-({s}\sigma)^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\,{s}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\,\beta}\,\chi^{\beta}$
(2.34)

where
$({s}\sigma)^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
is the inverse of
$(s\sigma)^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}={s}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\gamma}\sigma^{\gamma}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$.
On the substitution of this solution in (2.33) the rest of relations
constitute $m_{0}-m_{1}$ independent gauge conditions

$\Big{(}\delta^{\alpha}_{\,\beta}-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}({s}\sigma)^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}\,}{s}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}\Big{)}\chi^{\beta}=0.$
(2.35)

on $n$ original fields $\phi^{i}$ as it should be. Indeed, what stands here as
a matrix coefficient of $\chi^{\beta}$ is the projector
$T^{\alpha}_{\>\beta}(\sigma,s)$ in the space of gauge indices having $m_{1}$
right zero eigenvalue eigenvectors
$\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$ and
$m_{1}$ left zero covectors
$s^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\;\alpha}$,

$\displaystyle
T^{\alpha}_{\>\beta}(\sigma,s)=\delta^{\alpha}_{\,\beta}-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}({s}\sigma)^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}{s}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta},\quad\;\;T^{\alpha}_{\>\beta}(\sigma,s)\,\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=0,\quad\;\;s^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\;\alpha}T^{\alpha}_{\>\beta}(\sigma,s)=0,$
(2.36) $\displaystyle
T^{\alpha}_{\>\beta}(\sigma,s)\,T^{\beta}_{\>\gamma}(\sigma,s)=T^{\alpha}_{\>\gamma}(\sigma,s),\quad\;\;{\rm
rank}\,T^{\alpha}_{\>\beta}(\sigma,s)=m_{0}-m_{1},$ (2.37)

which explains why Eq.(2.35) comprises $m_{0}-m_{1}$ gauge conditions rather
than $m_{0}$ ones. It is remarkable that despite the presence of an auxiliary
element – an arbitrary matrix
${s}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}$ of rank
$m_{1}$ – these gauge conditions and expressions for extra ghosts (2.34) are
$s$-independent. This easily follows on shell from the variational equation

$\delta_{{s}}\big{[}\,({s}\sigma)^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}{s}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}\chi^{\beta}\,\big{]}=({s}\sigma)^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\,\;\;\;\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}\,(\delta{{s}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}_{\,\gamma}})\,T^{\gamma}_{\>\beta}(\sigma,s)\,\chi^{\beta}=0.$
(2.38)

Finally, consider the on-shell Hessian of the action in the zero ghost number
sector of fields $\varPhi_{0}=(\phi^{i}$,
$\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}})$ which
is actually the Hessian of
$S[\,\phi\,]-\tfrac{1}{2}X^{\alpha}{\varkappa}_{\alpha\beta}X^{\beta}$.
Bearing in mind the on-shell value of $X^{\alpha}=0$ we get the block-matrix
operator

$\left.\frac{\delta^{2}S^{\scriptscriptstyle{FP}}}{\delta\varPhi_{0}\,\delta\varPhi_{0}}\right|_{\,\delta
S^{\scriptscriptstyle{FP}}/\delta\varPhi_{\rm
red}=0}\\!\\!\\!\\!=\left[\begin{array}[]{cc}S_{,{i}{j}}{-}X^{\alpha}_{,{i}}{\varkappa}_{\alpha\beta}X^{\beta}_{,{j}}&\,\,-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\,{\varkappa}_{\alpha\beta}X^{\beta}_{,{i}}\vphantom{\Big{|}}\\\
-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\,{\varkappa}_{\alpha\beta}X^{\beta}_{,{i}}&\,\,-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}{\varkappa}_{\alpha\beta}\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\vphantom{\Big{|}}\end{array}\right].$
(2.39)

In view of invertibility of $\varkappa_{\alpha\beta}$ one can introduce the
nondegenerate $m_{1}\times m_{1}$ operator
$\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
and its inverse

$\displaystyle\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}{\varkappa}_{\alpha\beta}\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}},\quad\;\;{\kappa}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\equiv\big{(}\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\big{)}^{-1}.$
(2.40)

Then one can factorize the determinant of (2.39) as the product of
determinants of two operators

$\displaystyle\left.\det\,\frac{\delta^{2}S^{\scriptscriptstyle{FP}}}{\delta\varPhi_{0}\,\delta\varPhi_{0}}\right|_{\,\delta
S^{\scriptscriptstyle{FP}}/\delta\varPhi_{\rm red}=0}\\!\\!\\!\\!=\det
F_{ij}\,\det{\big{(}{-}\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\big{)}},$
(2.41)

where the new operator $F_{ij}$ is obviously the gauge-fixed inverse
propagator of fields $\phi^{i}$

$\displaystyle F_{{i}{j}}\equiv
S_{,{i}{j}}-X^{\alpha}_{,{i}}\,\varPi_{\alpha\beta}X^{\beta}_{,{j}},$ (2.42)
$\displaystyle\varPi_{\alpha\beta}\equiv{\varkappa}_{\alpha\beta}-{\varkappa}_{\alpha\gamma}\sigma^{\gamma}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}{\kappa}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\sigma^{{\,\delta}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}{\varkappa}_{{\,\delta}\beta},$
(2.43)

in which the gauge-fixing term is built in terms of the gauge matrices
$X^{\alpha}_{,{i}}$ and the gauge-fixing matrix $\varPi_{\alpha\beta}$. The
latter, however, does not coincide with the original matrix
$\varkappa_{\alpha\beta}$ in the gauge fermion (2.17), but rather converted to
the projector form with the following properties

$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\varPi_{\alpha\beta}=\varPi_{\alpha\beta}\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=0,\qquad\varPi_{\alpha\beta}\varkappa^{\beta\gamma}\varPi_{\gamma{\,\delta}}=\varPi_{\alpha{\,\delta}}.$
(2.44)

This projection is fully consistent with the fact that the rank
$n-m_{0}+m_{1}$ of $S_{,ij}$ should be raised up to $n$ by adding nonzero
eigenvalues not in $m_{0}$-dimensional subspace, but in the
$(m_{0}-m_{1})$-dimensional one. In particular, with the projector (2.43) one
can rewrite on-shell gauge conditions (2.35) on fields $\phi^{i}$ as
$\varPi_{\alpha\beta}\chi^{\beta}=0$ . The equivalence of this to (2.35) is
established by choosing
${s}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\beta}=m^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}{\varkappa}_{\alpha\beta}$
and noting that the projector
$\delta^{\alpha}_{\,\beta}-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}({s}\sigma)^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}{s}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}$
does not depend on a nondegenerate matrix factor
$m^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$.

###  2.3 One-loop contribution to the generating functional

From (2.23) the one-loop contribution to the generating functional (its pre-
exponential factor) reads

$\displaystyle Z^{1-\rm
loop}=\,\frac{\det\rho^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}{\big{(}\\!\det\varkappa^{\alpha\beta}\big{)}^{1/2}}\,\left(\mathrm{sdet}\,\frac{\delta^{(l)}\delta^{(r)}S^{\scriptscriptstyle{FP}}}{\delta\varPhi_{\rm
red}\,\delta\varPhi_{\rm red}}\,\right)^{-1/2}$ (2.45)

where the Hessian in the sector of reduced fields has in virtue of (2.24) the
following block matrix structure

$\displaystyle\frac{\delta^{(l)}\delta^{(r)}S^{\scriptscriptstyle{FP}}}{\delta\varPhi_{\rm
red}\,\delta\varPhi_{\rm red}}$ $\displaystyle=$
$\displaystyle\left[\begin{array}[]{c:c:c}{\displaystyle\frac{\delta^{2}S^{\scriptscriptstyle{FP}}}{\delta\varPhi_{0}\,\delta\varPhi_{0}}}&\begin{array}[]{cc}0&\,\,0\\\
0&\,\,0\vphantom{\big{|}_{|}}\\\ \end{array}&\begin{array}[]{cc}0&\,\,0\\\
0&\,\,0\\\ \end{array}\\\ \hdashline\begin{array}[]{cc}0&\,\,0\\\ 0&\,\,0\\\
\end{array}&\begin{array}[]{cc}\;\,0&-F^{\beta}_{\;\alpha}\vphantom{\big{|}^{|}}\\\
F^{\alpha}_{\;\beta}\vphantom{\big{|}_{|}}\\!\\!\\!&0\\\
\end{array}&\begin{array}[]{cc}0&\,\,0\\\ 0&\,\,0\\\ \end{array}\\\
\hdashline\begin{array}[]{cc}0&\,\,0\\\ 0&\,\,0\\\
\end{array}&\begin{array}[]{cc}0&\,\,0\\\ 0&\,\,0\\\
\end{array}&\begin{array}[]{cc}0&\\!F^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\vphantom{\big{|}^{|}}\\\
\\!F^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\vphantom{\big{|}_{|}}&0\\\
\end{array}\\\ \end{array}\right]\,.$ (2.65)

Left column and upper row here corresponds to the zero ghost number sector
$\varPhi_{0}=(\phi^{i},\mathcal{C}^{\prime{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}})$,
middle column and row corresponds to the odd sector
$(\mathcal{C}^{\beta},\bar{\mathcal{C}}_{\beta})$ of gauge ghost and antighost
fields, right bottom
$(\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}},\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}})$
sector corresponds to the even sector of ghost-for-ghost and antighost-for-
ghosts fields.

In virtue of the factorization property in the reduced fields block (2.41) the
calculation of the full superdeterminant finally gives the one-loop
contribution to the generating functional of the first-stage reducible gauge
theory

$\displaystyle Z^{1-\rm
loop}\;=\;\,\frac{\det\rho^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}{\big{(}\\!\det\varkappa^{\alpha\beta}\big{)}^{1/2}}\,\frac{\det
F^{\alpha}_{\;\beta}}{\big{(}\\!\det
F_{{i}{j}}\big{)}^{1/2}\big{(}\\!\det\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\big{)}^{1/2}\det
F^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}$
(2.66)

Let us assemble together all numerous ingredients of this expression which
were introduced above in the course of derivation of this formula. Here the
inverse propagator of the original gauge field $F_{ij}$ (2.42), the ghost
operator $F^{\alpha}_{\;\beta}$ (2.27) and the ghosts-for-ghosts operator
$F^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
(2.28) are respectively

$\displaystyle F_{{i}{j}}$ $\displaystyle\equiv$ $\displaystyle
S_{,{i}{j}}-X^{\alpha}_{,{i}}\varPi_{\alpha\beta}X^{\beta}_{,{j}},$ (2.67)
$\displaystyle F^{\alpha}_{\;\beta}$ $\displaystyle\equiv$ $\displaystyle
X^{\alpha}_{,{i}}\mathcal{R}^{i}_{\beta}-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}{\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\,\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta},$
(2.68) $\displaystyle
F^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
$\displaystyle\equiv$
$\displaystyle\omega^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\gamma}\mathcal{Z}^{\gamma}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}.$
(2.69)

Gauge field and ghost operators (2.67), (2.68) are both gauge fixed with the
aid of gauge conditions matrices $X^{\alpha}_{,{i}}$,
$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$ and
$\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}$ and the
gauge-fixing matrices $\varPi_{\alpha\beta}$ and
${\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
– kernels of bilinear terms in ($X^{\alpha}_{,{i}}$,
$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$,
$\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}$).
Whereas these matrices of gauge conditions for ghosts
$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$ and
$\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}$ are a
part of the originally chosen gauge fermion (2.17), a similar matrix in the
gauge fields sector is a special projector $\varPi_{\alpha\beta}$,
$\varPi_{\alpha\beta}\varkappa^{\beta\gamma}\varPi_{\gamma{\,\delta}}=\varPi_{\alpha{\,\delta}}$,
on the direction in the space of gauge indices orthogonal to the “vielbein”
$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$,
$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\varPi_{\alpha\beta}=\varPi_{\alpha\beta}\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=0$,

$\displaystyle\varPi_{\alpha\beta}\equiv{\varkappa}_{\alpha\beta}-{\varkappa}_{\alpha\gamma}\sigma^{\gamma}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}{\kappa}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\sigma^{{\,\delta}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}{\varkappa}_{{\,\delta}\beta},$
(2.70)
$\displaystyle\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}{\varkappa}_{\alpha\beta}\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}},\qquad{\kappa}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\equiv(\kappa^{-1})^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}.$
(2.71)

This type of “orthogonality” is determined with respect to the symmetric
metric $\varkappa_{\alpha\beta}$ originally introduced in the gauge fermion
(2.17). The projection of this metric onto the space of reducible generator
indices $a$ with the aid of the vielbein
$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$ gives
rise to the gauge-fixing matrix
$\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
in the space of these indices. The determinants of all gauge-fixing matrices
$\varkappa_{\alpha\beta}$
$\rho^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
and
$\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
appropriately enter the final algorithm for the one-loop generating functional
(2.66).

Ranks of all the above gauge matrices are maximal and determined by the range
of their indices, and the main criterion of their choice is the invertibility
of the full set of gauge and ghost operators (2.67)-(2.69).

The final important comment is the definition of the on-shell condition for
the obtained algorithm (2.66). All ghosts, ghosts for ghosts and their anti-
ghost fields are zero on shell (2.26)
$\mathcal{C}^{\alpha}=\bar{\mathcal{C}}_{\alpha}=\mathcal{C}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=\bar{\mathcal{C}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=0$.
The exception is the only “non-classical” field $\mathcal{C}^{\prime\,a}$
which has a zero ghost number, though originally it was called the extra ghost
[15]. On shell it is generically non-vanishing and is given by the expression
(2.34), so that the gauge conditions matrix equals

$X^{\alpha}_{,{i}}=\chi^{\alpha}_{,{i}}-\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}},{i}}\,({s}\sigma)^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\,{s}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\,\beta}\,\chi^{\beta}$
(2.72)

and coincides with $\chi^{\alpha}_{,i}$ only for $\phi$-independent
$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}$ or
vanishing on shell
${s}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\,\beta}\chi^{\beta}$.
For the original gauge field $\phi^{i}$ on-shell restriction and gauge
conditions (2.32)-(2.33) equivalently read $S_{,{i}}=0$,
$\varPi_{\alpha\beta}\chi^{\beta}=0$ (note that the original gauge functions
$\chi^{\alpha}(\phi)$ introduced in the gauge fermion (2.17) does not
necessarily vanish – only their projection vanishes on shell).

###  2.4 Ward identities and gauge independence of the effective action

One-loop effective action corresponding to (2.66) reads

$\displaystyle i\varGamma^{1-\rm
loop}{}=-\tfrac{1}{2}\mathop{\mathrm{Tr}}{\ln{F_{{i}{j}}}}+\mathop{\mathrm{Tr}}{\ln{F^{\alpha}_{\;\beta}}}-\mathop{\mathrm{Tr}}{\ln{F^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}}-\tfrac{1}{2}\mathop{\mathrm{Tr}}{\ln\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}-\tfrac{1}{2}\mathop{\mathrm{Tr}}{\ln{\varkappa^{\alpha\beta}}}+\mathop{\mathrm{Tr}}{\ln{\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}}$
(2.73)

and includes gauge field, ghost, ghost-for-ghost contributions and the
contribution of three gauge-fixing matrices. On shell this expression should
not depend on all gauge-fixing entities $\chi^{\alpha}(\phi)$,
$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}(\phi)$,
$\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}(\phi)$,
$\varkappa^{\alpha\beta}$,
$\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
as it is dictated by the general BV theory. It is worth checking this property
and revealing the details of the perturbative mechanism of such gauge
independence. In the one-loop approximation this mechanism is based on the on-
shell Ward identities for all tree-level propagators of the theory:
$F^{-1\,{i}{j}}$, $F^{-1\,\alpha}_{\;\;\;\;\;\;\,\beta}$ and
$F^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$.

Ward identity for the Green’s function of the gauge field operator $F_{ij}$
follows from the sequence of on-shell relations

$\displaystyle\mathcal{R}^{{i}}_{\gamma}=F^{-1\,{i}{j}}F_{{j}{k}}\mathcal{R}^{k}_{\gamma}=-F^{-1\,{i}{j}}X^{\alpha}_{,{j}}\varPi_{\alpha\beta}X^{\beta}_{\,,{k}}\mathcal{R}^{k}_{\gamma}=-F^{-1\,{i}{j}}X^{\alpha}_{,{j}}\varPi_{\alpha\beta}F^{\beta}_{\;\gamma},$
(2.74)

where we used the fact that on shell $\mathcal{R}^{k}_{\gamma}$ is a zero
vector of $S_{,{j}{k}}$ since
$S_{,{j}{k}}\mathcal{R}^{j}_{\gamma}=-S_{,{j}}\mathcal{R}^{j}_{\gamma,{k}}$,
and
$\varPi_{\alpha\beta}X^{\beta}_{,{k}}\mathcal{R}^{k}_{\gamma}=\varPi_{\alpha\beta}F^{\beta}_{\;\gamma}$
in view of
$\varPi_{\alpha\beta}\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=0$.
Thus, contracting this relation with
$F^{-1\,\gamma}_{\;\;\;\;\;\;{\,\delta}}$, we get on shell

$\displaystyle
F^{-1\,ij}X^{\alpha}_{,{j}}\varPi_{\alpha\beta}\,\big{|}_{\,S_{,{i}}=0}=-\mathcal{R}^{{i}}_{\gamma}F^{-1\,\gamma}_{\;\;\;\;\;\;\beta},$
(2.75)

whence it follows the effective action independence on the choice of gauge
matrix $X^{\alpha}_{,{i}}$ and the gauge condition $\chi^{\alpha}(\phi)$

$\displaystyle i\,\delta_{\chi}\varGamma^{\rm
1-loop}=i\,\delta_{X}\varGamma^{\rm
1-loop}=\big{(}F^{-1\,{i}{j}}X^{\alpha}_{,{i}}\varPi_{\alpha\beta}+\mathcal{R}^{i}_{\alpha}F^{-1\,\alpha}_{\;\;\;\;\;\;\beta}\big{)}\,\delta
X^{\beta}_{,{i}}=0.$ (2.76)

Similarly, from the relation
${\mathcal{Z}}^{\gamma}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}=F^{-1\,\gamma}_{\;\;\;\;\;\;\,\alpha}F^{\alpha}_{\;\beta\,}\mathcal{Z}^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}=-F^{-1\,\gamma}_{\;\;\;\;\;\;\,\alpha}\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,}{\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}F^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}$
it follows the Ward identity relating the ghost and ghosts-for-ghosts
propagators,

$\displaystyle
F^{-1\,\gamma}_{\;\;\;\;\;\;\,\alpha}\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,}{\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\,\big{|}_{\,S_{,{i}}=0}=-{\mathcal{Z}}^{\gamma}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}F^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{c}}}}_{\;\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}},$
(2.77)

and the effective action turns out to be on-shell independent of
$\omega^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\,\alpha}$,

$\displaystyle i\,\delta_{\omega}\varGamma^{\rm
1-loop}=-\big{(}F^{-1\,\beta}_{\;\;\;\;\;\;\,\alpha}\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\,{\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}+\mathcal{Z}^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\,F^{-1\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\;\;\;\;\;\;\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\big{)}\,\delta\omega^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\,\beta}=0.$
(2.78)

The dependence of $\varGamma^{\rm 1-loop}$ on
$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
involves three terms of (2.73) the sum of which also turns out to be zero on
shell

$\displaystyle i\,\delta_{\sigma}\varGamma^{\rm
1-loop}=-\tfrac{1}{2}\,\delta_{\sigma}{\rm Tr}\,\ln F_{ij}+\delta_{\sigma}{\rm
Tr}\ln F^{\alpha}_{\;\beta}-\tfrac{1}{2}\,\delta_{\sigma}{\rm
Tr}\ln\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=0.$
(2.79)

To prove this one should use in the variation of the operator (2.67) the
variation of the projector $\varPi_{\alpha\beta}$,
$\delta_{\sigma}\varPi_{\alpha\beta}=-2\,\delta\sigma^{\gamma}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\,\kappa^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\,\varPi_{\gamma(\alpha}\,\varkappa_{\beta){\,\delta}}\,\sigma^{\,\delta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$,
the variation of
$\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$,
$\delta_{\sigma}\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=2\,\varkappa_{\alpha\beta}\,\sigma^{\beta}_{({{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\delta\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}})}$,
and the corollaries of (2.75) and (2.77),

$\displaystyle
X^{\gamma}_{\,,{i}}\,F^{-1\,{i}{j}}X^{\beta}_{\,,{j}}\,\varPi_{\beta\alpha}\,\big{|}_{\,S_{,{i}}=0}=-X^{\gamma}_{\,,{i}}\mathcal{R}^{i}_{\beta}\,F^{-1\,\beta}_{\;\;\;\;\;\;\,\alpha},$
(2.80) $\displaystyle
X^{\gamma}_{\,,{i}}\mathcal{R}^{i}_{\beta}\,F^{-1\,\beta}_{\;\;\;\;\;\;\,\alpha}=\big{(}F^{\gamma}_{\;\beta}+\sigma^{\gamma}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,}{\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}\,}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}\big{)}\,F^{-1\,\beta}_{\;\;\;\;\;\;\,\alpha}=\delta^{\gamma}_{\,\alpha}+\,\sigma^{\gamma}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,}{\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}\,}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta\,}F^{-1\,\beta}_{\;\;\;\;\;\;\,\alpha},$
(2.81)
$\displaystyle\omega^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\,\beta\,}F^{-1\,\beta}_{\;\;\;\;\;\;\,\alpha}\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\,\big{|}_{\,S_{,{i}}=0}=-\rho^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}.$
(2.82)

The $\varkappa^{\alpha\beta}$-variation of the effective action also vanishes
on shell because of the variations
$\delta_{\varkappa}\varPi_{\alpha\beta}=-\varPi_{\alpha{\,\delta}}\delta\varkappa^{{\,\delta}\gamma}\varPi_{\gamma\beta}$,
$\delta_{\\!\varkappa}\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\delta{\varkappa}_{\alpha\beta}\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$,
Eq.(2.81) and its corollary
$\varPi_{\alpha\gamma}X^{\gamma}_{\,,{i}}F^{-1\,{i}{j}}X^{{\,\delta}}_{\,,{j}}\varPi_{{\,\delta}\beta}=-\varPi_{\alpha\beta}$
(which in its turn is provided by the orthogonality relation
$\varPi_{\alpha\beta}\sigma^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=0$),

$\displaystyle i\,\delta_{\varkappa}\varGamma^{\rm 1-loop}$ $\displaystyle=$
$\displaystyle-\tfrac{1}{2}F^{-1\,{j}{i}}\delta_{\\!\varkappa}F_{{i}{j}}-\tfrac{1}{2}{\kappa}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\delta_{\\!\varkappa}\kappa_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}-\tfrac{1}{2}{\varkappa}_{\beta\alpha}\delta\varkappa^{\alpha\beta}=0.$
(2.83)

Finally, on-shell independence of the gauge-fixing matrix
$\rho^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
follows by direct variation and the use of the corollary (2.82) of the Ward
identity (2.77) for the ghost propagator,

$\displaystyle i\,\delta_{\rho}\varGamma^{\rm
1-loop}=-\big{(}\omega^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}\,F^{-1\,\beta}_{\;\;\;\;\;\;\alpha}\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}+\rho^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\big{)}\,\delta{{\rho^{-1}}^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}=0.$
(2.84)

##  3 Reducible gauge structure of restricted gauge theories

Reducible structure of a gauge theory can be induced by the procedure of
restriction of the originally irreducible gauge theory. To see this consider a
generic gauge theory with the action
$\hat{S}[\,\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\,]$ subject
to a closed gauge algebra of _irreducible_ generators
$\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}$

$\displaystyle\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}=0,\quad\;\;\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\hat{\mathcal{R}}^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\beta}-\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\beta,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\hat{\mathcal{R}}^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\alpha}=\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\gamma}\hat{C}^{\gamma}_{\alpha\beta}.$
(3.1)

Irreducibility of the generators
$\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}$
implies that the rank of their matrices coincides with the range of the
indices $\alpha$ which in its turn is lower than the range of the indices
${\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ enumerating the original gauge
fields $\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$,
$\operatorname{\,\mathrm{rank}\,}{\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}}={\rm
range}\;\alpha=m_{0}<\hat{n}={\rm
range}\;{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$. We will call such a
theory the parental one.

Restricted gauge theory originating from the parental one is the theory whose
configuration space variables are kinematically constrained by the equations

$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}})=0,$
(3.2)

where we will consider the functions
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}})$
to be functionally independent, that is characterized by the full rank
condition of their gradient matrix,

$\operatorname{\,\mathrm{rank}\,}{\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}}=m_{1}\equiv{\rm
range}\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}.$ (3.3)

For simplicity we assume that these functions are either ultralocal
(algebraic) in spacetime or the surface of these constraint functions can be
parameterized in terms of local independent fields which will be denoted in
what follows by $\phi^{i}$.

###  3.1 Two representations of a restricted theory

Such a restricted theory can be described in two equivalent ways. One way is
to represent it in terms of the Lagrange multipliers action

$S^{\lambda}[\,\varphi,\lambda\,]=\hat{S}[\,\varphi\,]-\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\,\varphi\,),$
(3.4)

whose classical equations of motion obtained by varying both its gauge fields
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ and Lagrange
multipliers $\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ read

$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}-\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}=0,\quad\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=0.$
(3.5)

Another representation is the reduced theory, when one solves first the
constraints (3.2) with respect to
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ as functions of the
reduced set of fields $\phi^{i}$ and formulates the theory in terms of the
reduced action $S^{\rm red}[\,\phi\,]$. The latter is obtained by substituting
in the parental theory action the functions
$e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}(\phi)$ of embedding the
$\phi^{i}$-subspace into the space of original gauge fields
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$,

$\displaystyle\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}=e^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}(\phi),\quad\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\big{(}e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}(\phi)\big{)}\equiv
0,$ (3.6) $\displaystyle S^{\rm red}[\,\phi\,]=\hat{S}[\,e(\phi)\,].$ (3.7)

The equations of motion in both formulations are obviously
equivalent666Classical equivalence of two theories can be defined by
equivalence of spaces of solutions of their equations of motion. In particular
it can be established by local projecting correspondence (3.8) and
complementary observation that reduced fields did not carry additional degrees
of freedom. The latter is the case here since above assumptions on
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi)$ imply that
such reducibility fields exist. All this is the involved rephrasing of the
equivalence between theories related by reduction w.r.t. (true) auxiliary
variables. Reducing theories restricted with Lagrange multipliers method
equivalence between unreduced and reduced representation is even easier and
guaranteed by finding rank $m_{1}$ combinations of fields
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ which are expressible
from (3.2). because the contraction of the first of equations (3.5) results in

$e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}\,\big{(}\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}-\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\big{)}=e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}\,\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\equiv\frac{\delta
S^{\rm
red}[\,\phi\,]}{\delta\phi^{i}},\;\quad\;e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}\equiv\frac{\delta
e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}(\phi)}{\delta\phi^{i}},$ (3.8)

where we took into account that in view of (3.6) the covector
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$
is orthogonal to the surface of constraints
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=0$,

$\frac{\delta}{\delta\phi^{i}}\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\big{(}\varphi(\phi)\big{)}=\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\,\big{|}_{\,\theta=0}\,e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}=0.$
(3.9)

Important question is the relation of the original _parental_ theory and the
_restricted_ one from the point of view their physical equivalence. Solutions
of the system of equations (3.5) are obviously inequivalent to those of the
parental theory, $\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}=0$,
when on shell the Lagrange multipliers
$\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ are nonvanishing.
Note that in view of linearity of the $\lambda$-action (3.4) in
$\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ the variational
equations with respect to Lagrange multipliers do not allow one to express
them from the full set of equations of motion. There is another way to write
down their equations of motion by contracting the first set of equations (3.5)
with the parental theory generators
$\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}$ and
use the Noether identities (3.1). Then we get

$\displaystyle\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}=0,$
(3.10)

where
$Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}=Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}(\varphi)$
is the _gauge restriction operator_ which will be very important in what
follows,

$Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}\equiv\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\,}\hat{\mathcal{R}}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\alpha}.$
(3.11)

This equation for $\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$
has a unique solution
$\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=0$ only when the
rank of $Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}$ is
maximal, that is it coincides with the range of the index
${{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ enumerating the constraint
functions $\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$. In this
case the meaning of the constraints
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi)=0$ is
nothing but a partial gauge fixing of gauge invariance of the parental theory
(${\rm
range}\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=m_{1}<m_{0}={\rm
range}\;\alpha$).

On the contrary, when the rank of gauge restriction operator
$Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}$ is lower than
$m_{1}$

$\displaystyle{\rm
rank}\,Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}=m_{1}-m_{2}$
(3.12)

then $m_{2}$ Lagrange multipliers can be freely specified, the rest
$m_{1}-m_{2}$ of them being fixed as unique functions of the former free
ones.777Rank relation (3.12) supposes off-shell constantness of the rank of
the gauge restriction operator
$Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}$. This is the
substantial regularity condition which allows to avoid various irregularities
and to simplify presentation. In particular together with (3.3) it allows
Lagrange multipliers
$\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ to be gauge
invariants in restricted theory and thus imply trivial action of gauge
generators on these fields. However note that certain weakenings of the rank
constantness assumption for
$Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}$ are possible
without serious complications of the whole construction. For example when the
rank of $Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}$
increases off the restriction surface
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=0$ this generates
nontrivial gauge transformation of Lagrange multipliers but still allows to
construct the key ingredient of our approach – constant-rank projector which
will be discussed in Sect.3.2. This implies, in particular, the existence of
$m_{2}$ gauge invariants of the parental theory constrained to be zero in
restricted theory. To see this note that the rank deficiency implies that
operator $Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}$ has
$m_{2}$ left zero vectors
$Y^{\\!{\mathchoice{A}{A}{\scriptscriptstyle{A}}{A}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$,

$Y^{\\!{\mathchoice{A}{A}{\scriptscriptstyle{A}}{A}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,}Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}=0,\quad{\rm
range}\;{\mathchoice{A}{A}{\scriptscriptstyle{A}}{A}}=m_{2}\leq m_{1},$ (3.13)

and this, according to the definition of this operator, implies the existence
of $m_{2}$ parental gauge invariants

$\theta^{\mathchoice{A}{A}{\scriptscriptstyle{A}}{A}}(\varphi)\equiv
Y^{\\!{\mathchoice{A}{A}{\scriptscriptstyle{A}}{A}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}(\varphi)\,\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi),\quad\theta^{\mathchoice{A}{A}{\scriptscriptstyle{A}}{A}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\,}\hat{\mathcal{R}}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\alpha}\big{|}_{\,\theta=0}=0,$
(3.14)

which are constrained to vanish. So generically the restricted theory (3.4) is
_inequivalent_ to the parental gauge theory
$\hat{S}[\,\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\,]$, because
the latter does not a priori impose any restrictions on its gauge invariant
objects. In what follows we will consider such restricted theories which
incorporate new physics beyond their parental ones.

There is another informative representation of the solution for Lagrange
multipliers. If one constructs dual set
$\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}$
to covariant vectors
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$

$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\,\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}=\delta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}.$
(3.15)

then, contracting the equation of motion (3.5) with
$\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}$
one obtains

$\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}.$
(3.16)

For local in spacetime restriction functions888Remind that we consider either
ultralocal (algebraic) restriction functions
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi)$ or those
for which $m_{1}={\rm
range}\;{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ ultralocal
independent combinations of
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ can be expressed from
restriction conditions
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi)=0$. In such
case useful criteria for the existence of local (in fact ultralocal) vectors
$\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}$
dual to
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$
(3.15) – is the existence of the ultralocal nondegenerate linear combination
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}c^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}$
with local vectors
$c^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}$
(in particular – the presence of ultralocal minor of maximal $m_{1}$ rank in
matrix
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$).
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ this is a local
expression for Lagrange multipliers in terms of
$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$ – the left hand
side of equations of motion of the parental theory. Resolving Lagrange
multipliers $\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ in
restricted equations of motion (3.5) leads to the projected set of equations

$\hat{S}_{,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}(\delta^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}-\theta^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}})=0\;\;\quad\big{(}\Leftrightarrow\quad\hat{S}_{,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}=0\;\big{)}.$
(3.17)

The projector
$(\delta^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}-\theta^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}})$
keeps only part of parental equations of motion
$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$ along directions
tangential to restriction surface
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=0$. Which is
equivalent to (3.8) and reinstates the observation that along tangential
directions equations of motion for restricted theory coincides with that for
parental theory.

On the contrary, along complementary (normal) directions, equations of motion
for these two theories differ. Contraction of the parental equations
$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}=0$ and restricted
theory equations
$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}-\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}=0$
with
$\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$
gives correspondingly
$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=0$
and
$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$.
Normal subset of the parental equations in addition to equations (3.17)
further restricts classical configurations of fields
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ in parental theory.
While normal subset of the restricted theory equations merely expresses
$\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$ in terms of
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ and thus does not
additionally constrain the latter.999 Comparing the sets of equations of
motion which define dynamics of
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ in parental and
restricted theories one finds that in restricted theory $m_{1}$ constraints
$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi)=0$
are “removed” while $m_{1}$ new constraints
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi)=0$ are
applied. This adds a bit of intuition but by itself does not predicts number
of degrees of freedom in restricted theory or its gauge symmetries, etc, since
this requires deeper analysis of interrelation of restriction conditions (3.2)
and dynamical and gauge structures of the parental theory. However due to
gauge invariance of the parental theory (3.1) not all dynamical restrictions
on $\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ in the right hand
side of (3.16) are effectively removed. The linear dependence
$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}=0$
between $\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$ may express
certain part of
$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$
in (3.16) as linear combinations of tangential equations 3.17 which are still
freezed to $0$ within restricted equations of motion. Obviously such linear
combinations are found by first contracting (3.16) with
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$
and then with
$\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}$
which leads to
$\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\alpha}=\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\alpha}$.
The same structure
$\hat{S}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\alpha}$
appears in the left hand side of “tangential” equations 3.17 after contracting
with
$\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}$ and
thus _vanishes_ on-shell. This reinstates constraints on Lagrange multipliers
(3.10),
$\lambda_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}=0$.

###  3.2 Gauge symmetry and reducibility

We assume that the restricted theory does not acquire local gauge symmetries
beyond those of the original parental action. On the other hand, not all gauge
transformations of the the parental theory
$\delta_{\xi}\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}=\hat{\mathcal{R}}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\alpha}\xi^{\alpha}$
generate symmetries of the reduced action (3.4). These symmetries are only
those which _preserve_ the constraints

$\delta_{\xi}\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,\big{|}_{\,\theta=0}\equiv\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\hat{\mathcal{R}}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\alpha}\xi^{\alpha}\,\big{|}_{\,\theta=0}=0,$
(3.18)

so that with the definition (3.11) the allowed gauge transformation parameters
$\xi^{\alpha}_{\rm red}$ in the restricted theory should be solutions of the
linear equation
$Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}\xi^{\alpha}_{\rm
red}=0$, i.e. right zero vectors of the matrix
$Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}$. The subspace
of reduced gauge parameters can be obtained by projecting gauge parameters of
the parental theory $\xi^{\alpha}$ with a projector $T^{\alpha}_{\>\beta}$
which is orthogonal in the space of gauge indices to
$Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}$,

$\xi^{\alpha}_{\rm
red}=T^{\alpha}_{\>\beta}\,\xi^{\beta},\;\quad\;Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha\,}T^{\alpha}_{\>\beta}=0.$
(3.19)

On the other hand, gauge symmetries of the restricted theory can formulated in
the parental field space of
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ with free (non-
reduced) gauge parameters $\xi^{\alpha}$. This happens if instead of
projecting $\xi^{\alpha}$ we project with respect to the gauge index $\alpha$
the original parental generators
$\hat{\mathcal{R}}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\alpha}$.
Thus we get the reducible set of gauge generators

$\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta}\equiv\hat{\mathcal{R}}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\alpha\,}T^{\alpha}_{\>\beta}.$
(3.20)

This formulation is covariant from the viewpoint of the parental gauge theory
because the original multiplets of parental field indices
${\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ and gauge indices $\alpha$
remain unsplitted, but the price paid for this covariance is the reducibility
of the generators
$\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta}$. They
are indeed reducible because the projector $T^{\alpha}_{\>\beta}$ in view of
rank deficiency (following from (3.19)) possess right zero vectors
$k^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$ which become
also the right zero vectors of
$\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta}$

$T^{\alpha}_{\>\beta\,}k^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=0,\;\quad\;\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta\,}k^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=0.$
(3.21)

Despite reducibility an important advantage of such projected representation
is that it generates gauge transformations with arbitrary _unrestricted_ gauge
parameters $\xi^{\alpha}$. In the course of subsequent Batalin-Vilkovisky
extension of configuration space it will give rise to gauge ghosts equivalent
to that of the parental theory. This property is achieved at the cost of
_reducibility_ of the projected gauge structure (3.20) and introduction of
ghost for ghosts fields.

The projector $T^{\alpha}_{\>\beta}$ may be explicitly constructed in terms of
the left and right kernel bases (3.19),(3.21)

$T^{\alpha}_{\>\beta}\,=\,\delta^{\alpha}_{\,\beta}-k^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\,{(Qk)^{-1}}\vphantom{|}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\,Q^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\beta}\,\equiv\,T^{\alpha}_{\>\beta}(k,Q).$
(3.22)

Left kernel $Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}$
(3.11) is fixed by the restriction conditions and the choice of parental gauge
generators. Thus (3.22) may be considered as the family parameterized by its
right kernel basis
$k^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$. The
structure of $T^{\alpha}_{\>\beta}(k,Q)$, which in what follows will be
referred without specification of kernels, is analogous to the set of
projectors $T^{\alpha}_{\>\beta}(\sigma,s)$ (2.36) introduced above.

Here we emphasize the property that the matrix
$(Qk)^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\equiv
Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\gamma}\,k^{\gamma}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
is not directly invertible. Rather it could have rank equal or lower to that
of ${\rm
rank}\,Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\gamma}=m_{1}-m_{2}$.
The rank deficiency is dictated by the requirement of new physics incorporated
by the restricted theory (3.12) – physical inequivalence to the parental gauge
theory. As discussed in Appendix A, this extra difficulty can be circumvented
by the Moore-Penrose construction of the generalized inverse [21], provided
the following rank restriction conditions (3.23) are satisfied,

$\operatorname{\,\mathrm{rank}\,}{(Qk)^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}=\operatorname{\,\mathrm{rank}\,}{Q^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}}=\operatorname{\,\mathrm{rank}\,}{k^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}.$
(3.23)

This guarantees unambiguous definition of projector (3.22) and its correct
rank property,
$\operatorname{\,\mathrm{rank}\,}{T^{\alpha}_{\>\beta}}=m_{0}-m_{1}+m_{2}$.

Rank restrictions requirements (3.23) in particular are guaranteed when
$k^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$ is
parameterized in terms of nondegenerate two-forms $B^{\beta\alpha}(\varphi)$
and
$c_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}(\varphi)$
– metrics in spaces of $\alpha$-indices and
${{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$-indices respectively,

$k^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=B^{\beta\alpha}Q^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}c_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}},\qquad
T^{\alpha}_{\>\beta}=\delta^{\alpha}_{\,\beta}-B^{\alpha\gamma}Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\gamma}\big{(}\,QBQ^{T}\,\big{)}^{-1}_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{\,\beta},$
(3.24)

which will be used in unimodular gravity calculations in Sect.5. In fact in
this case the projector $T^{\alpha}_{\>\beta}$ is defined by
$Q^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\alpha}$ and
$B^{\beta\alpha}$ only because the dependence on
$c_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$
completely cancels out.101010The “symmetric” form (3.24) has the structure
analogous to that of symmetric projector $\varPi_{\alpha\beta}$ (2.43) (with
left index raised) when
$\varkappa_{\alpha\beta}\leftrightarrow(B^{\beta\alpha})^{-1}$ and
$\sigma^{\alpha}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\leftrightarrow
B^{\alpha\beta}Q^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}_{\,\beta}c_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}$.

The generators (3.20) of gauge transformations in restricted theory are thus
far defined in the parental theory field space of
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$. They determine the
gauge transformations $\delta_{\xi}^{\rm
red}\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}=\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta}\xi^{\beta}$
_tangential_ to the surface of
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi)=0$. As any
field on this surface can be parameterized by $\phi^{i}$ these gauge
transformations can be expressed via the gauge transformations of the reduced
restricted gauge theory $\delta_{\xi}^{\rm
red}\phi^{i}=\mathcal{R}^{i}_{\beta}\xi^{\beta}$, $\delta_{\xi}^{\rm
red}\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}=e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}\,\delta_{\xi}^{\rm
red}\phi^{i}$, so that the corresponding generators $\mathcal{R}^{i}_{\beta}$
of reduced space gauge fields $\phi^{i}$ are related to
$\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta}$ by
obvious pushforward relations

$\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\alpha}(e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}(\phi))\,=\,\frac{\delta
e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}(\phi)}{\delta\phi^{i}}\mathcal{R}^{{i}}_{\alpha}(\phi^{i})\,\equiv\,e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}\mathcal{R}^{{i}}_{\alpha}.$
(3.25)

Thus, the reduced space representation of the restricted gauge theory has
reducible generators $\mathcal{R}^{{i}}_{\alpha}$ and their first-stage
reducibility generators
$\mathcal{Z}^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$,

$\frac{\delta S^{\rm
red}[\,\phi\,]}{\delta\phi^{i}}\,\mathcal{R}^{{i}}_{\beta}(\phi)=0,\quad\mathcal{R}^{{i}}_{\beta\,}\mathcal{Z}^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}=0$
(3.26)

where
$\mathcal{Z}^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
exhaust linear combinations of right zero vectors of the projector
$T^{\alpha}_{\>\beta}$ and have the form

$\mathcal{Z}^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}=k^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}\,}\mu^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
(3.27)

with arbitrary invertible factor
$\mu^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$,
$\operatorname{\,\mathrm{rank}\,}{k^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\mu^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}}=\operatorname{\,\mathrm{rank}\,}{k^{\beta}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}}$.
Such natural choice imply off-shell first-stage reducibility (3.26) of
projected gauge generators.

Note that the projectors (3.22) generically are nonlocal depending on
structure of restriction functions and gauge generators involved in the
construction of gauge restriction operator
$Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}$.

###  3.3 Gauge algebra of a restricted theory

To proceed further we have to know the gauge algebra of the restricted theory
generators. A priori maximum that can be stated from the first of their
relations (3.26) (under certain assumptions of regularity of
$\mathcal{R}^{{i}}_{\beta}(\phi)$ [27]) is that they satisfy an open algebra
(2.3) with some structure functions and $S_{,{i}}$ replaced by $S^{\rm
red}_{,{i}}$. Quite remarkably, we will be able to derive this algebra first
by expressing via Eq.(3.25) $\mathcal{R}^{{i}}_{\beta}$ in terms of
$\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta}$ and
using the algebra of the projected (full space) generators (3.20),
$\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta}=\hat{\mathcal{R}}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\alpha}T^{\alpha}_{\>\beta}$.
The latter, in its turn, follows from the gauge algebra of the parental theory
and turns out to be also closed in the case of closure of the algebra of
$\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}$.
Thus finally we will show that the resulting algebra is also closed as long as
we start with the closed algebra of the parental theory.

Expressing $\mathcal{R}^{{i}}_{\beta}$ in terms of
$\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta}$ goes
with the aid of covariant vectors
$e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ dual to
$e^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{,{j}}$

$\mathcal{R}^{{i}}_{\alpha}(\phi)=e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\mathcal{R}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}\big{(}e(\phi)\big{)}.$
(3.28)

These vectors satisfy the biorthogonality relation
$e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\,e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{j}}=\delta^{i}_{\,{j}}$
and represent a part of the complete local basis
$(e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}},\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}})$
in the cotangent bundle of the manifold of
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$. Vectors
$e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ may be chosen orthogonal
to the set
$\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}$
dual to
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$
(3.15), so that
$(e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}},\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}})$
and
$(e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}},\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}})$
form biorthogonal pair of tangent and cotangent bases

$\displaystyle
e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\,e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{j}}=\delta^{i}_{\,{j}},\quad\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\,\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}=\delta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}},\quad
e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\,\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}=0,\quad\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\,e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{j}}=0.$
(3.30) $\displaystyle
e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}\,e^{i}_{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}+\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}\,\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}=\delta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}},$

the last equation here expressing completeness of this basis. 111111
Completing discussion of biorthogonal basis construction let us mention the
particular “ _symmetric_ ” case of such basis which may be constructed with
the aid of nondegenerate metric
$G_{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}$
on the manifold parameterized by
$\varphi^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$. The presence of such
metric induces metric $G_{{i}{j}}$ on the reduced $\phi^{i}$-space – the
surface of
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}(\varphi)=0$, and its
inverse
$G^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\equiv(G_{{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}})^{-1}$
induces the inverse metric along normal directions  $G_{{i}{j}}\equiv
e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}\,G_{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\,e^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{,{j}},\\!\quad\\!G^{{i}{j}}\equiv(G_{{j}{i}})^{-1},\,\quad\,G^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\equiv\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\,G^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\,\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}},\\!\quad\\!G_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}\equiv(G^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}})^{-1}.$
Starting from mutually orthogonal
$e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}$ and
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$
one can construct complementary basis elements
$e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}$ and
$\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}$
satisfying (3.30),(3.30) as
$e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}=G^{{i}{j}}e^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{,{j}}G_{{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}},\,\quad\,\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=G^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}_{,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}G_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}.$
For fixed $e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{j}}$ and
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$
there is a freedom of choosing complementary basis elements

$e^{{i}}_{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\to
e^{{i}}_{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}+b^{i}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}},\;\quad\;\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}\to\theta^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}-e^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{,{i}}b^{i}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}}$
(3.31)

which preserves (3.30),(3.30). However this freedom does not spoil expression
(3.28) since the shift ambiguity term in
$e^{{i}}_{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}$ is killed due to
defining property of restricted generators, (3.20)
$\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta}=\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{,{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}\hat{\mathcal{R}}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\alpha\,}T^{\alpha}_{\>\beta}=Q^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,\alpha}T^{\alpha}_{\>\beta}=0$,
thus making expression (3.28) for $\mathcal{R}^{{i}}_{\alpha}(\phi)$
unambiguous.

Using (3.28) we have

$\mathcal{R}^{{i}}_{[\,\alpha,\,{j}}\mathcal{R}^{j}_{\beta\,]}=e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\,\mathcal{R}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{[\,\alpha,\,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\mathcal{R}^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\beta\,]}+\mathcal{R}^{k}_{[\,\alpha}\,\mathcal{R}^{j}_{\beta\,]}e^{i}_{{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}},{j}}\,e^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{,{k}}=e^{i}_{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}\,\mathcal{R}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{[\,\alpha,\,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\mathcal{R}^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\beta\,]},$
(3.32)

where square brackets denote pairs of antisymmetrized indices and the second
term in the middle part vanishes because for dual
$e^{i}_{{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}$ and
$e^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{,{k}}$ (3.30)
$e^{i}_{{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}},{j}}e^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{,{k}}=-e^{i}_{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}e^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{,{k}{j}}\equiv-e^{i}_{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}\,\delta^{2}e^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}(\phi)/\delta\phi^{k}\delta\phi^{j}$
and so antisymmetrization in indices kills it. Thus the gauge algebra of
$\mathcal{R}^{i}_{\alpha}$ directly follows from that of
$\mathcal{R}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}$.

As shown in Appendix B the commutator of projected generators
$\mathcal{R}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha}$
replicates the algebraic relation of the parental theory. If the latter is
prescribed by

$\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\hat{\mathcal{R}}^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\beta}-\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\beta,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\hat{\mathcal{R}}^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\alpha}=\hat{\mathcal{R}}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\gamma}\hat{C}^{\gamma}_{\alpha\beta}+\hat{E}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}_{\alpha\beta}\hat{S}_{,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}},$
(3.33)

then the projected generators of the restricted gauge theory
$\mathcal{R}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\beta}=\hat{\mathcal{R}}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\alpha}T^{\alpha}_{\>\beta}$,
which are defined by Eqs.(3.20)-(3.22) both on and outside of the constraint
surface $\theta^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}=0$, are
subject to a similar gauge algebra

$\mathcal{R}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\alpha,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\mathcal{R}^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\beta}-\mathcal{R}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\beta,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}\mathcal{R}^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\alpha}=\mathcal{R}^{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}_{\gamma}C^{\gamma}_{\alpha\beta}+{E}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}_{\alpha\beta}\hat{S}_{,{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}$
(3.34)

with new structure functions $C^{\gamma}_{\alpha\beta}$ and
${E}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}_{\alpha\beta}$

$\displaystyle{C}^{\gamma}_{\alpha\beta}$ $\displaystyle\equiv$ $\displaystyle
T^{\gamma}_{\>\zeta}\hat{C}^{\zeta}_{{\,\delta}\epsilon}T^{{\,\delta}}_{\>\alpha}T^{\epsilon}_{\>\beta}+N^{\gamma}_{\,\beta\alpha}-N^{\gamma}_{\,\alpha\beta},$
(3.35)
$\displaystyle{E}^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}_{\alpha\beta}$
$\displaystyle\equiv$ $\displaystyle
D^{{\mathchoice{I}{I}{\scriptscriptstyle{I}}{I}}}_{\,{\mathchoice{K}{K}{\scriptscriptstyle{K}}{K}}}D^{{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}_{\,{\mathchoice{L}{L}{\scriptscriptstyle{L}}{L}}}\hat{E}^{{\mathchoice{K}{K}{\scriptscriptstyle{K}}{K}}{\mathchoice{L}{L}{\scriptscriptstyle{L}}{L}}}_{\gamma{\,\delta}}\,T^{\gamma}_{\>\alpha}T^{{\,\delta}}_{\>\beta},$
(3.36)

where

$\displaystyle N^{\gamma}_{\,\alpha\beta}\equiv
T^{\gamma}_{\>{\,\delta}}k^{{\,\delta}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}},{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}}{(Qk)^{-1}}\vphantom{|}^{{\color[rgb]{0.390625,0.1953125,0.0390625}{a}}}_{\,{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}Q^{{{\color[rgb]{0.390625,0.1953125,0.0390625}{b}}}}_{\,\alpha}\mathcal{R}^{\mathchoice{J}{J}{\scriptscriptstyle{J}}{J}}_{\,\beta},$