Higher-Stage Noether Identities and Second Noether Theorems

The direct and inverse second Noether theorems are formulated in a general case of reducible degenerate Grassmann-graded Lagrangian theory of even and odd variables on graded bundles. Such Lagrangian theory is characterized by a hierarchy of nontrivial higher-stageNoether identities which is described in the homology terms. If a certain homology regularity condition holds, one can associate with a reducible degenerate Lagrangian the exact Koszul–Tate chain complex possessing the boundary operator whose nilpotentness is equivalent to all complete nontrivial Noether and higher-stage Noether identities. The second Noether theorems associate with the above-mentioned Koszul–Tate complex a certain cochain sequence whose ascent operator consists of the gauge and higher-order gauge symmetries of a Lagrangian system. If gauge symmetries are algebraically closed, this operator is extended to the nilpotent BRST operator which brings the above-mentioned cochain sequence into the BRST complex and provides a BRST extension of an original Lagrangian.


Introduction
The second Noether theorems are well known to provide the correspondence between Noether identities (henceforth NI) and gauge symmetries of a Lagrangian system [1].We aim to formulate these theorems in a general case of reducible degenerate Lagrangian systems characterized by a hierarchy of nontrivial higher-stage NI [2,3].To describe this hierarchy, one needs to involve Grassmann-graded objects.In a general setting, we therefore consider Grassmann-graded Lagrangian systems of even and odd variables on a smooth manifold  (Section 5).
Lagrangian theory of even (commutative) variables on an -dimensional smooth manifold  conventionally is formulated in terms of smooth fibre bundles over  and jet manifolds of their sections [3][4][5] in the framework of general technique of nonlinear differential operators and equations [3,6,7].At the same time, different geometric models of odd variables either on graded manifolds or supermanifolds are discussed [8][9][10][11][12].Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras [12,13].However, graded manifolds are characterized by sheaves on smooth manifolds, while supermanifolds are constructed by gluing of sheaves on supervector spaces.Since nontrivial higher-stage NI of a Lagrangian system on a smooth manifold  form graded  ∞ ()-modules, we follow the well known Serre-Swan theorem extended to graded manifolds (Theorem 5) [12].It states that if a graded commutative  ∞ ()-ring is generated by a projective  ∞ ()-module of finite rank, it is isomorphic to a ring of graded functions on a graded manifold whose body is .Accordingly, we describe odd variables in terms of graded manifolds [3,12].
Let us recall that a graded manifold is a locally ringed space, characterized by a smooth body manifold  and some structure sheaf A of Grassmann algebras on  [12,13].Its sections form a graded commutative  ∞ ()-ring of graded functions on a graded manifold (, A).The differential calculus on a graded manifold is defined as the Chevalley-Eilenberg differential calculus over this ring (Section 2).By virtue of Batchelor's theorem (Theorem 4), there exists a vector bundle  →  with a typical fibre  such that the structure sheaf A of (, A) is isomorphic to a sheaf A  of germs of sections of the exterior bundle ∧ * of the dual  * of  whose typical fibre is the Grassmann algebra ∧ * [13].This Batchelor's isomorphism is not canonical.In applications, it however is fixed from the beginning.Therefore, we restrict our consideration to graded manifolds (, A  ), called the simple graded manifolds (Section 3).

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Lagrangian theory on fibre bundles  →  can be adequately formulated in algebraic terms of a variational bicomplex of exterior forms on the infinite order jet manifold  ∞  of sections of  → , without appealing to the calculus of variations [3][4][5]14].This technique is extended to Lagrangian theory on graded manifolds and bundles [2,12,15,16].It is phrased in terms of the Grassmanngraded variational bicomplex of graded exterior forms on a graded infinite order jet manifold ( ∞ , A  ∞  ) (Section 5).Lagrangians and the Euler-Lagrange operator are defined as elements (63) and the coboundary operator (64) of this bicomplex, respectively.
A problem is that any Euler-Lagrange operator satisfies NI, which therefore must be separated into the trivial and nontrivial ones.These NI can obey first-stage NI, which in turn are subject to the second-stage ones, and so on.Thus, there is a hierarchy of NI and higher-stage NI which must be separated into the trivial and nontrivial ones (Section 7).If certain homology regularity conditions hold (Condition 1), one can associate with a Lagrangian system the exact Koszul-Tate (henceforth KT) complex (123) possessing the boundary KT operator whose nilpotentness is equivalent to all complete nontrivial NI (99) and higher-stage NI (124) [2,12].
The inverse second Noether theorem formulated in homology terms (Theorem 33) associates with this KT complex (123) the cochain sequence (138) with the ascent operator (139), called the gauge operator, whose components are nontrivial gauge and higher-stage gauge symmetries of Lagrangian theory [2,12].Conversely, given these symmetries, the direct second Noether theorem (Theorem 34) states that the corresponding NI and higher-stage NI hold.
The gauge operator unlike the KT one is not nilpotent, and gauge symmetries need not form an algebra [17][18][19].Gauge symmetries are said to be algebraically closed if the gauge operator admits the nilpotent BRST extension (155).If this extension exists, the above-mentioned cochain sequence (138) is brought into the BRST complex (156).The KT and BRST complexes provide the BRST extension (177) of an original Lagrangian theory by antifields and ghosts [12,18].
The most physically relevant Yang-Mills gauge theory on principal bundles and gauge gravitation theory on natural bundles are irreducible degenerate Lagrangian systems which possess nontrivial Noether identities, but trivial first-stage ones [2,20].In Section 10, we analyze topological BF theory which exemplifies a finitely reducible degenerate Lagrangian model.

Grassmann-Graded Differential Calculus
Throughout this work, by the Grassmann gradation is meant the Z 2 -one, and a Grassmann-graded structure is called graded if there is no danger of confusion.The symbol [⋅] stands for the Grassmann parity.Let us recall the relevant basics of the graded algebraic calculus [12,13].
Let K be a commutative ring.A K-module  is called graded if it is endowed with a grading automorphism ,  2 = Id.A graded module falls into a direct sum of modules  =  0 ⊕  1 such that () = (−1) [] ,  ∈  [𝑞] .One calls  0 and  1 the even and odd parts of , respectively.In particular, by a real graded vector space  =  0 ⊕  1 is meant a graded R-module.
where  and   are gradedhomogeneous elements of A. Its even part A 0 is a subalgebra of A, and the odd one Hereafter, all algebras and vector spaces are assumed to be real.
Remark 2. Let  be a vector space and Λ = ∧ its exterior algebra.It is a graded commutative ring, called the Grassmann algebra, with respect to the Grassmann gradation Hereafter, Grassmann algebras of finite rank when  = R  only are considered.
Given a graded algebra A, a left graded A-module  is defined as a left A-module where Similarly, right graded A-modules are treated.If A is graded commutative, a graded A-module  is provided with a graded A-bimodule structure by letting  = (−1) [][] .Remark 3. A graded algebra g is called a Lie superalgebra if its product [⋅, ⋅], called the Lie superbracket, obeys the relations (2) A graded vector space  is a g-module if it is provided with an R-bilinear map Given a graded commutative ring A, the following are standard constructions of new graded modules from the old ones.
(i) The direct sum of graded modules and a graded factor module are defined just as those of modules over a commutative ring.
(ii) The tensor product  ⊗  of graded A-modules  and  is their tensor product as A-modules such that In particular, the tensor algebra ⊗ of a graded A-module  is defined just as that of a module over a commutative ring.Its quotient ∧ with respect to the ideal generated by elements is a bigraded exterior algebra of a graded module  provided with a graded exterior product (iii) A morphism Φ :  →  of graded A-modules seen as additive groups is said to be an even (resp., odd) graded morphism if Φ preserves (resp., changes) the Grassmann parity of all graded-homogeneous elements of  and if the relations hold.A morphism Φ :  →  of graded A-modules as additive groups is called a graded A-module morphism if it is represented by a sum of even and odd graded morphisms.A set Hom A (, ) of graded morphisms of a graded A-module  to a graded A-module  is naturally a graded A-module.
A graded A-module  * = Hom A (, A) is called the dual of .
Linear differential operators and the differential calculus over a graded commutative ring are defined similarly to those in commutative geometry [3].
Let A be a graded commutative ring and ,  graded A-modules.A vector space Hom(, ) of graded real space homomorphisms Φ :  →  admits two graded A-module structures An element Δ ∈ Hom(, ) is said to be a -valued graded differential operator of order  on  if   0 ∘ ⋅ ⋅ ⋅ ∘    Δ = 0 for any tuple of  + 1 elements  0 , . . .,   of A.
In particular, zero order graded differential operators are A-module morphisms  → .For instance, let  = A.
Any zero order -valued graded differential operator Δ on A is given by its value Δ(1).A first order -valued graded differential operator Δ on A obeys a condition It is called the -valued graded derivation of A if Δ(1) = 0; that is, the graded Leibniz rule holds.If  is a graded derivation of A, then  is so for any  ∈ A. Hence, graded derivations of A constitute a graded A-module d(A, ), called the graded derivation module.If  = A, a graded derivation module dA also is a real Lie superalgebra with respect to a superbracket Since dA is a Lie superalgebra, let us consider the Chevalley-Eilenberg complex  * [dA; A], where a graded commutative ring A is regarded as a dA-module [3,21].It is a complex where   [dA; A] = Hom(∧  dA, A) are dA-modules of real linear graded morphisms of graded exterior products ∧  dA to A. One can show that complex (13) contains a subcomplex O * [dA] of A-linear graded morphisms [3].The N-graded module O * [dA] is provided with the structure of a bigraded A-algebra with respect to the graded exterior product where and  1 , . . .,  + are gradedhomogeneous elements of dA.The Chevalley-Eilenberg coboundary operator  (13) and the exterior product ∧ (14) obey relations and thus they bring O * [dA] into a differential bigraded algebra (henceforth DBGA).It is called the graded differential calculus over a graded commutative ring A. In particular, we have

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One can extend this duality relation to any element  ∈ O * [dA] by the rules As a consequence, every graded derivation  ∈ dA of A yields a derivation is called the de Rham complex of a graded commutative ring A.

Graded Manifolds and Bundles
A graded manifold of dimension (, ) is defined as a localringed space (, A), where  is an -dimensional smooth manifold  and A = A 0 ⊕A 1 is a sheaf of Grassmann algebras Λ of rank  (Remark 2) such that [3,13] (i) there is the exact sequence of sheaves where  is a body epimorphism onto a sheaf  ∞  of smooth real functions on ; (ii) R/R 2 is a locally free sheaf of  ∞ modules of finite rank (with respect to pointwise operations), and the sheaf A is locally isomorphic to the exterior product A sheaf A is called the structure sheaf of a graded manifold (, A), and a manifold  is said to be its body.Sections of a sheaf A are called graded functions on a graded manifold (, A).They constitute a graded commutative  ∞ ()-ring A() called the structure ring of (, A).
By virtue of Batchelor's theorem [13,22], graded manifolds possess the following structure.Theorem 4. Let (, A) be a graded manifold.There exists a vector bundle  →  with an -dimensional typical fibre  so that the structure sheaf of (, A) is isomorphic to a sheaf A  of sections of the exterior bundle ∧ * whose typical fibre is a Grassmann algebra ∧ * .
Combining Theorem 4 and the above-mentioned classical Serre-Swan theorem leads to the following Serre-Swan theorem for graded manifolds [12].Theorem 5. Let  be a smooth manifold.A graded commutative  ∞ ()-algebra A is isomorphic to the structure ring of a graded manifold with a body  iff it is the exterior algebra of some projective  ∞ ()-module of finite rank.
As was mentioned above Batchelor's isomorphism in Theorem 4 is not canonical, and we agree to call (, A  ) in Theorem 4 the simple graded manifold modelled over a characteristic vector bundle  → .Accordingly, the structure ring A  () of (, A  ) is a structure module of sections of the exterior bundle ∧ * .
Remark 6.One can treat a local-ringed space (, A 0 =  ∞  ) as a trivial graded manifold.It is a simple graded manifold whose characteristic bundle is  =  × {0}.Its structure module is a ring  ∞ () of smooth real functions on .
Given a simple graded manifold (, A  ), every trivialization chart (;   ,   ) of a vector bundle  →  yields a splitting domain (;   ,   ) of (, A  ) where {  } is the corresponding local fibre basis for  * → .Graded functions on such a chart are Λ-valued functions where   1 ⋅⋅⋅  () are smooth functions on .One calls {  ,   } the local generating basis for a graded manifold (, A  ).
Transition functions   =    (  )  of bundle coordinates on  →  induce the corresponding transformation law   =    (  )  of the associated local generating basis for a graded manifold (, A  ).
Let us consider the graded derivation module dA() of a graded commutative ring A().It is a Lie superalgebra relative to superbracket (12).Its elements are called the graded vector fields on a graded manifold (, A).A key point is the following [3,23].Lemma 7. Graded vector fields  ∈ dA  on a simple graded manifold (, A  ) are represented by sections of some vector bundle V  which is locally isomorphic to ∧ * ⊗  (⊗  ).
Graded vector fields on a splitting domain (;   ,   ) of (, A  ) read where   ,   are local graded functions on  possessing a coordinate transformation law Graded vector fields act on graded functions  ∈ A  () (22) by the rule Given a structure ring A  of graded functions on a simple graded manifold (, A  ) and the Lie superalgebra dA  of its graded derivations, let us consider the graded differential calculus Accordingly, the graded Chevalley-Eilenberg coboundary operator  (13), called the graded exterior differential, reads where derivations   ,   act on coefficients of graded exterior forms by formula (25) A morphism of graded manifolds (, A) → (  , A  ) is defined as that of local-ringed spaces where  is a manifold morphism and Φ is a sheaf morphism of A  to the direct image  * A of A onto   .Morphism (32) of graded manifolds is called (i) a monomorphism if  is an injection and Φ is an epimorphism and (ii) an epimorphism if  is a surjection and Φ is a monomorphism.An epimorphism of graded manifolds (, A) → (  , A  ), where  →   is a fibre bundle, is called the graded bundle [24,25].In this case, a sheaf monomorphism Φ induces a monomorphism of canonical presheaves A  → A, which associates with each open subset  ⊂  the ring of sections of A  over ().Accordingly, there is a pull-back monomorphism of the structure rings A  (  ) → A() of graded functions on graded manifolds (  , A  ) and (, A).
In particular, let (, A) be a graded manifold whose body  =  is a fibre bundle  :  → .Let us consider a trivial graded manifold (,  ∞  ) (Remark 6).Then we have a graded bundle We agree to call the graded bundle (33) over a trivial graded manifold (,  ∞  ) the graded bundle over a smooth manifold.Let us denote it by (, , A).Given a graded bundle (, , A), the local generating basis for a graded manifold (, A) can be brought into a form (  ,   ,   ) where (  ,   ) are bundle coordinates of  → .Remark 9. Let  →  be a fibre bundle.Then a trivial graded manifold (,  ∞  ) together with a ring monomorphism  ∞ () →  ∞ () is the graded bundle (, ,  ∞  ) (33).
Remark 10.A graded manifold (, A) itself can be treated as the graded bundle (, , A) (33) associated with the identity smooth bundle  → .
Let  →  and   →   be vector bundles and Φ :  →   their bundle morphism over a morphism  :  →   .Then every section  * of the dual bundle   * →   defines the pull-back section Φ *  * of the dual bundle  * →  by the law It follows that a bundle morphism (Φ, ) yields a morphism of simple graded manifolds This is a pair (, Φ =  * ∘ Φ * ) of a morphism  of body manifolds and the composition  * ∘Φ * of the pull-back A   ∋  → Φ *  ∈ A  of graded functions and the direct image  * of a sheaf A  onto   .Relative to local bases (  ,   ) and (  ,   ) for (, A  ) and (  , A   ), morphism (35) of simple graded manifolds reads   = (), Φ(  ) = Φ   ()  .The graded manifold morphism (35) is a monomorphism (resp., epimorphism) if Φ is a bundle injection (resp., surjection).In particular, the graded manifold morphism (35) is a graded bundle if Φ is a fibre bundle.Let A   → A  be the corresponding pull-back monomorphism of the structure rings.By virtue of Lemma 8 it yields a monomorphism of the DBGAs Let (, A  ) be a simple graded manifold modelled over a vector bundle  → .This is a graded bundle (, , A  ) modelled over a composite bundle  →  → . (37) If  →  is a vector bundle, this is a particular case of graded vector bundles in [11,24] whose base is a trivial graded manifold.The structure ring of graded functions on a simple graded manifold (, A  ) is the graded commutative  ∞ ()-ring A  = ∧ * () (21).Let the composite bundle (37) be provided with adapted bundle coordinates (  ,   ,   ) possessing transition functions   (  ),   (  ,   ), and   =    (  ,   )  .Then the corresponding local generating basis for a simple graded manifold (, A  ) is (  ,   ,   ) together with transition functions   =    (  ,   )  .We call it the local generating basis for a graded bundle (, , A  ).

Graded Jet Manifolds
As was mentioned above, Lagrangian theory on a smooth fibre bundle  →  is formulated in terms of the variational bicomplex on jet manifolds  *  of .These are fibre bundles over  and, therefore, they can be regarded as trivial graded bundles (,   ,  ∞    ).Then let us describe their partners in the case of graded bundles (, A  ) → (,  ∞  ) as follows.Note that, given a graded manifold (, A) and its structure ring A, one can define the jet module ) is a simple graded manifold modelled over a vector bundle  → , the jet module  1 A  is a module of global sections of a jet bundle  1 (∧ * ).A problem is that  1 A  fails to be a structure ring of some graded manifold.For this reason, we have suggested a different construction of jets of graded manifolds, though it is applied only to simple graded manifolds [12,23].
Let (, A  ) be a simple graded manifold modelled over a vector bundle  → .Let us consider a -order jet manifold    of .It is a vector bundle over .Then let (, A    ) be a simple graded manifold modelled over    → .We agree to call (, A    ) the graded -order jet manifold of a simple graded manifold (, A  ).Given a splitting domain (;   ,   ) of a graded manifold (, A  ), we have a splitting domain (;   ,   ,    ,    1  2 , . . .,    1 ⋅⋅⋅  ) of a graded jet manifold (, A    ).
As was mentioned above, a graded manifold is a particular graded bundle over its body (Remark 10).Then the definition of graded jet manifolds is generalized to graded bundles over smooth manifolds as follows.Let (, , A  ) be a graded bundle modelled over the composite bundle (37).It is readily observed that a jet manifold    of  →  is a vector bundle    →    coordinated by (  ,   Λ ,   Λ ), 0 ≤ |Λ| ≤ .Let (  , A  = A    ) be a simple graded manifold modelled over this vector bundle.Its local generating basis is We call (  , A  ) the graded -order jet manifold of a graded bundle (, , A  ).
In particular, let  →  be a smooth bundle seen as a trivial graded bundle (, ,  ∞  ) modelled over a composite bundle  × {0} →  → .Then its graded jet manifold is a trivial graded bundle (,   ,  ∞    ), that is, a jet manifold    of .Thus, the above definition of jets of graded bundles is compatible with the conventional definition of jets of fibre bundles.
Jet manifolds  *  of a fibre bundle  →  form the inverse sequence of affine bundles   −1 .One can think of elements of its projective limit  ∞  as being infinite order jets of sections of  →  identified by their Taylor series at points of .A set  ∞  is endowed with the projective limit topology which makes  ∞  a paracompact Fréchet manifold [3,5].It is called the infinite order jet manifold.A bundle coordinate atlas (  ,   ) of  provides  ∞  with the adapted manifold coordinate atlas The inverse sequence (38) of jet manifolds yields the direct sequence of graded differential algebras O *  of exterior forms on finite order jet manifolds where   −1 * are the pull-back monomorphisms.Its direct limit consists of all exterior forms on finite order jet manifolds modulo the pull-back identification.The O * ∞ (41) is a differential graded algebra which inherits operations of the exterior differential  and exterior product ∧ of exterior algebras O *  .
Fibre bundles  +1  →    (38) and the corresponding bundles  +1  →    yield graded bundles ( +1 , A +1 ) → (  , A  ) including pull-back monomorphisms of structure rings of graded functions on graded manifolds (  , A  ) and ( +1 , A +1 ).As a consequence, we have the inverse sequence of graded manifolds One can think of its inverse limit ( ∞ , A  ∞  ) as the graded Fréchet manifold whose body is an infinite order jet manifold  ∞  and whose structure sheaf As a consequence, we have a direct system of DBGAs The DBGA S * ∞ [; ] that we associate with a graded bundle (, A  ) is defined as the direct limit as a graded bundle modelled over a composite bundle Let us consider the corresponding DBGA Then, in accordance with Remark 11, there are monomorphisms (51) of BGDAs (55)

Graded Lagrangian Formalism
Let (, , A  ) be a graded bundle modelled over a composite bundle (37) over an -dimensional smooth manifold , and let S * ∞ [; ] be the associated DBGA (46) of graded exterior forms on graded jet manifolds of (, , A  ).As was mentioned above, Grassmann-graded Lagrangian theory of even and odd variables on a graded bundle is formulated Accordingly, the graded exterior differential  on S * ∞ [; ] falls into a sum  =   +   of the vertical and total graded differentials where   are graded total derivatives.These differentials obey the nilpotent relations A DBGA S * ∞ [; ] also is provided with the graded projection endomorphism such that ∘  = 0, and with the nilpotent graded variational operator With these operators a DBGA S * ∞ [; ] is decomposed into the Grassmann-graded variational bicomplex [12,23].We restrict our consideration to the short variational subcomplex of this bicomplex and its subcomplex of one-contact graded forms They possess the following cohomology [12,16].
Decomposed into a variational bicomplex, the DBGA S * ∞ [; ] describes graded Lagrangian theory on a graded bundle (, , A  ).Its graded Lagrangian is defined as an element of the graded variational complex (61).Accordingly, a graded exterior form is said to be its graded Euler-Lagrange operator.Its kernel yields an Euler-Lagrange equation We call a pair (S 0, ∞ [; ], ) the graded Lagrangian system and S * ∞ [; ] its structure algebra.The following are corollaries of Theorem 13 [12,16,23].

Corollary 14. Any variationally trivial odd Lagrangian is 𝑑
Corollary 15.Given a graded Lagrangian , there is the global variational formula where local graded functions  obey relations  ]  = 0, The form Ξ  (67) provides a global Lepage equivalent of a graded Lagrangian .
Given a graded Lagrangian system (S where   and   denote the horizontal and vertical parts of  [16]. A glance at expression (71) shows that a contact graded derivation  is the infinite order jet prolongation  =  ∞  of its restriction to a graded commutative ring  0 [; ].We call  (72) the generalized graded vector field on a graded manifold (, A  ).This fails to be a graded vector field on (, A  ) in general because its component may depend on jets of elements of the local generating basis for (, A  ).
In particular, the vertical contact graded derivation (71) reads It is said to be nilpotent if for any horizontal graded form  ∈  0, * ∞ .It is nilpotent only if it is odd and iff the equality holds for all   [16].
Remark 18.If there is no danger of confusion, the common symbol  further stands for a generalized graded vector field  (72), the contact graded derivation  determined by , and the Lie derivative L  .We agree to call all these operators, simply, a graded derivation of the structure algebra of a graded Lagrangian system.
Remark 19.For the sake of convenience, right graded derivations ⃖  = ⃖     also are considered.They act on graded functions and differential forms  on the right by the rules Given a Lagrangian system (S * ∞ [; ], ), the contact graded derivation  (71) is called the variational symmetry of a Lagrangian  if a Lie derivative L   of  along  is  exact; that is, L   =   .Then the following is a corollary of the variational formula (66) [16].

Theorem 20. The Lie derivative of a graded Lagrangian along any contact graded derivation (71) admits the decomposition
where Ξ  is the Lepage equivalent (67) of a Lagrangian .
A glance at expression (77) shows the following.

Lemma 21. (i) A contact graded derivation 𝜗 is a variational symmetry only if it is projected onto 𝑋. (ii) It is a variational
symmetry iff its vertical part   (71) is well.

Gauge Symmetries
Treating gauge symmetries of Lagrangian theory, one usually follows Yang-Mills gauge theory on principal bundles.This notion of gauge symmetries has been generalized to Lagrangian theory on an arbitrary fibre bundle [18].Here, we extend it to Lagrangian theory on graded bundles.
Let (S * ∞ [; ], ) be a graded Lagrangian system on a graded bundle (, , A  ) with the local generating basis (  ).Let  =  0 ⊕  1 be a graded vector bundle over  possessing 10 Advances in Mathematical Physics an even part  0 →  and the odd one  1 → .We regard it as a composite bundle Given a Lagrangian  ∈ S 0, ∞ [; ], let us define its pullback and consider an extended Lagrangian system provided with the local generating basis (  ,   ).In view of the first condition in Definition 22, the variables   of the extended Lagrangian system (82) can be treated as gauge parameters of a gauge symmetry .Furthermore, we additionally assume that a gauge symmetry  is linear in gauge parameters   and their jets   Λ (see Remark 35).Then the generalized graded vector field  (72) reads In accordance with Remark 18, we also call it the gauge symmetry.By virtue of item (ii) of Lemma 21, the generalized vector field  (83) is a gauge symmetry iff its vertical part is so.Therefore, we can restrict our consideration to vertical gauge symmetries.

Noether and Higher-Stage Noether Identities
Without loss of generality, let a Lagrangian  be even and its Euler-Lagrange operator  (64) at least of first order.This operator takes its values into a graded vector bundle where  *  is the vertical cotangent bundle of  → .It however is not a vector bundle over .Therefore, we restrict our consideration to a case of the pull-back composite bundle  (37): where  1 →  is a vector bundle.
Remark 23.Let us introduce the following notation.Given the vertical tangent bundle  of a fibre bundle  → , by its density-dual bundle is meant a fibre bundle If  →  is a vector bundle, we have where  is called the density-dual of .Let be a graded vector bundle over .Its graded density-dual is defined to be  =  1 ⊕  0 with an even part  1 →  and the odd one  1 → .Given the graded vector bundle  (88), we consider a product (,  0 ×  , A ×   ) of graded bundles over product (53) of the composite bundles  →  0 →  and  (37) and the corresponding DBGA which we denote: In particular, we treat the composite bundle  (37) as a graded vector bundle over  possessing only an odd part.The density-dual (86) of the vertical tangent bundle  of  →  is  (84).If  is the pull-back bundle (85), then is a graded vector bundle over .It can be seen as product (53) of composite bundles and we consider the corresponding graded bundle (52) and the DBGA (54) which we denote of graded densities of antifield number ≤ 2. Its oneboundaries Φ, Φ ∈ P 0, ∞ [; ] 2 , by the very definition, vanish on-shell.

Any one-cycle
of complex ( 93) is a differential operator on a bundle  such that it is linear on fibres of  →  and its kernel contains the graded Euler-Lagrange operator  (64); that is, Then one can say that one-cycles (94) define the NI (95) of an Euler-Lagrange operator , which we agree to call the NI of a graded Lagrangian system (S * ∞ [; ], ) [2].In particular, one-chains Φ (94) are necessarily NI if they are boundaries.Therefore, these NI are called trivial.They are of the form Accordingly, nontrivial NI modulo trivial ones are associated with elements of the first homology  1 () of complex (93).A Lagrangian  is called degenerate if there are nontrivial NI.
Nontrivial NI can obey first-stage NI.In order to describe them, let us assume that a module  1 () is finitely generated.
Namely, there exists a graded projective  ∞ ()-module C (0) ⊂  1 () of finite rank possessing a local basis {Δ  }: such that any element Φ ∈  1 () factorizes as through elements (97) of C (0) .Thus, all nontrivial NI (95) result from the NI  93) is finitely generated in the above-mentioned sense, this complex can be extended to the one-exact chain complex (102) with a boundary operator whose nilpotency conditions are equivalent to the complete NI (99).
Proof.By virtue of Theorem 5, a graded module C (0) is isomorphic to that of sections of the density-dual  0 of some graded vector bundle  0 → .Let us enlarge P * ∞ [; ] to a DBGA which is nilpotent iff the complete NI (99) hold.Then  0 (101) is a boundary operator of a chain complex of graded densities of antifield number ≤ 3. Let  * ( 0 ) denote its homology.We have  0 ( 0 ) =  0 () = 0. Furthermore, any one-cycle Φ up to a boundary takes the form (98) and, therefore, it is a  0 -boundary Hence,  1 ( 0 ) = 0; that is, complex (102) is one-exact.
The first-stage NI (105) are trivial either if the two-cycle Φ (104) is a  0 -boundary or its summand  vanishes on-shell.Therefore, nontrivial first-stage NI fails to exhaust the second homology  2 ( 0 ) of complex (102) in general.
A degenerate Lagrangian system is called reducible if it admits nontrivial first-stage NI.
If the condition of Lemma 26 is satisfied, let us assume that nontrivial first-stage NI are finitely generated as follows.There exists a graded projective  ∞ ()-module C (1) ⊂  2 ( 0 ) of finite rank possessing a local basis {Δ  1 }: such that any element Φ ∈  2 ( 0 ) factorizes as through elements (112) of C (1) .Thus, all nontrivial first-stage NI (105) result from the equalities with a local basis {  ,   ,   ,   1 } where   1 are first-stage antifields of Grassmann parity [  1 ] = [Δ  1 ] + 1 and antifield number 3.This DBGA is provided with the odd right graded derivation which is nilpotent iff the complete NI (99) and the complete first-stage NI (114) hold.Then  1 (116) is a boundary operator of a chain complex of graded densities of antifield number ≤ 4. Let  * ( 1 ) denote its homology.We have Advances in Mathematical Physics 13 By virtue of expression (113), any two-cycle of the complex (117) is a boundary It follows that  2 ( 1 ) = 0; that is, complex (117) is two-exact.
If the third homology  3 ( 1 ) of complex (117) is not trivial, its elements correspond to second-stage NI which the complete first-stage ones satisfy, and so on.Iterating the arguments, we say that a degenerate graded Lagrangian system (S * ∞ [; ], ) is -stage reducible if it admits finitely generated nontrivial -stage NI, but no nontrivial ( + 1)stage ones.It is characterized as follows [2]: (i) There are graded vector bundles  0 , . . .,   over , and a DBGA P * ∞ [; ] is enlarged to a DBGA (ii) DBGA ( 120) is provided with a nilpotent right graded derivation of antifield number −1.The index  = −1 here stands for   .The nilpotent derivation  KT (121) is called the KT operator.
(iii) With this graded derivation, a module P which satisfies the following homology regularity condition.
(iv) The nilpotentness  2 KT = 0 of the KT operator (121) is equivalent to the complete nontrivial NI (99) and the complete nontrivial This item means the following.
are the -stage NI.Conversely, let condition (126) hold.It can be extended to a cycle condition as follows.It is brought into the form A glance at expression (122) shows that the term in its righthand side belongs to P 0, ∞ { − 2} +1 .It is a  −2 -cycle and, consequently, a  −1 -boundary  −1 Ψ in accordance with Condition 1. Then equality (126) is a c Σ −1 -dependent part of a cycle condition but   Ψ does not make a contribution to this condition.
Proof.The -stage NI (126) are trivial either if a   -cycle Φ (125) is a   -boundary or its summand  vanishes on-shell.Let us show that if the summand  of Φ (125) is -exact, then Φ is a   -boundary.If  = Ψ, one can write Hence, the   -cycle condition reads By virtue of Condition 1, any
It may happen that a graded Lagrangian system possesses nontrivial NI of any stage.However, we restrict our consideration to -reducible Lagrangians for a finite integer .In this case, the KT operator (121) and the gauge operator (139) contain finite terms.

Second Noether Theorems
Different variants of the second Noether theorem have been suggested in order to relate reducible NI and gauge symmetries [2,15,26].The inverse second Noether theorem (Theorem 33), which we formulate in homology terms, associates with the KT complex (123) of nontrivial NI the cochain sequence (138) with the ascent operator u (139) whose components are gauge and higher-stage gauge symmetries of a Lagrangian system.Let us start with the following notation.
Remark 31.Given the DBGA P * ∞ {} (120), we consider a DBGA Theorem 33.Given the KT complex (123), a module of graded densities  0, ∞ {} is decomposed into a cochain sequence Advances in Mathematical Physics 15 Equality (141) is split into a set of equalities where  = 1, . . ., .A glance at equality (142) shows that, by virtue of decomposition (77), the odd graded derivation of  0 ∞ {0} is a variational symmetry of a graded Lagrangian .Every equality (143) falls into a set of equalities graded by the polynomial degree in antifields.Let us consider the equalities which are linear in antifields   −2 .We have This equality is brought into the form Using relation (134), we obtain the equality The variational derivative of both of its sides with respect to   −2 leads to the relation which the odd graded derivation satisfies.Graded derivations  (144) and  () (149) constitute the ascent operator (139).
A glance at the variational symmetry  (144) shows that it is a derivation of a ring  0 ∞ [0] which satisfies Definition 22. Consequently,  (144) is a gauge symmetry of a graded Lagrangian  associated with the complete nontrivial NI (99).Therefore, it is a nontrivial gauge symmetry.
Turn now to relation (148).For  = 1, it takes a form of a first-stage gauge symmetry condition on-shell which the nontrivial gauge symmetry  (144) satisfies.Therefore, one can treat the odd graded derivation as a first-stage gauge symmetry associated with the complete first-stage NI Iterating the arguments, one comes to relation (148) which provides a -stage gauge symmetry condition which is associated with the complete nontrivial -stage NI (124).The odd graded derivation  () (149) is called the -stage gauge symmetry.
Thus, components of the ascent operator u (139) in Theorem 33 are nontrivial gauge and higher-stage gauge symmetries.Therefore, we agree to call this operator the gauge one.
With the gauge operator (139), the extended Lagrangian   (140) takes a form where  * 1 is a term of polynomial degree in antifields exceeding 1.
The correspondence of gauge and higher-stage gauge symmetries to NI and higher-stage NI in Theorem 33 is unique due to the following direct second Noether theorem.
Theorem 34.(i) If  (144) is a gauge symmetry, the variational derivative of the   -exact density   E   (142) with respect to ghosts   leads to the equality Remark 35.One can consider gauge symmetries which need not be linear in ghosts.However, direct second Noether Theorem 34 is not relevant to these gauge symmetries because, in this case, an Euler-Lagrange operator satisfies the identities depending on ghosts.

Lagrangian BRST Theory
In contrast with the KT operator (121), the gauge operator u (138) need not be nilpotent.Let us study its extension to a nilpotent graded derivation of ghost number 1 by means of antifield-free terms  () of higher polynomial degree in ghosts    and their jets    Λ , 0 ≤  < .We call b (155) the BRST operator, where -stage gauge symmetries are extended to -stage BRST transformations acting both on ( − 1)-stage and -stage ghosts [18].If a BRST operator exists, sequence (138) is brought into a BRST complex There is the following necessary condition of the existence of such a BRST extension.
Theorem 36.The gauge operator (138) admits the nilpotent extension (155) only if the gauge symmetry conditions (148) and the higher-stage NI (124) are satisfied off-shell.
Proof.It is easily justified that if the graded derivation b (155) is nilpotent, then the right-hand sides of equalities (148) equal zero; that is, Using relations (134)-(137), one can show that, in this case, the right-hand sides of the higher-stage NI (124) also equal zero [2] It follows at once from equalities (157) that the higherstage gauge operator  HS = u −  =  (1) Let us denote where  () () are terms of polynomial degree 2 ≤  ≤  + 1 in ghosts.Then the nilpotent property (159) of b falls into a set of equalities of ghost polynomial degrees 1, 2, and 3 ≤  ≤  + 3, respectively.Equalities (161) are exactly the gauge symmetry conditions (157) in Theorem 36.
Let us consider a pull-back composite bundle where   →  is a vector bundle.Let us regard it as an odd graded vector bundle over .The density-dual  of the vertical tangent bundle  of  →  is a graded vector bundle Graded densities of this DBGA are endowed with the antibracket Then one associates with any (even) Lagrangian L the odd vertical graded derivations such that  L (L  ) = {L, L  }.
such that b =   is the graded derivation defined by the Lagrangian   (177).
The Lagrangian   (177) is said to be the BRST extension of an original Lagrangian .

Example: Topological BF Theory
We address the topological BF theory of two exterior forms  and  of form degree || + || = dim  − 1 on a smooth manifold  [20,27]

)
which reproduces the complete NI (99) by means of relation (137).(ii)Given the -stage gauge symmetry condition (148), the variational derivative of equality (147) with respect to ghosts    leads to the equality, reproducing the -stage NI (124) by means of relations (135)-(137).

𝑧 𝐴 , 𝑐 𝑎 ) is locally generated by graded one-forms 𝑑𝑧 𝐴 , 𝑑𝑐 𝑖 such that
)over A  where S 0 [; ] = A  .Since the graded derivation module dA  is isomorphic to the structure module of sections of a vector bundle V  →  in Lemma 7, elements of S * [; ] are represented by sections of the exterior bundle ∧V  of the A  -dual V  →  of V  .With respect to the dual fibre bases {  } for  *  and {  } for  * , sections of V  take a coordinate form  =     +     ,     + (−1) [  ]     .
→  0 → (78)and consider a graded bundle (,  0 , A  ) modelled over it.Then we define product (52) of graded bundles (, , A  ) and (,  0 , A  ) over product (53) of the composite bundles  (78) and  (37).It reads (,  0 × ] , ) Let us consider the density-dual  (90) of the vertical tangent bundle  → , and let us enlarge an original DBGA S * ∞ [; ] to the DBGA P * ∞ [; ] (92) with the local generating basis (  ,   ), [  ] = [] + 1.Following the terminology of Lagrangian BRST theory [15, 19], we agree to call its elements   the antifields of antifield number Ant[  ] = 1.A DBGA P * ∞ [; ] is endowed with the nilpotent right graded derivation  = ⃖   E  , where E  are the variational derivatives (64).Then we have a chain complex 0 ← Im called the complete NI.Note that factorization (98) is independent of specification of a local basis {Δ  } and, being representatives of  1 (), graded densities Δ   (97) are not -exact.A Lagrangian system whose nontrivial NI are finitely generated is called finitely degenerate.Hereafter, degenerate Lagrangian systems only of this type are considered.If the homology  1 () of complex ( called the complete first-stage NI.Note that, by virtue of the condition of Lemma 26, the first summands of the graded densities Δ  1  (112) are not -exact.A degenerate Lagrangian system is called finitely reducible if it admits finitely generated nontrivial first-stage NI.
Lemma 27.The one-exact complex (102) of a finitely reducible Lagrangian system is extended to the two-exact one (117) with a boundary operator whose nilpotency conditions are equivalent to the complete NI (99) and the complete first-stage NI (114).Proof.By virtue of Theorem 5, a graded module C (1) is isomorphic to that of sections of the density-dual  1 of some graded vector bundle  1 → .Let us enlarge the DBGA P * ∞ {0} (100) to a DBGA with a local generating basis (  ,   ,   ,   1 , . . .,    ) where    are -stage antifields of antifield number Ant[   ] =  + 2.

)
graded in ghost number.Its ascent operator u (139) is an odd graded derivation of ghost number 1 where  (144) is a variational symmetry of a graded Lagrangian  and the graded derivations  () (149),  = 1, . . ., , obey relations (148).It is readily observed that the KT operator  KT is an exact symmetry of the extended Lagrangian   ∈ P 0, ∞ {} (140).Since the graded derivation  KT is vertical, it follows from decomposition (77) that . It follows that the summand    of each cocycle Δ   (122) is  −1 -closed.Then its summand ℎ   also is  −1closed and, consequently,  −2 -closed.Hence it is  −1 -exact by virtue of Condition 1.Therefore, Δ   contains only the term    linear in antifields.
The antibracket of a Lagrangian L is   -exact; that is, The graded derivation  L (174) is nilpotent.Equality (175) is called the classical master equation.A solution of the master equation (175) is called nontrivial if both derivations (173) do not vanish.Being an element of the DBGA P * ∞ {} (133), an original Lagrangian  obeys the master equation (175) and yields the graded derivations   = 0,   =  (173); that is, it is a trivial solution of the master equation.However, its extension   (153) need not satisfy the master equation.Therefore, let us consider its extension  =   +   =  +  1 +  2 + ⋅ ⋅ ⋅(176) by means of even densities   ,  ≥ 2, of zero antifield number and polynomial degree  in ghosts.Then the following is a corollary of Theorem 37. Corollary 38.A Lagrangian  is extended to a proper solution   (176) of the master equation iff the gauge operator u (138) admits a nilpotent extension   (174).However, one can say something more [2, 12].