# Higher-Stage Noether Identities and Second Noether Theorems

Authors: G. Sardanashvily
Journal: Advances in Mathematical Physics
 (2015)

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## Abstract

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-stage Noether 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.


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## Body

## 1. 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 X (Section 5).Lagrangian theory of even (commutative) variables on ann-dimensional smooth manifold X conventionally is formulated in terms of smooth fibre bundles over X and jet manifolds of their sections [3–5] 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 [8–12]. 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 X form graded C∞(X)-modules, we follow the well known Serre–Swan theorem extended to graded manifolds (Theorem 5) [12]. It states that if a graded commutative C∞(X)-ring is generated by a projective C∞(X)-module of finite rank, it is isomorphic to a ring of graded functions on a graded manifold whose body is X. 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 manifoldZ and some structure sheaf A of Grassmann algebras on Z [12, 13]. Its sections form a graded commutative C∞(Z)-ring of graded functions on a graded manifold (Z,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 E→Z with a typical fibre V such that the structure sheaf A of (Z,A) is isomorphic to a sheaf AE of germs of sections of the exterior bundle ∧E∗ of the dual E∗ of E whose typical fibre is the Grassmann algebra ∧V∗ [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 (Z,AE), called the simple graded manifolds (Section 3).Lagrangian theory on fibre bundlesY→X can be adequately formulated in algebraic terms of a variational bicomplex of exterior forms on the infinite order jet manifold J∞Y of sections of Y→X, without appealing to the calculus of variations [3–5, 14]. This technique is extended to Lagrangian theory on graded manifolds and bundles [2, 12, 15, 16]. It is phrased in terms of the Grassmann-graded variational bicomplex of graded exterior forms on a graded infinite order jet manifold (J∞Y,AJ∞F) (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 (Section7). 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 (Theorem33) 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 [17–19]. 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.Remark 1.
Smooth manifolds throughout are assumed to be Hausdorff, second-countable, and, consequently, paracompact. Given a smooth manifoldX, its tangent and cotangent bundles TX and T∗X are endowed with bundle coordinates (xλ,x˙λ) and (xλ,x˙λ) with respect to holonomic frames {∂λ} and {dxλ}, respectively. Given a coordinate chart (U;xλ) of X, a multi-index Λ of length |Λ|=k denotes a collection of indices (λ1⋯λk) modulo permutations. By λ+Λ is meant a multi-index (λλ1⋯λk). We use the compact notation ∂Λ=∂λk∘⋯∘∂λ1.


## 2. Grassmann-Graded Differential Calculus

Throughout this work, by the Grassmann gradation is meant theZ2-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].LetK be a commutative ring. A K-module Q is called graded if it is endowed with a grading automorphism γ, γ2=Id. A graded module falls into a direct sum of modules Q=Q0⊕Q1 such that γ(q)=(-1)[q]q, q∈Q[q]. One calls Q0 and Q1 the even and odd parts of Q, respectively. In particular, by a real graded vector space B=B0⊕B1 is meant a graded R-module.AK-algebra A is called graded if it is a graded K-module such that [aa′]=[a]+[a′], where a and a′ are graded-homogeneous elements of A. Its even part A0 is a subalgebra of A, and the odd one A1 is an A0-module. If A is a graded ring with the unit 1, then [1]=0. A graded algebra A is called graded commutative if aa′=(-1)aa′a′a.Hereafter, all algebras and vector spaces are assumed to be real.Remark 2.
LetV be a vector space and Λ=∧V its exterior algebra. It is a graded commutative ring, called the Grassmann algebra, with respect to the Grassmann gradation(1)Λ=Λ0⊕Λ1,Λ0=R⨁1≤k⋀2kV,Λ1=⨁1≤k⋀2k-1V.Hereafter, Grassmann algebras of finite rank when V=RN only are considered.Given a graded algebraA, a left graded A-module Q is defined as a left A-module where [aq]=[a]+[q]. Similarly, right graded A-modules are treated. If A is graded commutative, a graded A-module Q is provided with a graded A-bimodule structure by letting qa=(-1)aqaq.Remark 3.
A graded algebrag is called a Lie superalgebra if its product [·,·], called the Lie superbracket, obeys the relations(2)ɛ,ɛ′=--1ɛɛ′ɛ′,ɛ,-1ɛɛ′′ɛ,ɛ′,ɛ′′+-1ɛ′ɛɛ′,ɛ′′,ɛ+-1ɛ′′ɛ′ɛ′′,ɛ,ɛ′=0.A graded vector space P is a g-module if it is provided with an R-bilinear map(3)g×P∋ɛ,p⟶ɛp∈P,ɛp=ɛ+p,ɛ,ɛ′p=ɛ∘ɛ′--1ɛɛ′ɛ′∘ɛp.Given a graded commutative ringA, 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 productP⊗Q of graded A-modules P and Q is their tensor product as A-modules such that (4)p⊗q=p+q,ap⊗q=-1papa⊗q=-1pap⊗aq.In particular, the tensor algebra ⊗P of a graded A-module P is defined just as that of a module over a commutative ring. Its quotient ∧P with respect to the ideal generated by elements (5)p⊗p′+-1pp′p′⊗p,p,p′∈P,is a bigraded exterior algebra of a graded module P provided with a graded exterior product (6)p∧p′=--1pp′p′∧p.(iii) A morphismΦ:P→Q 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 P and if the relations (7)Φap=-1ΦaaΦp,p∈P,a∈A,hold. A morphism Φ:P→Q 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 HomA(P,Q) of graded morphisms of a graded A-module P to a graded A-module Q is naturally a graded A-module. A graded A-module P∗=HomA(P,A) is called the dual of P.Linear differential operators and the differential calculus over a graded commutative ring are defined similarly to those in commutative geometry [3].LetA be a graded commutative ring and P, Q graded A-modules. A vector space Hom(P,Q) of graded real space homomorphisms Φ:P→Q admits two graded A-module structures (8)aΦp=aΦp,Φ⋆ap=Φap,a∈A,p∈P.Let us put(9)δaΦ=aΦ--1aΦΦ⋆a,a∈A.An element Δ∈Hom(P,Q) is said to be a Q-valued graded differential operator of order s on P if δa0∘⋯∘δasΔ=0 for any tuple of s+1 elements a0,…,as of A.In particular, zero order graded differential operators areA-module morphisms P→Q. For instance, let P=A. Any zero order Q-valued graded differential operator Δ on A is given by its value Δ(1). A first order Q-valued graded differential operator Δ on A obeys a condition (10)Δab=Δab+-1aΔaΔb--1b+aΔabΔ1,a,b∈A.It is called the Q-valued graded derivation of A if Δ(1)=0; that is, the graded Leibniz rule (11)Δab=Δab+-1aΔaΔb,a,b∈A,holds. If ∂ is a graded derivation of A, then a∂ is so for any a∈A. Hence, graded derivations of A constitute a graded A-module d(A,Q), called the graded derivation module. If Q=A, a graded derivation module dA also is a real Lie superalgebra with respect to a superbracket(12)u,u′=u∘u′--1uu′u′∘u,u,u′∈A.SincedA is a Lie superalgebra, let us consider the Chevalley–Eilenberg complex C∗[dA;A], where a graded commutative ring A is regarded as a dA-module [3, 21]. It is a complex(13)0⟶R⟶A⟶dC1dA;A⟶d⋯CkdA;A⟶d⋯,where Ck[dA;A]=Hom(∧kdA,A) are dA-modules of real linear graded morphisms of graded exterior products ∧kdA 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(14)ϕ∧ϕ′u1,…,ur+s=∑i1<⋯<ir;j1<⋯<jsSgn1⋯r+si1⋯irj1⋯jsϕui1,…,uir·ϕ′uj1,…,ujs,where ϕ∈Or[dA],  ϕ′∈Os[dA], and u1,…,ur+s are graded-homogeneous elements of dA. The Chevalley–Eilenberg coboundary operator d (13) and the exterior product ∧ (14) obey relations(15)ϕ∧ϕ′=-1ϕϕ′+ϕϕ′ϕ′∧ϕ,dϕ∧ϕ′=dϕ∧ϕ′+-1ϕϕ∧dϕ′,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(16)O1dA=HomAdA,A=dA∗.One can extend this duality relation to any element ϕ∈O∗[dA] by the rules (17)ubda=-1ubbua,uϕ∧ϕ′=uϕ∧ϕ′+-1ϕ+ϕuϕ∧uϕ′,a,b∈A.As a consequence, every graded derivation u∈dA of A yields a derivation(18)Luϕ=udϕ+duϕ,Luϕ∧ϕ′=Luϕ∧ϕ′+-1uϕϕ∧Luϕ′,called the graded Lie derivative, of the DBGA O∗[dA].The minimal graded differential calculusO∗A⊂O∗[dA] over a graded commutative ring A consists of the monomials a0da1∧⋯∧dak, ai∈A. The corresponding complex(19)0⟶R⟶A⟶dO1A⟶d⋯OkA⟶d⋯is called the de Rham complex of a graded commutative ring A.


## 3. Graded Manifolds and Bundles

A graded manifold of dimension(n,m) is defined as a local-ringed space (Z,A), where Z is an n-dimensional smooth manifold Z and A=A0⊕A1 is a sheaf of Grassmann algebras Λ of rank m (Remark 2) such that [3, 13] (i) there is the exact sequence of sheaves(20)0⟶R⟶A⟶σCZ∞⟶0,R=A1+A12,where σ is a body epimorphism onto a sheaf CZ∞ of smooth real functions on Z; (ii) R/R2 is a locally free sheaf of CZ∞-modules of finite rank (with respect to pointwise operations), and the sheaf A is locally isomorphic to the exterior product ∧CZ∞(R/R2).A sheafA is called the structure sheaf of a graded manifold (Z,A), and a manifold Z is said to be its body. Sections of a sheaf A are called graded functions on a graded manifold (Z,A). They constitute a graded commutative C∞(Z)-ring A(Z) called the structure ring of (Z,A).By virtue of Batchelor’s theorem [13, 22], graded manifolds possess the following structure.Theorem 4.
Let(Z,A) be a graded manifold. There exists a vector bundle E→Z with an m-dimensional typical fibre V so that the structure sheaf of (Z,A) is isomorphic to a sheaf AE of sections of the exterior bundle ∧E∗ whose typical fibre is a Grassmann algebra ∧V∗.Combining Theorem4 and the above-mentioned classical Serre–Swan theorem leads to the following Serre–Swan theorem for graded manifolds [12].Theorem 5.
LetZ be a smooth manifold. A graded commutative C∞(Z)-algebra A is isomorphic to the structure ring of a graded manifold with a body Z iff it is the exterior algebra of some projective C∞(Z)-module of finite rank.As was mentioned above Batchelor’s isomorphism in Theorem4 is not canonical, and we agree to call (Z,AE) in Theorem 4 the simple graded manifold modelled over a characteristic vector bundle E→Z. Accordingly, the structure ring AE(Z) of (Z,AE) is a structure module(21)AE=AEZ=∧E∗Zof sections of the exterior bundle ∧E∗.Remark 6.
One can treat a local-ringed space(Z,A0=CZ∞) as a trivial graded manifold. It is a simple graded manifold whose characteristic bundle is E=Z×{0}. Its structure module is a ring C∞(Z) of smooth real functions on Z.Given a simple graded manifold(Z,AE), every trivialization chart (U;zA,qa) of a vector bundle E→Z yields a splitting domain (U;zA,ca) of (Z,AE) where {ca} is the corresponding local fibre basis for E∗→X. Graded functions on such a chart are Λ-valued functions(22)f=∑k=0m1k!fa1⋯akzca1⋯cak,where fa1⋯ak(z) are smooth functions on U. One calls {zA,ca} the local generating basis for a graded manifold (Z,AE). Transition functions q′a=ρba(zA)qb of bundle coordinates on E→Z induce the corresponding transformation law c′a=ρba(zA)cb of the associated local generating basis for a graded manifold (Z,AE).Let us consider the graded derivation moduledA(Z) of a graded commutative ring A(Z). It is a Lie superalgebra relative to superbracket (12). Its elements are called the graded vector fields on a graded manifold (Z,A). A key point is the following [3, 23].Lemma 7.
Graded vector fieldsu∈dAE on a simple graded manifold (Z,AE) are represented by sections of some vector bundle VE which is locally isomorphic to ∧E∗⊗Z(E⊗ZTZ).Graded vector fields on a splitting domain(U;zA,ca) of (Z,AE) read (23)u=uA∂A+ua∂a,∂a∘∂b=-∂b∘∂a,∂A∘∂a=∂a∘∂A,where uA,ua are local graded functions on U possessing a coordinate transformation law (24)u′A=uA,u′a=ρjauj+uA∂Aρjacj.Graded vector fields act on graded functions f∈AE(U) (22) by the rule(25)ufa⋯bca⋯cb=uA∂Afa⋯bca⋯cb+ukfa⋯b∂kca⋯cb.Given a structure ringAE of graded functions on a simple graded manifold (Z,AE) and the Lie superalgebra dAE of its graded derivations, let us consider the graded differential calculus(26)S∗E;Z=O∗dAEover AE where S0[E;Z]=AE. Since the graded derivation module dAE is isomorphic to the structure module of sections of a vector bundle VE→Z in Lemma 7, elements of S∗[E;Z] are represented by sections of the exterior bundle ∧V¯E of the AE-dual V¯E→Z of VE. With respect to the dual fibre bases {dzA} for T∗Z and {dcb} for E∗, sections of V¯E take a coordinate form (27)ϕ=ϕAdzA+ϕadca,ϕa′=ρ-1abϕb,ϕA′=ϕA+ρ-1ab∂Aρjaϕbcj.The duality isomorphism S1[E;Z]=dAE∗ (16) is given by the graded interior product (28)uϕ=uAϕA+-1ϕauaϕa.Elements of S∗[E;Z] are called graded exterior forms on a graded manifold (Z,AE). In particular, elements of S0[E;Z] are graded functions on (Z,AE).Seen as anAE-algebra, the DBGA S∗[E;Z] (26) on a splitting domain (U;zA,ca) is locally generated by graded one-forms dzA, dci such that (29)dzA∧dci=-dci∧dzA,dci∧dcj=dcj∧dci.Accordingly, the graded Chevalley–Eilenberg coboundary operator d (13), called the graded exterior differential, reads (30)dϕ=dzA∧∂Aϕ+dca∧∂aϕ,where derivations ∂λ, ∂a act on coefficients of graded exterior forms by formula (25), and they are graded commutative with graded forms dzA and dca. Formulas (15)–(18) hold.Lemma 8.
The DBGAS∗[E;Z] (26) is a minimal differential calculus over AE; that is, it is generated by elements df, f∈AE [3, 23].The de Rham complex (19) of the minimal graded differential calculus S∗[E;Z] reads(31)0⟶R⟶AE⟶dS1E;Z⟶d⋯SkE;Z⟶d⋯.Given the differential graded algebra O∗(Z) of exterior forms on Z, there exists a canonical cochain monomorphism O∗(Z)→S∗[E;Z] of the de Rham complex O∗(Z) to complex (31).A morphism of graded manifolds(Z,A)→(Z′,A′) is defined as that of local-ringed spaces(32)ϕ:Z⟶Z′,Φ^:A′⟶ϕ∗A,where ϕ is a manifold morphism and Φ^ is a sheaf morphism of A′ to the direct image ϕ∗A of A onto Z′. Morphism (32) of graded manifolds is called (i) a monomorphism if ϕ is an injection and Φ^ is an epimorphism and (ii) an epimorphism if ϕ is a surjection and Φ^ is a monomorphism.An epimorphism of graded manifolds(Z,A)→(Z′,A′), where Z→Z′ 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 U⊂Z the ring of sections of A′ over ϕ(U). Accordingly, there is a pull-back monomorphism of the structure rings A′(Z′)→A(Z) of graded functions on graded manifolds (Z′,A′) and (Z,A).In particular, let(Y,A) be a graded manifold whose body Z=Y is a fibre bundle π:Y→X. Let us consider a trivial graded manifold (X,CX∞) (Remark 6). Then we have a graded bundle(33)Y,A⟶X,CX∞.We agree to call the graded bundle (33) over a trivial graded manifold (X,CX∞) the graded bundle over a smooth manifold. Let us denote it by (X,Y,A). Given a graded bundle (X,Y,A), the local generating basis for a graded manifold (Y,A) can be brought into a form (xλ,yi,ca) where (xλ,yi) are bundle coordinates of Y→X.Remark 9.
LetY→X be a fibre bundle. Then a trivial graded manifold (Y,CY∞) together with a ring monomorphism C∞(X)→C∞(Y) is the graded bundle (X,Y,CY∞) (33).Remark 10.
A graded manifold(X,A) itself can be treated as the graded bundle (X,X,A) (33) associated with the identity smooth bundle X→X.LetE→Z and E′→Z′ be vector bundles and Φ:E→E′ their bundle morphism over a morphism ϕ:Z→Z′. Then every section s∗ of the dual bundle E′∗→Z′ defines the pull-back section Φ∗s∗ of the dual bundle E∗→Z by the law (34)vzΦ∗s∗z=Φvzs∗φz,vz∈Ez.It follows that a bundle morphism (Φ,ϕ) yields a morphism of simple graded manifolds(35)Z,AE⟶Z′,AE′.This is a pair (ϕ,Φ^=ϕ∗∘Φ∗) of a morphism ϕ of body manifolds and the composition ϕ∗∘Φ∗ of the pull-back AE′∋f→Φ∗f∈AE of graded functions and the direct image ϕ∗ of a sheaf AE onto Z′. Relative to local bases (zA,ca) and (z′A,c′a) for (Z,AE) and (Z′,AE′), morphism (35) of simple graded manifolds reads z′=ϕ(z), Φ^(c′a)=Φba(z)cb.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 AE′→AE be the corresponding pull-back monomorphism of the structure rings. By virtue of Lemma 8 it yields a monomorphism of the DBGAs(36)S∗E′;Z′⟶S∗E;Z.Let(Y,AF) be a simple graded manifold modelled over a vector bundle F→Y. This is a graded bundle (X,Y,AF) modelled over a composite bundle(37)F⟶Y⟶X.If Y→X 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 (Y,AF) is the graded commutative C∞(X)-ring AF=∧F∗(Y) (21). Let the composite bundle (37) be provided with adapted bundle coordinates (xλ,yi,qa) possessing transition functions x′λ(xμ), y′i(xμ,yj), and q′a=ρba(xμ,yj)qb. Then the corresponding local generating basis for a simple graded manifold (Y,AF) is (xλ,yi,ca) together with transition functions c′a=ρba(xμ,jj)cb. We call it the local generating basis for a graded bundle (X,Y,AF).


## 4. Graded Jet Manifolds

As was mentioned above, Lagrangian theory on a smooth fibre bundleY→X is formulated in terms of the variational bicomplex on jet manifolds J∗Y of Y. These are fibre bundles over X and, therefore, they can be regarded as trivial graded bundles (X,JkY,CJkY∞). Then let us describe their partners in the case of graded bundles (Y,AF)→(X,CX∞) as follows.Note that, given a graded manifold(X,A) and its structure ring A, one can define the jet module J1A of a C∞(X)-ring A [3]. If (X,AE) is a simple graded manifold modelled over a vector bundle E→X, the jet module J1AE is a module of global sections of a jet bundle J1(∧E∗). A problem is that J1AE 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(X,AE) be a simple graded manifold modelled over a vector bundle E→X. Let us consider a k-order jet manifold JkE of E. It is a vector bundle over X. Then let (X,AJkE) be a simple graded manifold modelled over JkE→X. We agree to call (X,AJkE) the graded k-order jet manifold of a simple graded manifold (X,AE). Given a splitting domain (U;xλ,ca) of a graded manifold (Z,AE), we have a splitting domain (U;xλ,ca,cλa,cλ1λ2a,…,cλ1⋯λka) of a graded jet manifold (X,AJkE).As was mentioned above, a graded manifold is a particular graded bundle over its body (Remark10). Then the definition of graded jet manifolds is generalized to graded bundles over smooth manifolds as follows. Let (X,Y,AF) be a graded bundle modelled over the composite bundle (37). It is readily observed that a jet manifold JrF of F→X is a vector bundle JrF→JrY coordinated by (xλ,yΛi,qΛa), 0≤|Λ|≤r. Let (JrY,Ar=AJrF) be a simple graded manifold modelled over this vector bundle. Its local generating basis is (xλ,yΛi,cΛa), 0≤|Λ|≤r. We call (JrY,Ar) the graded r-order jet manifold of a graded bundle (X,Y,AF).In particular, letY→X be a smooth bundle seen as a trivial graded bundle (X,Y,CY∞) modelled over a composite bundle Y×{0}→Y→X. Then its graded jet manifold is a trivial graded bundle (X,JrY,CJrY∞), that is, a jet manifold JrY of Y. Thus, the above definition of jets of graded bundles is compatible with the conventional definition of jets of fibre bundles.Jet manifoldsJ∗Y of a fibre bundle Y→X form the inverse sequence(38)Y⟵πJ1Y⟵⋯Jr-1Y⟵πr-1rJrY⟵⋯,of affine bundles πr-1r. One can think of elements of its projective limit J∞Y as being infinite order jets of sections of Y→X identified by their Taylor series at points of X. A set J∞Y is endowed with the projective limit topology which makes J∞Y a paracompact Fréchet manifold [3, 5]. It is called the infinite order jet manifold. A bundle coordinate atlas (xλ,yi) of Y provides J∞Y with the adapted manifold coordinate atlas(39)xλ,yΛi,yλ+Λ′i=∂xμ∂x′λdμyΛ′i,dλ=∂λ+yλi∂i+∑0<Λyλ+Λi∂iΛ.The inverse sequence (38) of jet manifolds yields the direct sequence of graded differential algebras Or∗ of exterior forms on finite order jet manifolds(40)O∗X⟶π∗O∗Y⟶π01∗O1∗⟶⋯Or-1∗⟶πr-1r∗Or∗⟶⋯,where πr-1r∗ are the pull-back monomorphisms. Its direct limit(41)O∞∗=lim→Or∗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 d and exterior product ∧ of exterior algebras Or∗.Fibre bundlesJr+1Y→JrY (38) and the corresponding bundles Jr+1F→JrF yield graded bundles (Jr+1Y,Ar+1)→(JrY,Ar) including pull-back monomorphisms of structure rings(42)Sr0F;Y⟶Sr+10F;Yof graded functions on graded manifolds (JrY,Ar) and (Jr+1Y,Ar+1). As a consequence, we have the inverse sequence of graded manifolds (43)Y,AF⟵J1Y,AJ1F⟵⋯Jr-1Y,AJr-1F⟵JrY,AJrF⟵⋯.One can think of its inverse limit (J∞Y,AJ∞F) as the graded Fréchet manifold whose body is an infinite order jet manifold J∞Y and whose structure sheaf AJ∞F is a sheaf of germs of graded functions on graded manifolds (J∗Y,AJ∗F) [12, 23].By virtue of Lemma8, the differential calculus Sr∗[F;Y] is minimal. Therefore, the monomorphisms of structure rings (42) yield the pull-back monomorphisms (36) of DBGAs(44)πrr+1∗:Sr∗F;Y⟶Sr+1∗F;Y.As a consequence, we have a direct system of DBGAs(45)S∗F;Y⟶π∗S1∗F;Y⟶⋯Sr-1∗F;Y⟶πr-1r∗Sr∗F;Y⟶⋯.The DBGA S∞∗[F;Y] that we associate with a graded bundle (Y,AF) is defined as the direct limit(46)S∞∗F;Y=lim→Sr∗F;Yof the direct system (45). It consists of all graded exterior forms ϕ∈S∗[Fr;JrY] on graded manifolds (JrY,Ar) modulo monomorphisms (44). Its elements obey relations (15).The cochain monomorphismsOr∗→Sr∗[F;Y] provide a monomorphism of the direct system (40) to the direct system (45) and, consequently, a cochain monomorphism O∞∗→S∞∗[F;Y].One can think of elements ofS∞∗[F;Y] as being graded differential forms on an infinite order jet manifold J∞Y in the sense that S∞∗[F;Y] is a submodule of the structure module of sections of some sheaf on J∞Y [12, 23]. In particular, one can restrict S∞∗[F;Y] to the coordinate chart (39) of J∞Y so that S∞∗[F;Y] as an O∞0-algebra is locally generated by the elements (47)cΛa,dxλ,θΛa=dcΛa-cλ+Λadxλ,θΛi=dyΛi-yλ+Λidxλ,0≤Λ,where cΛa, θΛa are odd and dxλ, θΛi are even. We agree to call (yi,ca) the local generating basis for S∞∗[F;Y]. Let the collective symbol sA stand for its elements. We further denote [A]=[sA].Remark 11.
Let(X,Y,AF) and (X,Y′,AF′) be graded bundles modelled over composite bundles F→Y→X and F′→Y′→X, respectively. Let F→F′ be a fibre bundle over a fibre bundle Y→Y′ over X. Then we have a graded bundle (X,Y,AF)→(X,Y′,AF′) together with the pull-back monomorphism (36) of DBGAs(48)S∗F′;Y′⟶S∗F;Y.Let (X,JrY,AJrF) and (X,JrY′,AJrF′) be graded bundles modelled over composite bundles JrF→JrY→X and JrF′→JrY′→X, respectively. Since JrF→JrF′ is a fibre bundle over a fibre bundle JrY→JrY′ over X, we also get a graded bundle (49)X,JrY,AJrF⟶X,JrY′,AJrF′together with the pull-back monomorphism of DBGAs(50)Sr∗F′;Y′⟶Sr∗F;Y.Monomorphisms (48)–(50),  r=1,2,…, provide a monomorphism of the direct limits(51)S∞∗F′;Y′⟶S∞∗F;Yof DBGAs Sr∗[F′;Y′] and Sr∗[F;Y], r=0,1,2,….Remark 12.
Let(X,Y,AF) and (X,Y′,AF′) be graded bundles modelled over composite bundles F→Y→X and F′→Y′→X, respectively. We define their product(52)X,Y,AF×XX,Y′,AF′=X,Y×XY′,AF×XF′as a graded bundle modelled over a composite bundle(53)F×XF′=F×Y×Y′F′⟶Y×XY′⟶X.Let us consider the corresponding DBGA(54)S∞∗F×XF′;Y×XY′.Then, in accordance with Remark 11, there are monomorphisms (51) of BGDAs(55)S∞∗F;Y⟶S∞∗F×XF;Y×XY′,S∞∗F′;Y′⟶S∞∗F×XF;Y×XY′.


## 5. Graded Lagrangian Formalism

Let(X,Y,AF) be a graded bundle modelled over a composite bundle (37) over an n-dimensional smooth manifold X, and let S∞∗[F;Y] be the associated DBGA (46) of graded exterior forms on graded jet manifolds of (X,Y,AF). As was mentioned above, Grassmann-graded Lagrangian theory of even and odd variables on a graded bundle is formulated in terms of the variational bicomplex in which the DBGA S∞∗[F;Y] is split in [2, 16, 23].A DBGAS∞∗[F;Y] is decomposed into S∞0[F;Y]-modules S∞k,r[F;Y] of k-contact and r-horizontal graded forms together with the corresponding projections (56)hk:S∞∗F;Y⟶S∞k,∗F;Y,hm:S∞∗F;Y⟶S∞∗,mF;Y.Accordingly, the graded exterior differential d on S∞∗[F;Y] falls into a sum d=dV+dH of the vertical and total graded differentials(57)dV∘hm=hm∘d∘hm,dVϕ=θΛA∧∂AΛϕ,ϕ∈S∞∗F;Y,dH∘hk=hk∘d∘hk,dH∘h0=h0∘d,dHϕ=dxλ∧dλϕ,dλ=∂λ+∑sλ+ΛA∂AΛ,where dλ are graded total derivatives. These differentials obey the nilpotent relations (58)dH∘dH=0,dV∘dV=0,dH∘dV+dV∘dH=0.A DBGA S∞∗[F;Y] also is provided with the graded projection endomorphism(59)ϱ=∑k>01kϱ¯∘hk∘hn:S∞∗>0,nF;Y⟶S∞∗>0,nF;Y,ϱ¯ϕ=∑-1ΛθA∧dΛ∂AΛϕ,ϕ∈S∞∗>0,nF;Y,such that ϱ∘dH=0, and with the nilpotent graded variational operator(60)δ=ϱ∘d:S∞∗,nF;Y⟶S∞∗+1,nF;Y.With these operators a DBGA S∞∗[F;Y] is decomposed into the Grassmann-graded variational bicomplex [12, 23]. We restrict our consideration to the short variational subcomplex(61)0⟶R⟶S∞0F;Y⟶dHS∞0,1F;Y⋯⟶dHS∞0,nF;Y⟶δϱS∞1,nF;Yof this bicomplex and its subcomplex of one-contact graded forms(62)0⟶S∞1,0F;Y⟶dHS∞1,1F;Y⋯⟶dHS∞1,nF;Y⟶ϱϱS∞1,nF;Y⟶0.They possess the following cohomology [12, 16].Theorem 13.
Cohomology of complex (61) equals the de Rham cohomology of Y. Complex (62) is exact.Decomposed into a variational bicomplex, the DBGAS∞∗[F;Y] describes graded Lagrangian theory on a graded bundle (X,Y,AF). Its graded Lagrangian is defined as an element(63)L=Lω∈S∞0,nF;Y,ω=dx1∧⋯∧dxn,of the graded variational complex (61). Accordingly, a graded exterior form(64)δL=θA∧EAω=∑-1ΛθA∧dΛ∂AΛLω∈ϱS∞1,nF;Yis said to be its graded Euler–Lagrange operator. Its kernel yields an Euler–Lagrange equation(65)δL=0,EA=∑-1ΛθA∧dΛ∂AΛL=0.We call a pair (S∞0,n[F;Y],L) the graded Lagrangian system and S∞∗[F;Y] its structure algebra.The following are corollaries of Theorem13 [12, 16, 23].Corollary 14.
Any variationally trivial odd Lagrangian isdH-exact.Corollary 15.
Given a graded LagrangianL, there is the global variational formula(66)dL=δL-dHΞL,Ξ∈S∞n-1F;Y,(67)ΞL=L+∑s=0θνs⋯ν1A∧FAλνs⋯ν1ωλ,FAνk⋯ν1=∂Aνk⋯ν1L-dλFAλνk⋯ν1+σAνk⋯ν1,k=1,2,…,where local graded functions σ obey relations σAν=0, σA(νkνk-1)⋯ν1=0.The formΞL (67) provides a global Lepage equivalent of a graded Lagrangian L.Given a graded Lagrangian system(S∞∗[F;Y],L), by its infinitesimal transformations are meant graded derivations of a graded commutative ring S∞0[F;Y]. These derivations constitute a S∞0[F;Y]-module dS∞0[F;Y] which is a real Lie superalgebra with respect to the Lie superbracket (12). The following holds [16].Theorem 16.
A derivation moduledS∞0[F;Y] is isomorphic to the S∞0[F;Y]-dual S∞1[F;Y]∗ of the module of graded one-forms S∞1[F;Y].In particular, it follows that the DBGAS∞∗[F;Y] is minimal differential calculus over a graded commutative ring S∞0[F;Y]. Restricted to the coordinate chart (39) of J∞Y, an algebra S∞∗[F;Y] is a free S∞0[F;Y]-module generated by one-forms dxλ, θΛA.Due to the isomorphism in Theorem16, any graded derivation ϑ∈dS∞0[F;Y] takes a form(68)ϑ=ϑλ∂λ+ϑA∂A+∑0<ΛϑΛA∂AΛ.Given ϑ∈dS∞0[F;Y] and ϕ∈S∞1[F;Y], let ϑϕ denote the corresponding interior product. Extended to the DBGA S∞∗[F;Y], it obeys a rule (69)ϑϕ∧σ=ϑϕ∧σ+-1ϕ+ϕϑϕ∧ϑσ,ϕ,σ∈S∞∗F;Y.Every graded derivationϑ (68) of a ring S∞0[F;Y] yields a Lie derivative (70)Lϑϕ=ϑdϕ+dϑϕ,Lϑϕ∧σ=Lϑϕ∧σ+-1ϑϕϕ∧Lϑσ,of a DBGA S∞∗[F;Y]. The graded derivation ϑ (68) is called contact if a Lie derivative Lϑ preserves the ideal of contact graded forms of S∞∗[F;Y] generated by contact one-forms.Lemma 17.
With respect to the local generating basis(sA) for the DBGA S∞∗[F;Y], any of its contact graded derivations takes a form(71)ϑ=ϑH+ϑV=υλdλ+υA∂A+∑Λ>0dΛυA-sμAυμ∂AΛ,where ϑH and ϑV denote the horizontal and vertical parts of ϑ [16].A glance at expression (71) shows that a contact graded derivation ϑ is the infinite order jet prolongation ϑ=J∞υ of its restriction(72)υ=υλ∂λ+υA∂A=υH+υV=υλdλ+uA∂A-sλA∂Aλto a graded commutative ring S0[F;Y]. We call υ (72) the generalized graded vector field on a graded manifold (Y,AF). This fails to be a graded vector field on (Y,AF) in general because its component may depend on jets of elements of the local generating basis for (Y,AF).In particular, the vertical contact graded derivation (71) reads (73)ϑ=υA∂A+∑Λ>0dΛυA∂AΛ.It is said to be nilpotent if (74)LϑLϑϕ=∑υΣB∂BΣυΛA∂AΛ+-1sBυAυΣBυΛA∂BΣ∂AΛϕ=0for any horizontal graded form ϕ∈S∞0,∗. It is nilpotent only if it is odd and iff the equality (75)LϑυA=∑υΣB∂BΣυA=0holds for all υA [16].Remark 18.
If there is no danger of confusion, the common symbolυ further stands for a generalized graded vector field υ (72), the contact graded derivation ϑ determined by υ, and the Lie derivative Lϑ. 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 derivationsυ←=∂←AυA also are considered. They act on graded functions and differential forms ϕ on the right by the rules (76)υ←ϕ=dϕυ←+dϕυ←,υ←ϕ∧ϕ′=-1ϕ′υ←ϕ∧ϕ′+ϕ∧υ←ϕ′.Given a Lagrangian system(S∞∗[F;Y],L), the contact graded derivation ϑ (71) is called the variational symmetry of a Lagrangian L if a Lie derivative LϑL of L along ϑ is dH-exact; that is, LϑL=dHσ. 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(77)LϑL=υVδL+dHh0ϑΞL+dVυHωL,where ΞL is the Lepage equivalent (67) of a Lagrangian L.A glance at expression (77) shows the following.Lemma 21.
(i) A contact graded derivationϑ is a variational symmetry only if it is projected onto X. (ii) It is a variational symmetry iff its vertical part ϑV (71) is well.


## 6. 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∞∗[F;Y],L) be a graded Lagrangian system on a graded bundle (X,Y,AF) with the local generating basis (sA). Let E=E0⊕E1 be a graded vector bundle over X possessing an even part E0→X and the odd one E1→X. We regard it as a composite bundle(78)E⟶E0⟶Xand consider a graded bundle (X,E0,AE) modelled over it. Then we define product (52) of graded bundles (X,Y,AF) and (X,E0,AE) over product (53) of the composite bundles E (78) and F (37). It reads (X,E0×XY,AE×XF). Let us consider the corresponding DBGA(79)S∞∗E×XF;E0×XYtogether with monomorphisms (55) of DBGAs (80)S∞∗F;Y⟶S∞∗E×XF;E0×XY,S∞∗E;E0⟶S∞∗E×XF;E0×XY.Given a LagrangianL∈S∞0,n[F;Y], let us define its pull-back(81)L∈S∞0,nF;Y⊂S∞∗E×XF;E0×XYand consider an extended Lagrangian system(82)S∞∗E×XF;E0×XY,Lprovided with the local generating basis (sA,cr).Definition 22.
A gauge transformation of the LagrangianL (81) is defined to be a contact graded derivation ϑ of the ring S∞0[E×XF;E0×XY] (79) such that a derivation ϑ equals zero on a subring S∞0[E;E0]⊂S∞0[E×XF;E0×XY]. A gauge transformation ϑ is called the gauge symmetry if it is a variational symmetry of the Lagrangian L (81).In view of the first condition in Definition22, the variables cr of the extended Lagrangian system (82) can be treated as gauge parameters of a gauge symmetry ϑ. Furthermore, we additionally assume that a gauge symmetry ϑ is linear in gauge parameters cr and their jets cΛr (see Remark 35). Then the generalized graded vector field υ (72) reads(83)υ=∑0≤Λ≤mυrλΛxμcΛr∂λ+∑0≤Λ≤mυrAΛxμ,sΣBcΛr∂A.In accordance with Remark 18, we also call it the gauge symmetry.By virtue of item (ii) of Lemma21, the generalized vector field υ (83) is a gauge symmetry iff its vertical part is so. Therefore, we can restrict our consideration to vertical gauge symmetries.


## 7. Noether and Higher-Stage Noether Identities

Without loss of generality, let a LagrangianL be even and its Euler–Lagrange operator δL (64) at least of first order. This operator takes its values into a graded vector bundle(84)VF¯=V∗F⨁F⋀nT∗X⟶F,where V∗F is the vertical cotangent bundle of F→X. It however is not a vector bundle over Y. Therefore, we restrict our consideration to a case of the pull-back composite bundle F (37):(85)F=Y×XF1⟶Y⟶X,where F1→X is a vector bundle.Remark 23.
Let us introduce the following notation. Given the vertical tangent bundleVE of a fibre bundle E→X, by its density-dual bundle is meant a fibre bundle(86)VE¯=V∗E⨁E⋀nT∗X.If E→X is a vector bundle, we have(87)VE¯=E¯×XE,E¯=E∗⨁X⋀nT∗X,where E¯ is called the density-dual of E. Let(88)E=E0⨁XE1be a graded vector bundle over X. Its graded density-dual is defined to be E¯=E¯1⊕E¯0 with an even part E¯1→X and the odd one E¯1→X. Given the graded vector bundle E (88), we consider a product (X,E0×XY,AE×XF) of graded bundles over product (53) of the composite bundles E→E0→X and F (37) and the corresponding DBGA which we denote: (89)P∞∗F×XE;Y=S∞∗F×XE;Y×XE0.In particular, we treat the composite bundle F (37) as a graded vector bundle over Y possessing only an odd part. The density-dual (86) of the vertical tangent bundle VF of F→X is VF¯ (84). If F is the pull-back bundle (85), then(90)VF¯=F¯1⨁YV∗Y⨁Y⋀nT∗X⨁YF1is a graded vector bundle over Y. It can be seen as product (53) of composite bundles (91)VF1¯=F¯1⨁XF1⟶F¯1⟶X,VY¯⟶Y⟶X,and we consider the corresponding graded bundle (52) and the DBGA (54) which we denote(92)P∞∗VF¯;Y=S∞∗VF¯;Y×XF¯1=S∞∗VF¯1×XVY¯;Y×XF¯1.Lemma 24.
One can associate with any graded Lagrangian system(S∞∗[F;Y],L) the chain complex (93) whose one-boundaries vanish on-shell.Proof.
Let us consider the density-dualVF¯ (90) of the vertical tangent bundle VF→F, and let us enlarge an original DBGA S∞∗[F;Y] to the DBGA P∞∗[VF¯;Y] (92) with the local generating basis (sA,s¯A), [s¯A]=[A]+1. Following the terminology of Lagrangian BRST theory [15, 19], we agree to call its elements s¯A the antifields of antifield number Ant[s¯A]=1. A DBGA P∞∗[VF¯;Y] is endowed with the nilpotent right graded derivation δ¯=∂←AEA, where EA are the variational derivatives (64). Then we have a chain complex(93)0⟵Im⁡δ¯⟵δ¯P∞0,nVF¯;Y1⟵δ¯P∞0,nVF¯;Y2of graded densities of antifield number ≤2. Its one-boundaries δ¯Φ, Φ∈P∞0,n[VF¯;Y]2, by the very definition, vanish on-shell.Any one-cycle(94)Φ=∑ΦA,Λs¯ΛAω∈P∞0,nVF¯;Y1of complex (93) is a differential operator on a bundle VF¯ such that it is linear on fibres of VF¯→F and its kernel contains the graded Euler–Lagrange operator δL (64); that is,(95)δ¯Φ=0,∑ΦA,ΛdΛEAω=0.Then one can say that one-cycles (94) define the NI (95) of an Euler–Lagrange operator δL, which we agree to call the NI of a graded Lagrangian system (S∞∗[F;Y],L) [2].In particular, one-chainsΦ (94) are necessarily NI if they are boundaries. Therefore, these NI are called trivial. They are of the form (96)Φ=∑TAΛBΣdΣEBs¯ΛAω,TAΛBΣ=--1ABTBΣAΛ.Accordingly, nontrivial NI modulo trivial ones are associated with elements of the first homology H1(δ¯) of complex (93). A Lagrangian L 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 moduleH1(δ¯) is finitely generated. Namely, there exists a graded projective C∞(X)-module C(0)⊂H1(δ¯) of finite rank possessing a local basis {Δrω}:(97)Δrω=∑ΔrA,Λs¯ΛAω,ΔrA,Λ∈S∞0F;Y,such that any element Φ∈H1(δ¯) factorizes as(98)Φ=∑Φr,ΞdΞΔrω,Φr,Ξ∈S∞0F;Y,through elements (97) of C(0). Thus, all nontrivial NI (95) result from the NI(99)δ¯Δr=∑ΔrA,ΛdΛEA=0,called the complete NI. Note that factorization (98) is independent of specification of a local basis {Δrω} and, being representatives of H1(δ¯), graded densities Δrω (97) are not δ¯-exact.A Lagrangian system whose nontrivial NI are finitely generated is called finitely degenerate. Hereafter, degenerate Lagrangian systems only of this type are considered.Lemma 25.
If the homologyH1(δ¯) of complex (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 Theorem5, a graded module C(0) is isomorphic to that of sections of the density-dual E¯0 of some graded vector bundle E0→X. Let us enlarge P∞∗[VF¯;Y] to a DBGA(100)P¯∞∗0=P∞∗VF¯×XE¯0;Y=S∞∗VF¯×XE¯0;E¯01×XF¯1×XYwith a local generating basis (sA,s¯A,c¯r) where c¯r are antifields of Grassmann parity [c¯r]=[Δr]+1 and antifield number Ant[c¯r]=2. DBGA (100) admits an odd right graded derivation(101)δ0=δ¯+∂←rΔrwhich is nilpotent iff the complete NI (99) hold. Then δ0 (101) is a boundary operator of a chain complex(102)0⟵Im⁡δ¯⟵δ¯P∞0,nVF¯;Y1⟵δ0P¯∞0,n02⟵δ0P¯∞0,n03of graded densities of antifield number ≤3. Let H∗(δ0) denote its homology. We have H0(δ0)=H0(δ¯)=0. Furthermore, any one-cycle Φ up to a boundary takes the form (98) and, therefore, it is a δ0-boundary (103)Φ=∑Φr,ΞdΞΔrω=δ0∑Φr,Ξc¯Ξrω.Hence, H1(δ0)=0; that is, complex (102) is one-exact.Let us consider the second homologyH2(δ0) of complex (102). Its two-chains read(104)Φ=G+H=∑Gr,Λc¯Λrω+∑HA,ΛB,Σs¯ΛAs¯ΣBω.Its two-cycles define the first-stage NI(105)δ0Φ=0,∑Gr,ΛdΛΔrω=-δ¯H.Conversely, let equality (105) hold. Then it is a cycle condition of the two-chain (104).Note that this definition of first-stage NI is independent of specification of a generating moduleC(0) up to chain isomorphisms between complexes (102).The first-stage NI (105) are trivial either if the two-cycle Φ (104) is a δ0-boundary or its summand G vanishes on-shell. Therefore, nontrivial first-stage NI fails to exhaust the second homology H2(δ0) of complex (102) in general.Lemma 26.
Nontrivial first-stage NI modulo trivial ones are identified with elements of the homologyH2(δ0) iff any δ¯-cycle ϕ∈P¯∞0,n{0}2 is a δ0-boundary.Proof.
It suffices to show that if a summandG of a two-cycle Φ (104) is δ¯-exact, then Φ is a boundary. If G=δ¯Ψ, let us write(106)Φ=δ0Ψ+δ¯-δ0Ψ+H.Hence, cycle condition (105) reads (107)δ0Φ=δ¯δ¯-δ0Ψ+H=0.Since any δ¯-cycle ϕ∈P¯∞0,n{0}2, by assumption, is δ0-exact, then (δ¯-δ0)Ψ+H is a δ0-boundary. Consequently, Φ (106) is δ0-exact. Conversely, let Φ∈P¯∞0,n{0}2 be a δ¯-cycle; that is, (108)δ¯Φ=2ΦA,ΛB,Σs¯ΛAδ¯s¯ΣBω=2ΦA,ΛB,Σs¯ΛAdΣEBω=0.It follows that Φ(A,Λ)(B,Σ)δ¯s¯ΣB=0 for all indices (A,Λ). Omitting a δ¯-boundary term, we obtain (109)ΦA,ΛB,Σs¯ΣB=GA,Λr,ΞdΞΔr.Hence, Φ takes a form (110)Φ=G′A,Λr,ΞdΞΔrs¯ΛAω.Then there exists a three-chain Ψ=G′(A,Λ)(r,Ξ)c¯Ξrs¯ΛAω such that(111)δ0Ψ=Φ+σ=Φ+G′′A,Λr,ΞdΛEAc¯Ξrω.Owing to the equality δ¯Φ=0, we have δ0σ=0. Thus, σ in expression (111) is a δ¯-exact δ0-cycle. By assumption, it is δ0-exact; that is, σ=δ0ψ. Thus, a δ¯-cycle Φ is a δ0-boundary.A degenerate Lagrangian system is called reducible if it admits nontrivial first-stage NI.If the condition of Lemma26 is satisfied, let us assume that nontrivial first-stage NI are finitely generated as follows. There exists a graded projective C∞(X)-module C(1)⊂H2(δ0) of finite rank possessing a local basis {Δr1ω}:(112)Δr1ω=∑Δr1r,Λc¯Λrω+hr1ω,such that any element Φ∈H2(δ0) factorizes as(113)Φ=∑Φr1,ΞdΞΔr1ω,Φr1,Ξ∈S∞0F;Y,through elements (112) of C(1). Thus, all nontrivial first-stage NI (105) result from the equalities(114)∑Δr1r,ΛdΛΔr+δ¯hr1=0,called the complete first-stage NI. Note that, by virtue of the condition of Lemma 26, the first summands of the graded densities Δr1ω (112) are not δ¯-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 Theorem5, a graded module C(1) is isomorphic to that of sections of the density-dual E¯1 of some graded vector bundle E1→X. Let us enlarge the DBGA P¯∞∗{0} (100) to a DBGA(115)P¯∞∗1=P∞∗VF¯×XE¯0×XE¯1;Ywith a local basis {sA,s¯A,c¯r,c¯r1} where c¯r1 are first-stage antifields of Grassmann parity [c¯r1]=[Δr1]+1 and antifield number 3. This DBGA is provided with the odd right graded derivation(116)δ1=δ0+∂←r1Δr1which is nilpotent iff the complete NI (99) and the complete first-stage NI (114) hold. Then δ1 (116) is a boundary operator of a chain complex(117)0⟵Im⁡δ¯⟵δ¯P∞0,nVF¯;Y1⟵δ0P¯∞0,n02⟵δ1P¯∞0,n13⟵δ1P¯∞0,n14of graded densities of antifield number ≤4. Let H∗(δ1) denote its homology. We have (118)H0δ1=H0δ¯,H1δ1=H1δ0=0.By virtue of expression (113), any two-cycle of the complex (117) is a boundary (119)Φ=∑Φr1,ΞdΞΔr1ω=δ1∑Φr1,Ξc¯Ξr1ω.It follows that H2(δ1)=0; that is, complex (117) is two-exact.If the third homologyH3(δ1) 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∞∗[F;Y],L) is N-stage reducible if it admits finitely generated nontrivial N-stage NI, but no nontrivial (N+1)-stage ones. It is characterized as follows [2]:(i)
There are graded vector bundlesE0,…,EN over X, and a DBGA P∞∗[VF¯;Y] is enlarged to a DBGA(120)P¯∞∗N=P∞∗VF¯×XE¯0×X⋯×XE¯N;Ywith a local generating basis(sA,s¯A,c¯r,c¯r1,…,c¯rN) where c¯rk are k-stage antifields of antifield number Ant[c¯rk]=k+2.(ii)
DBGA (120) is provided with a nilpotent right graded derivation(121)δKT=δN=δ¯+∑∂←rΔrA,Λs¯ΛA+∑1≤k≤N∂←rkΔrk,(122)Δrkω=∑Δrkrk-1,Λc¯Λrk-1ω+∑hrkrk-2,ΣA,Ξc¯Σrk-2s¯ΞA+⋯ω∈P¯∞0,nk-1k+1,of antifield number −1. The indexk=-1 here stands for s¯A. The nilpotent derivation δKT (121) is called the KT operator.(iii)
With this graded derivation, a moduleP¯∞0,n{N}≤N+3 of densities of antifield number ≤(N+3) is decomposed into the exact KT chain complex(123)0⟵Im⁡δ¯⟵δ¯P∞0,nVF¯;Y1⟵δ0P¯∞0,n02⟵δ1P¯∞0,n13⋯⟵δN-1P¯∞0,nN-1N+1⟵δKTP¯∞0,nNN+2⟵δKTP¯∞0,nNN+3which satisfies the following homology regularity condition.Condition 1.
Anyδk<N-cycle ϕ∈P¯∞0,n{k}k+3⊂P¯∞0,n{k+1}k+3 is a δk+1-boundary. (iv)
The nilpotentnessδKT2=0 of the KT operator (121) is equivalent to the complete nontrivial NI (99) and the complete nontrivial (k≤N)-stage NI(124)∑Δrkrk-1,ΛdΛ∑Δrk-1rk-2,Σc¯Σrk-2=-δ¯∑hrkrk-2,ΣA,Ξc¯Σrk-2s¯ΞA.This item means the following.Lemma 28.
Theδk-cocycles Φ∈P∞0,n{k}k+2 are k-stage NI, andvice versa.Proof.
Any(k+2)-chain Φ∈P∞0,n{k}k+2 takes a form(125)Φ=G+H=∑Grk,Λc¯Λrkω+∑HA,Ξrk-1,Σs¯ΞAc¯Σrk-1+⋯ω.If it is a δk-cycle, then(126)∑Grk,ΛdΛ∑Δrkrk-1,Σc¯Σrk-1+δ¯∑HA,Ξrk-1,Σs¯ΞAc¯Σrk-1=0are the k-stage NI. Conversely, let condition (126) hold. It can be extended to a cycle condition as follows. It is brought into the form(127)δk∑Grk,Λc¯Λrk+∑HA,Ξrk-1,Σs¯ΞAc¯Σrk-1=-∑Grk,ΛdΛhrk+∑HA,Ξrk-1,Σs¯ΞAdΣΔrk-1.A glance at expression (122) shows that the term in its right-hand side belongs to P∞0,n{k-2}k+1. It is a δk-2-cycle and, consequently, a δk-1-boundary δk-1Ψ in accordance with Condition 1. Then equality (126) is a c¯Σrk-1-dependent part of a cycle condition (128)δk∑Grk,Λc¯Λrk+∑HA,Ξrk-1,Σs¯ΞAc¯Σrk-1-Ψ=0,but δkΨ does not make a contribution to this condition.Lemma 29.
Trivialk-stage NI are δk-boundaries Φ∈P∞0,n{k}k+2.Proof.
Thek-stage NI (126) are trivial either if a δk-cycle Φ (125) is a δk-boundary or its summand G vanishes on-shell. Let us show that if the summand G of Φ (125) is δ¯-exact, then Φ is a δk-boundary. If G=δ¯Ψ, one can write(129)Φ=δkΨ+δ¯-δkΨ+H.Hence, the δk-cycle condition reads (130)δkΦ=δk-1δ¯-δkΨ+H=0.By virtue of Condition 1, any δk-1-cycle ϕ∈P¯∞0,n{k-1}k+2 is δk-exact. Then (δ¯-δk)Ψ+H is a δk-boundary. Consequently, Φ (125) is δk-exact.Lemma 30.
All nontrivialk-stage NI (126), by assumption, factorize as (131)Φ=∑Φrk,ΞdΞΔrkω,Φr1,Ξ∈S∞0F;Y,through the complete ones (124).It may happen that a graded Lagrangian system possesses nontrivial NI of any stage. However, we restrict our consideration toN-reducible Lagrangians for a finite integer N. In this case, the KT operator (121) and the gauge operator (139) contain finite terms.


## 8. 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 DBGAP¯∞∗{N} (120), we consider a DBGA(132)P∞∗N=P∞∗F×XE0×X⋯×XEN;Y,possessing the local generating basis (sA,cr,cr1,…,crN), [crk]=[c¯rk]+1, and a DBGA(133)P∞∗N=P∞∗VF¯×XE0×X⋯×XEN×XE¯0×X⋯×XE¯N;Ywith a local generating basis (sA,s¯A,cr,cr1,…,crN,c¯r,c¯r1,…,c¯rN). Their elements crk are called k-stage ghosts of ghost number gh[crk]=k+1 and antifield number Ant[crk]=-(k+1). A C∞(X)-module C(k) of k-stage ghosts is the density-dual of a module C(k+1) of (k+1)-stage antifields. In accordance with Remark 11, the DBGAs P¯∞∗{N} (120) and the BGDA P∗(N) (132) are subalgebras of the DBGA P∞∗{N} (133). The KT operator δKT (121) naturally is extended to a graded derivation of a DBGA P∞∗{N}.Remark 32.
Any graded differential formϕ∈S∞∗[F;Y] and any finite tuple (fΛ), 0≤|Λ|≤k, of local graded functions fΛ∈S∞0[F;Y] obey the following relations:(134)∑0≤Λ≤kfΛdΛϕ∧ω=∑-1ΛdΛfΛϕ∧ω+dHσ,(135)∑0≤Λ≤k-1ΛdΛfΛϕ=∑0≤Λ≤kηfΛdΛϕ,(136)ηfΛ=∑0≤Σ≤k-Λ-1Σ+ΛΣ+Λ!Σ!Λ!dΣfΣ+Λ,(137)ηηfΛ=fΛ.Theorem 33.
Given the KT complex (123), a module of graded densities P∞0,n{N} is decomposed into a cochain sequence(138)0⟶S∞0,nF;Y⟶uP∞0,nN1⟶uP∞0,nN2⟶u⋯,(139)u=u+u1+⋯+uN=uA∂∂sA+ur∂∂cr+⋯+urN-1∂∂crN-1,graded in ghost number. Its ascent operator u (139) is an odd graded derivation of ghost number 1 where u (144) is a variational symmetry of a graded Lagrangian L and the graded derivations u(k) (149), k=1,…,N, obey relations (148).Proof.
Given the KT operator (121), let us extend an original Lagrangian L to a Lagrangian(140)Le=L+L1=L+∑0≤k≤NcrkΔrkω=L+δKT∑0≤k≤Ncrkc¯rkωof zero antifield number. It is readily observed that the KT operator δKT is an exact symmetry of the extended Lagrangian Le∈P∞0,n{N} (140). Since the graded derivation δKT is vertical, it follows from decomposition (77) that(141)δ←Leδs¯AEA+∑0≤k≤Nδ←Leδc¯rkΔrkω=υAEA+∑0≤k≤NυrkδLeδcrkω=dHσ,υA=δ←Leδs¯A=uA+wA=∑cΛrηΔrAΛ+∑1≤i≤N∑cΛriη∂←AhriΛ,υrk=δ←Leδc¯rk=urk+wrk=∑cΛrk+1ηΔrk+1rkΛ+∑k+1<i≤N∑cΛriη∂←rkhriΛ.Equality (141) is split into a set of equalities(142)δ←crΔrδs¯AEAω=uAEAω=dHσ0,(143)δ←crkΔrkδs¯AEA+∑0≤i<kδ←crkΔrkδc¯riΔriω=dHσk,where k=1,…,N. A glance at equality (142) shows that, by virtue of decomposition (77), the odd graded derivation(144)u=uA∂A,uA=∑cΛrηΔrAΛ,of P∞0{0} is a variational symmetry of a graded Lagrangian L. 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 c¯rk-2. We have(145)δ←δs¯Acrk∑hrkrk-2,ΣA,Ξc¯Σrk-2s¯ΞAEAω+δ←δc¯rk-1crk∑Δrkrk-1′,Σc¯Σrk-1′∑Δrk-1rk-2,Ξc¯Ξrk-2ω=dHσk.This equality is brought into the form (146)∑-1ΞdΞcrk∑hrkrk-2,ΣA,Ξc¯Σrk-2EAω+urk-1∑Δrk-1rk-2,Ξc¯Ξrk-2ω=dHσk.Using relation (134), we obtain the equality(147)∑crkhrkrk-2,ΣA,Ξc¯Σrk-2dΞEAω+urk-1∑Δrk-1rk-2,Ξc¯Ξrk-2ω=dHσk′.The variational derivative of both of its sides with respect to c¯rk-2 leads to the relation(148)∑dΣurk-1∂∂cΣrk-1urk-2=δ¯αrk-2,αrk-2=-∑ηhrkrk-2A,ΞΣdΣcrks¯ΞA,which the odd graded derivation(149)uk=urk-1∂∂crk-1=∑cΛrkηΔrkrk-1Λ∂∂crk-1,k=1,…,N,satisfies. Graded derivations u (144) and u(k) (149) constitute the ascent operator (139).A glance at the variational symmetryu (144) shows that it is a derivation of a ring P∞0[0] which satisfies Definition 22. Consequently, u (144) is a gauge symmetry of a graded Lagrangian L associated with the complete nontrivial NI (99). Therefore, it is a nontrivial gauge symmetry.Turn now to relation (148). For k=1, it takes a form (150)∑dΣur∂rΣuA=δ¯αAof a first-stage gauge symmetry condition on-shell which the nontrivial gauge symmetry u (144) satisfies. Therefore, one can treat the odd graded derivation (151)u1=ur∂r,ur=∑cΛr1ηΔr1rΛ,as a first-stage gauge symmetry associated with the complete first-stage NI (152)∑Δr1r,ΛdΛ∑ΔrA,Σs¯ΣA=-δ¯∑hr1B,ΣA,Ξs¯ΣBs¯ΞA.Iterating the arguments, one comes to relation (148) which provides a k-stage gauge symmetry condition which is associated with the complete nontrivial k-stage NI (124). The odd graded derivation u(k) (149) is called the k-stage gauge symmetry.Thus, components of the ascent operatoru (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 Le (140) takes a form(153)Le=L+u∑0≤k≤Ncrk-1c¯rk-1ω+L1∗+dHσ,where L1∗ 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 Theorem33 is unique due to the following direct second Noether theorem.Theorem 34.
(i) Ifu (144) is a gauge symmetry, the variational derivative of the dH-exact density uAEAω (142) with respect to ghosts cr leads to the equality(154)δruAEAω=∑-1ΛdΛurAΛEA=∑-1ΛdΛηΔrAΛEA=∑-1ΛηηΔrAΛdΛEA=0,which reproduces the complete NI (99) by means of relation (137).
(ii) Given thek-stage gauge symmetry condition (148), the variational derivative of equality (147) with respect to ghosts crk leads to the equality, reproducing the k-stage NI (124) by means of relations (135)–(137).Remark 35.
One can consider gauge symmetries which need not be linear in ghosts. However, direct second Noether Theorem34 is not relevant to these gauge symmetries because, in this case, an Euler–Lagrange operator satisfies the identities depending on ghosts.


## 9. 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(155)b=u+γ=u+∑1≤k≤N+1γk=u+∑1≤k≤N+1γrk-1∂∂crk-1=uA∂∂sA+γr∂∂cr+∑0≤k≤N-1urk∂∂crk+γrk+1∂∂crk+1of ghost number 1 by means of antifield-free terms γ(k) of higher polynomial degree in ghosts cri and their jets cΛri, 0≤i<k. We call b (155) the BRST operator, where k-stage gauge symmetries are extended to k-stage BRST transformations acting both on (k-1)-stage and k-stage ghosts [18]. If a BRST operator exists, sequence (138) is brought into a BRST complex(156)0⟶S∞0,nF;Y⟶bP∞0,nN1⟶bP∞0,nN2⟶b⋯.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 derivationb (155) is nilpotent, then the right-hand sides of equalities (148) equal zero; that is,(157)uk+1uk=0,0≤k≤N-1,u0=u.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 that the summand Grk of each cocycle Δrk (122) is δk-1-closed. Then its summand hrk also is δk-1-closed and, consequently, δk-2-closed. Hence it is δk-1-exact by virtue of Condition 1. Therefore, Δrk contains only the term Grk linear in antifields.It follows at once from equalities (157) that the higher-stage gauge operator (158)uHS=u-u=u1+⋯+uNis nilpotent, and u(u)=u(u). Therefore, the nilpotency condition for the BRST operator b (155) takes a form(159)bb=u+γu+u+uHS+γγ=0.Let us denote(160)γ0=0,γk=γ2k+⋯+γk+1k,k=1,…,N+1,γirk-1=∑k1+⋯+ki=k+1-i∑0≤Λkjγirk1,…,rkirk-1,Λk1,…,ΛkicΛk1rk1⋯cΛkirki,γN+2=0,where γ(i)(k) are terms of polynomial degree 2≤i≤k+1 in ghosts. Then the nilpotent property (159) of b falls into a set of equalities(161)uk+1uk=0,0≤k≤N-1,(162)u+γ2k+1uk+uHSγ2k=0,0≤k≤N+1,(163)γik+1uk+uγi-1k+uHSγik+∑2≤m≤i-1γmγi-m+1k=0,i-2≤k≤N+1,of ghost polynomial degrees 1, 2, and 3≤i≤N+3, respectively.Equalities (161) are exactly the gauge symmetry conditions (157) in Theorem 36.Equality (162) for k=0 reads(164)u+γ1u=0,∑dΛuA∂AΛuB+dΛγrurB,Λ=0.It takes a form of the Lie antibracket(165)u,u=-2γ1u=-2∑dΛγrurB,Λ∂Bof an odd gauge symmetry u. Its right-hand side factorizes through u, but it is nonlinear in ghosts.Equalities (162)-(163) for k=1 take a form(166)u+γ22u1+u1γ1=0,γ32u1+u+γ1γ1=0.In particular, if a Lagrangian system is irreducible, that is, u(k)=0, the BRST operator reads(167)b=u+γ1=uA∂A+γr∂r=∑urA,ΛcΛr∂A+∑γpqr,Λ,ΞcΛpcΞq∂r.In this case, the nilpotency conditions (166) are reduced to the equality(168)u+γ1γ1=0.Furthermore, let a gauge symmetry u be affine in fields sA and their jets. It follows from the nilpotency condition (164) that the BRST term γ(1) is independent of original fields and their jets. Then relation (168) takes a form of the Jacobi identity(169)γ1γ1=0for coefficient functions γpqr,Λ,Ξ(x) in the Lie antibracket (165).Relations (165) and (169) motivate us to think of equalities (162)-(163) in a general case of reducible gauge symmetries as beingsui generis generalized commutation relations and Jacobi identities of gauge symmetries, respectively [18]. Therefore, one can say that gauge symmetries are algebraically closed (in the terminology of [19]) if the gauge operator u (139) admits the nilpotent BRST extension b (155).The DBGAP∞∗{N} (133) is a particular field-antifield theory of the following type [2, 15, 19].Let us consider a pull-back composite bundle(170)W=Z×XZ′⟶Z⟶X,where Z′→X is a vector bundle. Let us regard it as an odd graded vector bundle over Z. The density-dual VW¯ of the vertical tangent bundle VW of W→X is a graded vector bundle (171)VW¯=Z′¯⨁ZV∗Z⨁Z⋀nT∗X⨁YZ′over Z (cf. (90)). Let us consider the DBGA P∞∗[VW¯;Z] (92) with the local generating basis (za,z¯a), [z¯a]=[za]+1. Its elements za and z¯a are called fields and antifields, respectively.Graded densities of this DBGA are endowed with the antibracket(172)Lω,L′ω=δ←Lδz¯aδL′δza+-1L′L′+1δ←L′δz¯aδLδzaω.Then one associates with any (even) Lagrangian Lω the odd vertical graded derivations(173)υL=E←a∂a=δ←Lδz¯a∂∂za,υ¯L=∂←aEa=∂←∂z¯aδLδza,(174)ϑL=υL+υ¯Ll=-1a+1δLδz¯a∂∂za+δLδza∂∂z¯a,such that ϑL(L′ω)={Lω,L′ω}.Theorem 37.
The following conditions are equivalent [2, 12].(i)
The antibracket of a LagrangianLω is dH-exact; that is,(175)Lω,Lω=2δ←Lδz¯aδLδzaω=dHσ.(ii)
The graded derivationϑL (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 DBGAP∞∗{N} (133), an original Lagrangian L obeys the master equation (175) and yields the graded derivations υL=0, υ¯L=δ¯ (173); that is, it is a trivial solution of the master equation. However, its extension Le (153) need not satisfy the master equation. Therefore, let us consider its extension(176)LE=Le+L′=L+L1+L2+⋯by means of even densities Li, i≥2, of zero antifield number and polynomial degree i in ghosts. Then the following is a corollary of Theorem 37.Corollary 38.
A LagrangianL is extended to a proper solution LE (176) of the master equation iff the gauge operator u (138) admits a nilpotent extension ϑE (174).However, one can say something more [2, 12].Theorem 39.
If the gauge operatoru (138) can be extended to the BRST operator b (155), then the master equation has a nontrivial proper solution(177)LE=Le+∑1≤k≤Nγrk-1c¯rk-1ω=L+b∑0≤k≤Ncrk-1c¯rk-1ω+dHσ,such that b=υE is the graded derivation defined by the Lagrangian LE (177).The LagrangianLE (177) is said to be the BRST extension of an original Lagrangian L.


## 10. Example: Topological BF Theory

We address the topological BF theory of two exterior formsA and B of form degree |A|+|B|=dim⁡X-1 on a smooth manifold X [20, 27]. It is reducible degenerate Lagrangian theory which satisfies homology regularity condition (Condition 1). Its dynamic variables A and B are sections of a fibre bundle (178)Y=⋀pT∗X⊕⋀qT∗X,p+q=n-1>1,coordinated by (xλ,Aμ1⋯μp,Bν1⋯νq). Without loss of generality, let q be even and q≥p. The corresponding differential graded algebra is O∞∗ (41).There are canonicalp- and q-forms (179)A=Aμ1⋯μpdxμ1∧⋯∧dxμp,B=Bν1⋯νqdxν1∧⋯∧dxνqon Y. A Lagrangian of topological BF theory reads(180)LBF=A∧dHB=ϵμ1⋯μnAμ1⋯μpdμp+1Bμp+2⋯μnω,where ϵ is the Levi–Civita symbol. It is a reduced first order Lagrangian. Its first order Euler–Lagrange operator(181)δL=EAμ1⋯μpdAμ1⋯μp∧ω+EBνp+2⋯νndBνp+2⋯νn∧ω,EAμ1⋯μp=ϵμ1⋯μndμp+1Bμp+2⋯μn,EBμp+2⋯μn=-ϵμ1⋯μndμp+1Aμ1⋯μpsatisfies the Noether identities(182)dμ1EAμ1⋯μp=0,dν1EBν1⋯νq=0.Given a family of vector bundles(183)Ek=⋀p-k-1T∗X×X⋀q-k-1T∗X,0≤k<p-1,Ek=R×X⋀q-pT∗X,k=p-1,Ek=⋀q-k-1T∗X,p-1<k<q-1,Eq-1=X×R,let us enlarge an original differential graded algebra O∞∗ to the BGDA P∞∗{q-1} (133) which is(184)P∞∗q-1=P∞∗VY¯⨁YE0⊕⋯⨁YEq-1⨁YE¯0⨁Y⋯⨁YE¯q-1;Y.It possesses a local generating basis(185)Aμ1⋯μp,Bν1⋯νq,εμ2⋯μp,…,εμp,ɛ,ξν2⋯νq,…,ξνq,ξ,A¯μ1⋯μp,B¯ν1⋯νq,ɛ¯μ2⋯μp,…,ɛ¯μp,ɛ¯,ξ¯ν2⋯νq,…,ξ¯νq,ξ¯of Grassmann parity(186)εμk⋯μp=ξνk⋯νq=k+1mod⁡2,ɛ=pmod⁡2,ξ=0,ɛ¯μk⋯μp=ξ¯νk⋯νq=kmod⁡2,ɛ¯=p+1mod⁡2,ξ¯=1,of ghost number(187)ghεμk⋯μp=ghξνk⋯νq=k,ghɛ=p+1,ghξ=q+1,and of antifield number(188)AntA¯μ1⋯μp=AntB¯νp+1⋯νq=1,Antɛ¯μk⋯μp=Antξ¯νk⋯νq=k+1,Antɛ¯=p,Antɛ¯=q.One can show that homology regularity condition (Condition1) holds ([20, Lemma 4.5.5]), and the DBGA P∞∗{q-1} (184) is endowed with the Koszul–Tate operator(189)δKT=∂←∂A¯μ1⋯μpEAμ1⋯μp+∂←∂B¯ν1⋯νqEBν1⋯νq+∑2≤k≤p∂←∂ɛ¯μk⋯μpΔAμk⋯μp+∂←∂ɛ¯dμpɛ¯μp+∑2≤k≤q∂←∂ξ¯νk⋯νqΔBνk⋯νq+∂←∂ξ¯dνqξ¯νq,ΔAμ2⋯μp=dμ1A¯μ1⋯μp,ΔAμk+1⋯μp=dμkɛ¯μkμk+1⋯μp,2≤k<p,ΔBν2⋯νq=dν1B¯ν1⋯νq,ΔBνk+1⋯νq=dνkξ¯νkνk+1⋯νq,2≤k<q.Its nilpotentness provides the complete Noether identities (182) and the (k-1)-stage ones (190)dμkΔAμk⋯μp=0,k=2,…,p,dνkΔBνk⋯νq=0,k=2,…,q.It follows that the topological BF theory is (q-1)-reducible.Applying inverse second Noether Theorem33, one obtains the gauge operator (139) which reads(191)u=dμ1εμ2⋯μp∂∂Aμ1μ2⋯μp+dν1ξν2⋯νq∂∂Bν1ν2⋯νq+dμ2εμ3⋯μp∂∂εμ2μ3⋯μp+⋯+dμpɛ∂∂εμp+dν2ξν3⋯νq∂∂ξν2ν3⋯νq+⋯+dνqξ∂∂ξνq.In particular, a gauge symmetry of the Lagrangian LBF (180) is (192)u=dμ1εμ2⋯μp∂∂Aμ1μ2⋯μp+dν1ξν2⋯νq∂∂Bν1ν2⋯νq.It also is readily observed that the gauge operatoru (191) is nilpotent. Thus, it is the BRST operator b=u. As a result, the Lagrangian LBF is extended to the proper solution of the master equation LE=Le (177) which reads(193)Le=LBF+εμ2⋯μpdμ1A¯μ1⋯μp+∑1<k<pεμk+1⋯μpdμkɛ¯μk⋯μp+ɛdμpɛ¯μp+ξν2⋯νqdν1B¯ν1⋯νq+∑1<k<qξνk+1⋯νqdνkξ¯νk⋯μq+ξdνqξ¯νq.



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