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S JHEP00(2002)000 JHEP00(2002)000 = 10 0 two d (1.1) (1.2) has the b z p a g in superspace . + a z 0 · · · d xterior differential + ∧ 1 b a z g 0 a rm d z with the coordinate b 0 g a d )= al in superspace with coordinates z p 1) . 0 ∧ a d b − ...a . z ( 1 z 0 a 0 0 a z d d φ b b +1) , ( -form z z b a b b +1 g p g b g g g a a a g g g +1)( )= 1) 1) a , g 1) g a – 1 – ( − − p z and satisfying the permutation relations − 0 1) a d ...a = ( = ( ( g 1 − = ( 0 a g b b b φ ≡ z z 1 z = ( a a ) a 0 z z b a z d 0 z z 0 d 0 ( d ∧ g d a z ∧ 0 a d z 0 ∧···∧ d a p g a z . In this case the symmetry property of an exterior product of 1) 0 a d − ( = 0 ∂/∂z Φ ≡ a to be equal to zero z ∂ a z ∂ a having Grassmann parities z 0 a There exist two possibilities to definez an exterior differenti 1. Introduction: two definitions of an exterior differential Contents 1. Introduction: two definitions of2. an exterior differential Poisson in brackets related superspace with the3. exterior differentials Linear Poisson 1 brackets related with4. semi-simple Lie Lie groups superalgebra for the5. BRST and anti-BRST Conclusion charges 6 3 8 d The first one realized when we set the Grassmann parity of the e Note that relation (1.1) can be rewritten in the following fo where form and a permutation relation of the exterior differential By defining an exterior product of a differential differentials is JHEP00(2002)000 1 (1.4) (1.3) (1.5) , p ...a 1 0 a , φ q b has to be z ...b ∂ b 1 q p 0 b b b z a and leads to the z g g ψ 0 p p d + + 1) ...a ∧ 1 . . − . · · · 0 a 1 · · · ) ) ( a a 0 0 φ z z + 1 1 + ordinate 1 0 Ψ Ψ a fferentials is defined as 1 0 0 d 1 d a z b ...a a e arises when the Grassmann d d ˜ 0 ∧ z g ential is adopted p ( ( g b 1 d 0 a z e exterior product of two forms d ∧ ∧ φ 1 ˜ b . ∧ 0 0 )= d z )= b . a b q p 0 ∂ z Φ z g ∧···∧ b 1 1 p Φ ...b p z d 1) d ...a +Φ 1 ∧···∧ a 0 1) b 1 0 as follows b . ∧ 0 z − d 1 a p z ψ 0 − ( 0 0 a b +1) φ Ψ ( b d g ∧ z ( g Ψ + 1 p 0 +1 p ∧ b + ( d a a pq g +1) is chosen to be equal to unit 1) ) 0 g z a a 0 ∧ +1)( 0 g g 1) a − Ψ a ( Φ z d 1 g , g − 0 b 1) -form Φ ∂ ( – 2 – , g ∧ q 1) )= d z a p = ( p − a ( 0 ) 1) z − ...b q 0 p z d = ( 1 1 ...a 1 − 0 d b Φ 1 0 1) = ( d ...a 0 1 a = ( ψ ( for the exterior multiplication. ∧···∧ 1 Ψ b − d = φ 1 g 0 a = ( b z b 1 ˜ 1 ∧ 1 z ∧ 1 φ a b z a a d 1 d 0 z z z 0 ∧···∧ a z 0 1 1 d ˜ ∧ ) = ( ) = ( 1 z Φ -form q d d d a 0 0 0 b d p ˜ ˜ z ∧ ∧ z d Ψ Ψ = 1 a 0 d z d 0 ∧ ∧ a 1 g Φ pq 0 d 0 ∧···∧ ˜ 0 ∧··· 1) p d 1) q (Φ (Φ b a ∧···∧ 0 0 − − z z ( d d 0 p 1 a d d z = ( 0 = = 0 d 0 1 Ψ ∧ Ψ Φ 0 ∧ d 0 Φ = -form 0 q Φ 0 Another definition for the exterior differential in superspac d In this case we use another notation 1 as which differs from (1.3) with the absence of the grading factor and a we obtain the following symmetry property of this product By setting the exterior differential of the following form of the Leibnitz rule parity of the exterior differential and a rule for the permutation of such a differential with the co Then the symmetry property of the exterior product for two di we obtain the Leibnitz rule for the exterior differential of th Note that very often another definition for the exterior differ If the exterior product of the Relation (1.4) can be represented in the form JHEP00(2002)000 1 . p 1 of (2.1) (1.6) , ...a 1 a , 1 a z -form Φ q = 0 φ p b ...b z ǫ 1 ∂ 1 b b ψ z p 1 d q b ...a ˜ ∧ g 1 1 1 a a + φ z 1 1 1 a in the following form d · · · z . ˜ ∧ 1 1 mann parity ) e of the above mentioned 1 d + or the coordinate and Φ ˜ ∧ 1 ever, to the different results Ψ b . 1 erior product of a r product (1.4) coincide with 0 g ˜ a ∧··· d lished as a result of the direct ties, result in the same exterior z p ( ) and by putting the following 1 a , ˜ ∧ ˜ B ∧··· -form Φ d z p 1 )= p b 1 p b . q a z d z Φ 1 ...a 1 z p p , written in arbitrary non-canonical 1 ∂ 1 → ...b d a Φ ǫ 1 a 1 g d -forms Φ 1) ) 1 b ˜ ∧ φ p ˜ + 2. Thus, we proved that two definitions ∧ z 1 − ψ k +1) ( 1 ··· 2 ( b b Ψ p/ a g + ab ǫ z g 1 + ( 1 pq ω a 2] d 1 g =1 P +1)( 1) a p/ ˜ k ∧ [ + z a Ψ p − g , g – 3 – ˜ ( ∂ ∧ ← 1) q ) 1) 1 − 1) A = ( ˜ ...b ∧··· − Φ − 1 q 1 1 1 b b = = ( d z Ψ ψ = ( ǫ 1 p 1 = ( ˜ ∧ } b d p 1 b ) z ...a q z 1 Φ 1 ...a b ) = ( 1 d 0 a g 1 1 d A, B 1 a ˜ φ ∧ + a Ψ { φ z ··· 1 ˜ ∧ 1 a + 1 d 1 z b ˜ 1 ∧··· g (Φ q d 1 b + ˜ ∧ q d z ( 1 p d 1) = ˜ will be ∧··· − p 1 1 a z Ψ = ( 1 d 1 a ˜ ∧ Ψ z 1 2] denotes a whole part of the quantity ˜ ∧ d -form Ψ 1 p/ q = Φ -form q A Poisson bracket, having a Grassmann parity The equivalence of the exterior calculi obtained with the us 1 Φ 1 d and is defined in the following way then the symmetry property of this product is Then the Leibnitz rule for the exterior differential of the ext and a Let us define in this case the exterior differential of the Note that in this casethe the ones symmetry for properties the of usual the Grassmann exterio product of two differentials f another one. variables different definitions for the exteriorverification differential by can taking be into estab relations account between relations coefficients (1.1) of and the (1.5 corresponding where [ calculus. 2. Poisson brackets related with theNow exterior we differentials show that applicationunder of construction these differentials from leads, a how given Poisson bracket with a Grass for the exterior differential, differed with the Grassmann pari JHEP00(2002)000 (2.4) (2.5) (2.2) (2.3) (2.6) G. i . b ζ ζ y + ∂ → ǫ ) ǫ  ab H ǫ a ζ ζ ω y c , d z ≡ a ∂ z , a  ζ ζ z c . ζ d this composition has the ζ + ζ ( y . By taking the exterior ǫ d . owing binary composition 1 ) , + a ζ ab ǫ ǫ ǫ d y = , ) def , ω ǫ , H ∂ ← , ) } = 0 ζ x or ǫ ab ǫ ǫ + , , d 0 = 0 H ǫ } G, F ω a ζ ǫ b ( d (mod 2) ba ǫ ) z z H B, A } d bc ǫ ζ ( b ω ( { ∂ , which correspond to a Hamiltonian → ) z ω (mod 2) b ) + ǫ a ǫ ζ d z ǫ ∂ (mod 2) = z z + def ∂ + ab + ǫ b B,C ab ǫ ∂ ǫ + B g G ω ) ) under it designates a summation over ab ǫ { ) g ω g ǫ ǫ ǫ )( , we can establish the symmetry of the ad b ǫ ω + a ζ ǫ )( )( z ) y ǫ ω H ζ A, = + + ǫ abc + ζ ) ζ B + { ab ∂ ǫ ǫ a + ← b d ǫ + g d ) ǫ g G A ( a g ǫ + } ω ( ( ) g g g c + ǫ ǫ ∂ ( + ( ∂ + – 4 – g F and their differentials + 1) ζ + C + g )( 1) a A ab g ǫ ( ǫ ,H a − a g 1) g a ( g ( )( F ω − + z ǫ 1) ζ ( g z a − − ≡ g + { ≡ ≡ − ( − 1) A ( ≡ ) ǫ = g  + ( ǫ = − 1) ] ( − = } ζ H ) ab ǫ ) − a ab ǫ ǫ b ǫ . 1) + b +( ( = ω z } c ǫ ) ω dt z − H ) ∂ b dz ζ  ζ ζ ( ζ y A, B ) d abc g + ab ǫ d X { ( ∂ ǫ ( → ( ( ω ) and A, B ∂ ∂ F, G g { X ab ǫ ) ABC = [( ( ω g ab ǫ of the variables a, b a F, G ζ a ( ω z dt ζ G dz + d ∂ ← ǫ h 1) is one of the exterior differentials , ) and a sum with a symbol ( = ( F and A ) from the Hamilton equations (2.5), we obtain ( ), = a F ǫ = 0 g ζ z ζ d ζ ζ dt + ≡ d ( ǫ ( )= ) ǫ d ζ A d g H ( F, G g The Hamilton equations for the phase variables ( ( ǫ H differential where can be represented in the form has the following main properties: cyclic permutations of As a result offor two functions last equations we have by definition the foll where which lead to the corresponding relations for the matrix Grassmann parity composition (2.6) In consequence of the grading properties (2.2) for the matri By using the symmetry property (2.3) of JHEP00(2002)000 (2.7) (2.8) tion of -forms and the p , we come to the ab ǫ G ω rties for the Poisson ent form and rise low i initial bracket (2.1) is . a by means of the matrix z ∂ a → ζ = 0 ζ y on the case of the brackets ζ isson bracket results in the + ǫ a . + n parities. The details of this y ǫ e matrix ) ∂ ← . ζ ) + ζ ǫ (see also [3, 6, 7, 8, 9, 10, 11]) onto -forms so that being taken between can be treated as = 0 ) + ǫ p ζ a 1 n with Lie bracket of vector fields see in the + + y a . The bracket (2.6) is a generalization ) ǫ g 3 F, G . Thus, the application of the exterior z ) ( , , ( ( ζ a ˜ C , related with g ba 2 ǫ C = 0 ζ + E, = 0 + ω 1) ζ ( ǫ a ζ ǫ ) ǫ + − y ..., + ζ ǫ } ( ǫ b ( 1)-form ) + y ǫ – 5 – − − + according to the rule (2.7) then we come to the ζ = G q + g C, ǫ a a ζ ...,C ab ǫ a y )( y ...,C + { z ζ ( ω ∂ → of the variables ∂ p + a a ζ ǫ z p y + ∂ ← E h = g ( in relation (2.7). ˜ F C 1) ǫ = − ( ζ ) + ǫ ) X EFG ( ) into degrees = 1, due to relations (1.4), (1.6), the terms in the decomposi a 1 F, G ( ζ -form results in a ( ,y q a z ( F We see that the composition (2.6) satisfies all the main prope Note that by transition to the variables In the case Let us also note that if we take the bracket (2.6) in the compon It follows from the structure of the bracket (2.6) that if the There is no summation over Concerning Poisson bracket between 1-forms and its relatio -form and a 2 3 p ab ǫ Jacobi identities for this composition At last, taking into account relations (2.3) and (2.4) for th bracket with the Grassmann parity equal to brackets of the different Grassmann parities. differentials of opposite Grassamann parities to the given Po ω book [1]. the Poisson bracket (2.6) takes a canonical form a function that can be proved with the use of the Jacobi identities (2.4) bracket (2.6) can be considered as a Poisson bracket on a of the bracket introduced(2.1) in with [2, arbitrary 3] Grassmann on parities. the superspace case and indexes with the use of the matrix the cases of superspace andgeneralization the will brackets be of given diverse in Grassman a separate paper [12]. degenerate and possessed of a Casimir function generalizations of the Schouten-Nijenhuis brackets [4, 5] then the bracket (2.6) has as Casimir functions this one and also a function of the form JHEP00(2002)000 (3.1) (3.2) (3.3) which γ αβ c form (3.3) takes a G, ) β (3.2) a nilpotent θ ups 0 ∂ → C γ θ γ . αβ n bracket given in terms c α 6) the following linear odd cture constants = 0 θ n to the linear even and odd e considered, this bracket has 2 ∂ ← G. ) , ) 1 s apart from B. α ˜ + C x ( β = 0 β x ∂ → x 0 α ∂ → } ∂ → , . Θ 0 γ . , γ , ∂ ← γ γ x x γ = 0 = 0 γ x γ βλ − γ 1 δ βα c αβ ) ) ...,C αβ c α λ c 1 βα α c λγ { Θ ˜ c − c x C α αγ α β λ ∂ → x θ c – 6 – = , α = ∂ ← ∂ ← αβ x , we obtain from the bracket (3.1) in conformity γ = ..., a c 1 αβ ∂ =Θ ( ← z ) A d + g ( αβ β αβ α c β = F g θ x θ , αβγ X ( α 0 ∂ → = } αβ x (here γ g 1 ) β x = α θ γ x 0 A, B α 4 { C αβ x F, G c ( = α x 1 ˜ ∂ ← C ( F are Grassmann variables. In this case relation (2.7) has the = α 1 ) x 1 d is an inverse tensor to the Cartan–Killing metric are also Grassmann variables in term of which the odd bracket F, G = ( β αβ α g θ Let us take as an initial Poisson bracket (2.1) the linear eve Lie-Poisson-Kirillov bracket. 4 Here we apply the procedurebrackets described connected in with the a previous semi-simple sectio Lie group having stru 3. Linear Poisson brackets related with semi-simple Lie gro In the case of aa semi-simple Casimir Lie function group, which hereafter will b where of the commuting variables obey the usual conditions By using the odd exterior differential with the transition from the bracket (2.1) to the bracket (2. bracket where canonical form where Θ According to (2.8) the bracket (3.3) has as Casimir function quantity JHEP00(2002)000 (3.6) (3.5) (3.7) (3.9) (3.8) (3.11) (3.10) , according 1 d G, ) β , x ∂ → = 0 (3.4) ) = 0 γ α 2 ) θ x 2 1 γ , ∆ type differential second − = αβ a t (3.9) to the even linear c = 0 ket introduced in Refs. [18, (∆ z , Casimir function α 2 α ) x θ 1 , ∂ ∂ ← ) α C θ α ( B, tion which obey the commutation + ∂ S β α β 2 1 θ , α θ S ∂ → Z . ∂ 2 1 → , , + γ γ γ γ β = 0 γ β − α ≡ (in this case θ Z S T x θ θ . T 1 γ γ γ γ ∂ ∂ γ α α } γ γ θ S 1 ]= αβ αβ αβ αβ θ αβ x 1 c c c c γ c γ ) + ] = 0 ] = ( , α 1 β α α αβ θ α αβ ) – 7 – x c ]= ]= ]= c , θ T ∂ ...,C ← ,S ( ∂ β β ← β α { α α = θ = = A θ T ( , T ,S + ,Z . With the help of the odd differential [ α 1 ∂ α α α α α , ) θ = β S T λγ , T S x Z + ∂ [ [ 1 g [ α ∂ 1 → } λ αβγ θ − ) c ( , γ 1 γ αβ θ α ) c θ γ , x β A, B ( α θ { = αβ α x α c x ( θ ∂ α α [ x αβγ θ = 1 2 c ∂ ∂ [ ← 1 ( − 2 1 C F − = and 1 = β − θ 0 ) ∆= ∆ αβ g F, G = ( satisfy the relations α θ α Z The odd bracket (3.3) has two nilpotent Batalin-Vilkovisky Now let us take as an initial bracket (2.1) the linear odd brac and where are generators of the Lie group inrelations the co-adjoint representa order ∆-operators [13, 14] (see also [15, 16, 17]) Note that which has in the case of the semi-simple Lie group a nilpotent and and 19] and given in terms of Grassmann variables where to the transition frombracket of (2.1) the to form (2.6), we come from the bracke JHEP00(2002)000 (4.1) (4.2) (4.3) (3.13) (3.10) (3.12) respec- 1 nilpotent C son bracket form akes a canonical . . = 0 = 0 2 2 o second order nilpotent ) pondingly (see, e.g., [21]) 1 T and anti-BRST charges utation relations Lie superalgebra for them (∆ ir functions besides , Q , ) G. α ) (3.13) and ∆ (3.5). These oper- , S , α = 0 α nstead of this it has two θ ) 1 2 Q S 0 α ∂ → ) α 0 Z + α θ ˜ α Θ C 2 1 α , Z ∂ , T ← , γ − ( θ , defined on the Grassmann algebra with + α γ ..., with the help of the linear even (3.11) and + 1 θ ( α − ]= α α βα ] = 0 T 0 c Θ ) α ,θ , ∆]=0 β ]= , ∂ → ,Q α , 0 – 8 – T α α ) α ( α x θ θ , αβγ T T 1 2 ( and ∆ c α =Θ ∂ ← α α α γ 1 θ ( x α = ( T θ T [ [ F x α β } + θ x = ∆ 0 α ) − x 0 , ) 0 Q, α ) = { x , ( 0 α ˜ α F, G C x ( θ ( [ as representations for the ghosts and antighosts operators α and ∆ satisfy the following anticommutation relation: 2 1 α θ [ θ − Q ∂ 2 1 = = . 1 are commuting variables. Relation (2.7) in this case has the and β Q ∆ x α α θ θ , have been introduced in connection with the linear odd Pois 1 αβ α d g θ are Grassmann variables in term of which the bracket (3.11) t = = β α α x x The even bracket (3.11), in contrast to the odd bracket, has n odd (3.3) Poisson brackets we constructed the operators ators can be treated asif we the consider BRST and anti-BRST charges corres tively. The operators (3.9), which corresponds tohas a been given. semi-simple These Lierespectively. operators group, are and standard the terms in the BRS 4. Lie superalgebra for the BRSTThus, and in the anti-BRST superspace charges with coordinates generators In the papers [18, 19] the operators ∆ where two terms in the right-hand side of which, because of the comm In accordance with (2.8) the even bracket (3.11) has as Casim form for the Martin bracket [20] where Θ differential ∆-like operators. It is surprising enough that i differential operators of the first order where the function and JHEP00(2002)000 . α K Z α (4.8) (4.4) (4.5) (4.6) (4.7) (4.10) (4.11) Z and determined contains the α α T α Z α α Z T α Z Z , ∆, and Q α with the usual quadratic T and ∆: α α erator and the quantity Z Q T the co-adjoint representation , s ulation of the BRST operator antities N . α δ . The quantity θ Z , ∆, ∂ α α γ Q T Z θ , ∂ K . , , , , ∆ (4.9) α λγδ and − Q, θ − c ] = 0 ∂ λ α β N α ] = 0 ] = 0 ] = 0 T ]= θ α αβ α α – 9 – ∆]=0 = ,Z c ,Q , T ∆] = α = β α α α θ T S N,Z N,Q N, T α Z Z α N N, [ [ [ [ [ α [ θ T S [ 1 2 = K for the semi-simple Lie group. α Z α Z has the following permutation relations with N which can be written as α S α S Note that the Lie superalgebra for the quantities The operator and commutes with the central elements has the form by the relations (3.8),cohomologies (4.1)–(4.11) [22]. can be used for the calc Casimir operator We can add the commutation relations (3.8) for the generator are central elements of the Lie superalgebra formed by the qu where term can be considered as a representation for the ghost number op The relations (4.1)–(4.6) remain(3.6) valid an if arbitrary we representation take for instead the of generators JHEP00(2002)000 , (1956) 338. 59 ons and its arts of the work have re naturally connected the help of these differ- and related also with the th a generalization of the huizen, W. Siegel and B. on brackets, corresponding e brackets of an arbitrary α sions and for hospitality at e parities of the both initial n parity of another one. We (1955) 390. erski, M. Tonin and J. Wess and given respectively on the linear Poisson brackets which ,θ pectively at the University of tials with opposite Grassmann γ α 58 x (1940) 449. αβ c 43 (1961) 59. , Prentice Hall, Inc. Englewood Cliffs, N.J. 74 – 10 – variables, we come with the help of the Grassmann- α x Non-linear Poisson brackets. Geometry and quantization Proc. Kon. Ned. Akad. Wet. Amsterdam A. (1955) 390. Ann. Math. 17 , Tambov, 1989; Moscow, Nauka, 1990. and commuting Proc. Nederl. Acad. Wetensh., ser. A. Proceeding of the Conference “Theory of group representati Proc. Kon. Ned. Akad. Wet. Amsterdam A. Indag. Math. Lectures on differential geometry α θ 1964. applications in physics” Moscow, Nauka, 1991. Then we introduced a prescription for the construction with It is also indicated that the resulting bracket is related wi By applying the prescription to the linear odd and even Poiss We demonstrated that these resulting even and odd brackets a [6] A. Nijenhuis, [5] A. Nijenhuis, [8] K. Kodaira and D.C. Spencer, [3] M.V. Karasev and V.P. Maslov, [2] M.V. Karasev, [7] A. Frohlicher and A. Nijenhuis, [4] J.A. Schouten, [1] S. Sternberg, parities give the same exterior calculus. entials from a given Poissonshowed bracket that of the the parity definite ofbracket Grassman and the exterior resulting differential bracket used. depends on th Schouten-Nijenhuis bracket on theGrassmann superspace parity. case and on th to a semi-simple Lie group with the structure constants 5. Conclusion Thus, we illustrated that in superspace the exterior differen anticommuting odd exterior differential to the correspondingly even and odd are both defined on the superspace with the coordinates same semi-simple Lie group. with the BRST and anti-BRST charges respectively. Acknowledgments One of the authorsfor (V.A.S.) is useful grateful discussions toZumino and A.P. Isaev, for to J. stimulating S.J. Luki discussionsMaryland, Gates, SUNY and Jr., (Stony for Brook) P. hospitalitybeen and Van res performed. LBNL Nieuwen (Berkeley) V.A.S. whereSISSA the thanks (Trieste) p where L. this Bonora work for has been fruitful completed. discus References JHEP00(2002)000 , Mod. , (1981) (1999) 155 Nucl. , B 102 B 451 (1978) 61. Commun. Math. The lecture given 111 , lism Deformation theory and Phys. Lett. Phys. Lett. n Grassmann algebra, odules, , , er, heoretical Methods in Physics” ic Methods in Theoretical Commun. Math. Phys. , (1992) 342. Ann. Phys. 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