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SUSY-EXTENDED FIELD THEORY

GENNADI SARDANASHVILY1 Department of Theoretical Physics, Physics Faculty, Moscow State University, 117234 Moscow, Russia

A field model on fibre bundles can be extended in a standard way to the BRS-invariant SUSY field model which possesses the Lie supergroup ISp(2) of symmetries.

1 Introduction

Generalizing the BRS mechanics of E.Gozzi and M.Reuter [1-4], we show that any field model (independently of its physical symmetries) can be extended in a standard way to a SUSY field model. In comparison with the SUSY field theory in Ref. [5,6], this extension is formulated in terms of simple graded manifolds. From the physical viewpoint, the SUSY-extended field theory may describe odd deviations of physical fields, e.g., of a Higgs field. We follow the conventional geometric formulation of field theory where classical fields are represented by sections of a smooth fibre bundle Y X coordinated by (xλ,yi). A first order Lagrangian L of field theory is defined as a horizontal→ density

λ i i 1 n 1 n L = (x ,y ,y )ω :J Y T∗X, ω = dx dx ,n=dimX, (1) L λ →∧ ∧···∧ on the first order jet manifold J 1Y Y of sections of Y X [7-9]. The jet manifold 1 λ i i → → J Y coordinated by (x ,y ,yλ) plays the role of a finite-dimensional configuration space of fields. The corresponding Euler–Lagrange equations take the coordinate form

δ =(∂ d ∂λ) =0,d=∂+yi∂+yi∂µ. (2) iL i− λ i L λ λ λi λµ i Every Lagrangian L (1) yields the Legendre map

1 λ λ λ L : J Y Π,piL=πi=∂i, →Y ◦ L of J 1Y to the Legendre bundleb b

n Π= T∗X V∗Y TX, (3) ∧ ⊗Y ⊗Y 1E-mail: [email protected]

1 where V ∗Y is the vertical cotangent bundle of Y X. The Legendre bundle Π is equipped λ i λ → with the holonomic coordinates (x ,y ,pi ). It is seen as a momentum phase space of covariant Hamiltonian field theory where canonical momenta correspond to derivatives of fields with respect to all space-time coordinates [8-10]. Hamiltonian dynamics on Π is phrased in terms of Hamiltonian forms

H = pλdyi ω (xλ,yi,pλ)ω, ω = ∂ ω. (4) i ∧ λ −H i λ λc The corresponding covariant Hamilton equations read

yi = ∂i ,pλ= ∂ . (5) λ λH λi − iH Note that, if X = R, covariant Hamiltonian formalism provides the adequate Hamil- tonian formulation of time-dependent mechanics [4, 11]. This fact enables us to extend the above-mentioned BRS mechanics of E.Gozzi and M.Reuter to field theory. A preliminary step toward the desired SUSY extension of field theory is its extension to field theory on the vertical tangent bundle VY of Y X, coordinated by (xλ,yi,y˙i) [9]. Coordinatesy ˙i describe linear deviations of fields. The→ configuration and phase spaces of field theory on VY are

J 1VY VJ1Y, (xλ,yi,yi,y˙i,y˙i), (6) ∼= λ λ Π VΠ, (xλ,yi,pλ,y˙i,p˙λ). (7) VY ∼= i i Due to the isomorphisms (6) – (7), the corresponding vertical prolongation of the Lagran- gian L (1) reads i i λ LV =(˙y∂i+˙yλ∂i)L, (8) while that of the Hamiltonian form H (4) is

H =(˙pλdyi + pλdy˙i) ω ω, =(˙yi∂ +˙pλ∂i) . (9) V i i ∧ λ −HV HV i i λ H The corresponding Euler–Lagrange and Hamilton equations describe the Jacobi fields of solutions of the Euler–Lagrange equations (2) and the Hamilton equations (5) of the initial field model on Y . The SUSY-extended field theory is formulated in terms of simple graded manifolds whose characteristic vector bundles are the vector bundles VJ1VY J1VY in Lagran- → gian formalism and V ΠVY ΠVY in Hamiltonian formalism [12]. The SUSY extension adds to (yi, y˙i) the odd variables→ (ci, ci). The corresponding SUSY extensions of the La- grangian L (1) and the Hamiltonian form H (4) are constructed in order to be invariant under the BRS transformation ∂ u = ci∂ + iy˙i . (10) Q i ∂ci

2 Moreover, let us consider fibre bundles Y X characterized by affine transition functions of bundle coordinates yi (they are not necessarily→ affine bundles). Almost all field models are of this type. In this case, the transition functions of holonomic coordinatesy ˙i on VY are independent of yi,and(ci)and(ci) have the same transition functions. Therefore, BRS-invariant Lagrangians and Hamiltonian forms of the SUSY-extended field theory are also invariant under the Lie supergroup ISp(2) with the generators uQ (10) and ∂ ∂ ∂ ∂ ∂ u = ci∂ iy˙i ,u=ci ,u=ci ,u=ci ci . (11) Q i − ∂ci K ∂ci K ∂ci C ∂ci − ∂ci Note that, since the SUSY-extended field theory, by construction, is BRS-invariant, one hopes that its quantization may be free from some divergences.

2 Technical preliminaries

Given a fibre bundle Z X coordinated by (xλ,zi), the k-order jet manifold J kZ is → λ i endowed with the adapted coordinates (x ,zΛ), 0 Λ k, where Λ is a symmetric multi-index (λ ...λ ), Λ = m. Recall the canonical≤| morphism|≤ m 1 | | λ i 1 λ = dx (∂λ + zλ∂i):J Z, T∗X TZ. ⊗ →Y ⊗Z Exterior forms φ on a jet manifold J kZ, k =0,1,..., are naturally identified with their pull-backs onto J k+1Z. There is the exterior algebra homomorphism h : φ dxλ + φ dzi (φ + φΛyi )dxλ, (12) 0 λ i 7→ λ i λ called the horizontal projection, which sends exterior forms on J kZ onto the horizontal forms on J k+1Z X. Recall the operator of the total derivative → d = ∂ + zi ∂ + zi ∂µ + . λ λ λ i λµ i ··· A connection on a fibre bundle Z X is regarded as a global section → Γ=dxλ (∂ +Γi∂) ⊗ λ λ i of the affine jet bundle J 1Z Z. One says that a section s : X Z is an integral section of Γ if Γ s = J 1s,where→ J1sis the jet prolongation of s to→ a section of the jet bundle J 1Z X◦. Given the→ vertical tangent bundle VZ of a fibre bundle Z X, we will use the notation → ∂ ∂˙ = ,∂=˙zi∂. i ∂z˙i V i

3 3 The covariant Hamiltonian field theory

Let us summarize briefly the basics of the covariant Hamiltonian field theory on the Legendre bundle Π (3) [8-10]. Let Y X be a fibre bundle and L a Lagrangian (1) on J 1Y . The associated Poincar´e–Cartan→ form

H = πλdyi ω (πλyi )ω L i ∧ λ − i λ −L is defined as the horizontal Lepagean equivalent of L on J 1Y Y , i.e. L = h (H ), → 0 L where h0 is the horizontal projection (12). Every Poincar´e–Cartan form HL yields the bundle morphism of J 1Y to the homogeneous Legendre bundle

1? n 1 Z = J Y = T ∗Y ( − T ∗X) (13) Y ∧ ∧ λ i λ equipped with the holonomic coordinates (x ,y ,pi ,p). Because of the canonical isomor- n 1 − phism Π = V ∗Y ( T ∗X), we have the 1-dimensional affine bundle ∼ ∧Y ∧ π : Z Π. (14) ZΠ Y →

The homogeneous Legendre bundle ZY is provided with the canonical exterior n-form

λ i ΞY = pω + pi dy ωλ. (15)

Then a Hamiltonian form H (4) on the Legendre bundle Π is defined in an intrinsic way as the pull-back H = h∗ΞY of this canonical form by some section h of the fibre bundle (14). It is readily observed that the Hamiltonian form H (4) is the Poincar´e–Cartan form of the Lagrangian L = h (H)=(pλyi )ω (16) H 0 i λ−H on the jet manifold J 1Π, and the Hamilton equations (4) coincide with the Euler–Lagrange equations for this Lagrangian. Note that the Lagrangian LH plays a prominent role in the path integral formulation of Hamiltonian mechanics and field theory. The Legendre bundle Π (3) is provided with the canonical polysymplectic form

Ω = dpλ dyi ω ∂ . (17) Y i ∧ ∧ ⊗ λ Given a Hamiltonian form H (4), a connection

γ = dxλ (∂ + γi ∂ + γµ ∂i ) (18) ⊗ λ λ i λi µ 4 on Π X is called a Hamiltonian connection for H if it obeys the condition → γ Ω = dH, c Y γi = ∂i ,γλ= ∂ . (19) λ λH λi − iH Every integral section r : X Π of the Hamiltonian connection γ (19) is a solution of the Hamilton equations (5). Any→ Hamiltonian form H admits a Hamiltonian connection γH (19). A Hamiltonian form H (4) defines the Hamiltonian map

γH 1 1 i i H :Π J Π J Y, yλ = ∂λ , → →Y H which is the same for all Hamiltonianc connections associated with H. In the case of hyperregular Lagrangians, Lagrangian and covariant Hamiltonian for- malisms are equivalent. For any hyperregular Lagrangian L, there exists a unique Ha- miltonian form H such that the Euler–Lagrange equations (2) for L are equivalent to the Hamilton equations (5) for H. The case of degenerate Lagrangians is more intricate. One can state certain relations between solutions of Euler–Lagrange and Hamilton equations if a Hamiltonian form H satisfies the relations L H L = L, (20a) ◦ ◦ pλ∂i = H (20b) bi λcH−Hb b L◦ [8-10]. It is called associated with a Lagrangian L. If thec relation (20b) takes place on the Lagrangian constraint space L(J 1Y ) Π, one says that H is weakly associated with L. We will show that, if L and H are associated,⊂ then their vertical and SUSY prolongations are weakly associated. b

4 The vertical extension of field theory

As was mentioned above, the vertical extension of field theory on a fibre bundle Y X to the vertical tangent bundle VY X describes linear deviations of fields. → Let L be a Lagrangian on the configuration→ space J 1Y . Due to the isomorphism (6), its prolongation (8) onto the vertical configuration space J 1VY can be defined as the vertical tangent morphism

1 n L =pr VL:VJ Y T∗X, (21) V 2◦ →∧ = ∂ =(˙yi∂ +˙yi∂λ) , LV V L i λ i L to the morphism L (1). The corresponding Euler–Lagrange equations read δ˙ = δ =0, (22a) iLV iL δ = ∂ δ =0,∂=˙yi∂+˙yi∂λ+˙yi ∂µλ. (22b) iLV V iL V i λ i µλ i 5 The equations (22a) are exactly the Euler–Lagrange equations (2) for the Lagrangian L. In order to clarify the physical meaning of the equations (22b), let us suppose that Y X is a vector bundle. Given a solution s of the Euler–Lagrange equations (2), let δs→be a Jacobi field, i.e., s + εδs is also a solution of the same Euler–Lagrange equations modulo terms of order > 1 in a parameter ε. Then it is readily observed that the Jacobi field δs satisfies the Euler–Lagrange equations (22b). The Lagrangian (21) yields the vertical Legendre map

1 LV = V L : VJ Y VΠ, (23) VY→ pbλ = ∂˙λ b = πλ, p˙λ = ∂ πλ. (24) i i LV i i V i Due to the isomorphism (7), the vertical tangent bundle V ΠofΠ Xplays the role of the momentum phase space of field theory on VY, where the canonical→ conjugate pairs i λ i λ are (y , p˙i )and(˙y,pi ). In particular, V Π is endowed with the canonical polysymplectic form (17) which reads

Ω =∂ Ω =[dp˙λ dyi + dpλ dy˙i] ω ∂ . (25) VY V Y i ∧ i ∧ ∧ ⊗ λ

Let ZVY be the homogeneous Legendre bundle (13) over VY with the corresponding λ i i λ λ coordinates (x ,y ,y˙ ,pi ,qi ,p). It can be endowed with the canonical form ΞVY (15). Sections of the affine bundle Z VΠ, (26) VY → by definition, provide Hamiltonian forms on V Π. Let us consider the following particular case of these forms which are related to those on the Legendre bundle Π. Due to the fibre bundle

ζ : VZ Z , (27) Y → VY (xλ,yi,y˙i,pλ,qλ,p) ζ =(xλ,yi,y˙i,p˙λ,pλ,p˙), i i ◦ i i the vertical tangent bundle VZ of Z X is provided with the exterior form Y Y → λ i λ i Ξ = ζ∗Ξ =˙pω +(˙p dy + p dy˙ ) ω . V VY i i ∧ λ Given the affine bundle Z Π (14), we have the fibre bundle Y → Vπ :VZ VΠ, (28) ZΠ Y → where VπZΠ is the vertical tangent map to πZΠ. The fibre bundles (26), (27) and (28) form the commutative diagram. Let h be a section of the affine bundle Z Πand Y → H=h∗Ξ the corresponding Hamiltonian form (4) on Π. Then the section Vh of the

6 fibre bundle (28) and the corresponding section ζ Vh of the affine bundle (26) defines ◦ the Hamiltonian form HV =(Vh)∗ΞV (9) on V Π. One can think of this form as being a vertical extension of H. In particular, let γ (18) be a Hamiltonian connection on Π for the Hamiltonian form H. Then its vertical prolongation

Vγ=γ+dxµ [∂ γi ∂˙ + ∂ γλ ∂˙i ]. ⊗ V µ i V µi λ on V Π X is a Hamiltonian connection for the vertical Hamiltonian form H (9) with → V respect to the polysymplectic form ΩVY (25). 1 The Hamiltonian form HV (9) defines the Lagrangian LHV (16) on J V Π, which takes the form

L = h (H )=[˙pλ(yi ∂i ) y˙i(pλ + ∂ )+d (pλy˙i)]ω. HV 0 V i λ− λH − λi iH λ i The corresponding Hamilton equations contain the Hamilton equations (5) and the equa- tions

y˙i = ∂i = ∂ ∂i , p˙λ = ∂ = ∂ ∂ λ λHV V λH λi − iHV − V iH i i λ λ for Jacobi fields δy = εy˙ , δpi = εp˙i . The Hamiltonian form HV (9) on V Π yields the vertical Hamiltonian map

1 HV = V H : V Π VJ Y, (29) VY→ yci = ∂˙i c = ∂i , y˙i = ∂ ∂i . (30) λ λHV λH λ V λH

Let H be associated with a Lagrangian L.ThenHV is weakly associated with LV . Indeed, if the morphisms H and L obey the relation (20a), then the corresponding vertical tangent morphisms satisfy the relation c b V L V H V L = V L. ◦ ◦

The condition (20b) for HV reducesb to thec equalityb b

∂ (p)= (∂ H)(p),pL(J1Y), iH − iL◦ ∈ which is fulfilled if H is associated with Lc[8-10]. b

5 Geometry of simple graded manifolds

Following the BRS extension of Hamiltonian mechanics, we formulate the SUSY-extended field theory in terms of simple graded manifolds.

7 Let E Z be a vector bundle with an m-dimensional typical fibre V and E∗ Z the dual of→E. Let us consider the exterior bundle → m k E∗ = R ( E∗) (31) ∧ ⊕Z k⊕=1 ∧ whose typical fibre is the finite Grassmann algebra V ∗.By ismeantthesheafofits ∧ AE sections. The pair (Z, E) is a graded manifold with the body manifold Z coordinated by A A (z ) and the structure sheaf E [13-15]. We agree to call it a simple graded manifold with the characteristic vector bundleA E. This is not the terminology of [16] where this term is applied to all graded manifolds of finite rank in connection with Batchelor’s theorem. In accordance with this theorem, any graded manifold is isomorphic to a simple graded manifold though this isomorphism is not canonical [14, 17]. To keep the structure of a simple graded manifold, we will restrict transformations of (Z, E) to those induced by bundle automorphisms of E Z. A Global sections of the exterior→ bundle (31) are called graded functions. They make up a the Z2-graded ring E(Z). Let c be a basis for E∗ Z with respect to some bundle A {a } a a b → A i atlas with transition functions ρb , i.e., c0 = ρb (z)c . We will call (z ,c) the local basis for the simple graded manifold{ (Z,} ). With respect to this basis, graded functions read AE m 1 f = f ca1 cak, (32) k! a1...ak ··· kX=0 where fa1 ak are local functions on Z, and we omit the symbol of the exterior product of elements ···c. The coordinate transformation law of graded functions (32) is obvious. We will use the notation [.] of the Grassmann parity. Given a simple graded manifold (Z, ), by the sheaf Der of graded derivations of AE AE E is meant a subsheaf of endomorphisms of E such that any its section u over an open subsetA U Z is a graded derivation of the gradedA algebra (U)oflocalsectionsofthe ⊂ AE exterior bundle E∗ , i.e., ∧ |U [u][f] u(ff0)=u(f)f0+( 1) fu(f0) − for the homogeneous elements u (Der )(U) and f,f0 (U). Graded derivations ∈ AE ∈AE are called graded vector fields on a graded manifold (Z, E) (or simply on Z if there is no danger of confusion). They can be represented by sectionsA of some vector bundle as follows. Due to the canonical splitting VE ∼= E E, the vertical tangent bundle VE E can be provided with the fibre basis ∂/∂ca× dual of ca . This is the fibre basis→ for pr VE E. Then a graded vector field{ on a} trivialization{ } domain U reads 2 ∼= ∂ u = uA∂ + ua , A ∂ca

8 where uA,ua are local graded functions. It yields a derivation of (U)bytherule AE ∂ u(f ca cb)=uA∂ (f )ca cb +udf (ca cb). (33) a...b ··· A a...b ··· a...b ∂cd c ··· This rule implies the corresponding coordinate transformation law

A A a aj A aj u0 = u ,u0=ρju+u∂A(ρj)c of graded vector fields. It follows that graded vector fields on Z can be represented by sections of the vector bundle Z which is locally isomorphic to the vector bundle VE →

E U E∗ (pr2VE TZ) U, V | ≈∧ ⊗Z ⊕Z | and has the transition functions

A 1a1 1ak A z0 = ρ− ρ− z , i1...ik i1 ··· ik a1...ak k! i 1b1 1bk i j A i v0 = ρ− ρ− ρ v + z ∂ ρ j1...jk j1 jk j b1...bk b1...bk 1 A bk ··· " (k 1)! − # − of the bundle coordinates (zA ,vi ), k =0,...,m. These transition functions fulfill a1...ak b1...bk the cocycle relations. Graded vector fields on Z constitute a Lie superalgebra with respect to the bracket

[u][u ]+1 [u, u0]=uu0 +( 1) 0 u0u. − There is the exact sequence over Z of vector bundles

0 E∗ pr2VE E E∗ TZ 0. →∧ ⊗Z →V →∧ ⊗Z → Its splitting ∂ γ :˙zA∂ z˙A(∂ + γa ) (34) A7→ A A ∂ca is represented by a section e e ∂ γ = dzA (∂ + γa ) (35) ⊗ A A ∂ca of the vector bundle T ∗Z E e Z such that thee composition ⊗Z V →

γ Z T ∗Z E T ∗Z ( E∗ TZ) T∗Z TZ → ⊗Z V → ⊗Z ∧ ⊗Z → ⊗Z e 9 A is the canonical form dz ∂A on Z. The splitting (34) transforms every vector field τ on Z into a graded vector⊗ field ∂ τ = τ A∂ = τ A(∂ + γa ), (36) A 7→ ∇τ A A ∂ca which is a graded derivation of satisfying the Leibnize rule AE (sf)=(τ ds)f + s (f),f (U),sC∞(Z), U Z. ∇τ c ∇τ ∈AE ∈ ∀⊂ Therefore, one can think of the graded derivation (36) and, consequently, of the ∇τ splitting (34) as being a graded connection on the simple graded manifold (Z, E). In particular, this connection provides the corresponding decomposition A ∂ ∂ ∂ u = uA∂ + ua = uA(∂ + γa )+(ua uAγa) A ∂ca A A ∂ca − A ∂ca of graded vector fields on Z. Note that this notione of a gradede connection differs from that of connections on graded fibre bundles in Ref. [18]. In accordance with the well-known theorem on a splitting of an exact sequence of vector bundles, graded connections always exist. For instance, every linear connection

γ = dzA (∂ + γ a vb∂ ) ⊗ A A b a on the vector bundle E Z yields the graded connection → ∂ γ = dzA (∂ + γ a cb ) (37) S ⊗ A A b ∂ca such that, for any vector field τ on Z and any graded function f, the graded derivation τ (f) with respect to the connection (37) is exactly the covariant derivative of f relative to∇ the connection γ. Let now Z X be a fibre bundle coordinated by (xλ,zi), and let → γ =Γ+γ a vbdxλ ∂ λ b ⊗ a be a connection on E X which is a linear morphism over a connection Γ on Z X. Then we have the bundle→ monomorphism →

λ λ i a b ∂ γS : E∗ TX u ∂λ u (∂λ +Γλ∂i +γλ bc ) E ∧ ⊗Z 3 7→ ∂ca ∈V over Z, called a composite graded connection on Z X. It is represented by a section → ∂ γ =Γ+γ a cbdxλ (38) S λ b ⊗ ∂ca

10 of the fibre bundle T ∗X E Z such that the composition ⊗Z V →

γS Z T ∗X E T ∗X ( E∗ TZ) T∗X TX → ⊗Z V → ⊗Z ∧ ⊗Z → ⊗Z is the pull-back onto Z of the canonical form dxλ ∂ on X. ⊗ λ Given a graded manifold (Z, E), the dual of the sheaf Der E is the sheaf Der∗ E generated by the -module morphismsA A A AE φ :Der( (U)) (U). (39) AE →AE One can think of its sections as being graded exterior 1-forms on the graded manifold (Z, ). They are represented by sections of the vector bundle ∗ Z which is the A VE → E∗-dual of . This vector bundle is locally isomorphic to the vector bundle ∧ VE

E∗ U E∗ (pr2VE∗ T∗Z) U, V | ≈∧ ⊗Z ⊕Z | and has the transition functions

1a1 1ak 1a v0 = ρ− ρ− ρ− va ...a a, j1...jkj j1 ··· jk j 1 k k! 1b1 1bk j z0 = ρ− ρ− z + v ∂ ρ i1...ikA i1 ik b1...bkA b1...bk 1j A b ··· " (k 1)! − k # − of the bundle coordinates (za1...akA,vb1...bkj), k =0,...,m, with respect to the dual bases A b dz for T ∗Z and dc for pr V ∗E = E∗. Graded exterior 1-forms read { } { } 2 A a φ = φAdz + φadc . They have the coordinate transformation law

1b 1b a j φa0 = ρ− aφb,φA0=φA+ρ−a∂A(ρj)φbc. Then the morphism (39) can be seen as the interior product

u φ = uAφ +( 1)[φa]uaφ . (40) c A − a There is the exact sequence

0 E∗ T∗Z E∗ E∗ pr2VE∗ 0 →∧ ⊗Z →V →∧ ⊗Z → of vector bundles. Any graded connection γ (35) yields the splitting of this exact sequence, and defines the corresponding decomposition of graded 1-forms e φ = φ dzA + φ dca =(φ +φ γa)dzA + φ (dca γa dzA). A a A a A a − A 11 e e k Graded k-forms φ are defined as sections of the graded exterior bundle E∗ such that ∧Z V φ σ +[φ][σ] φ σ =( 1)| || | σ φ, ∧ − ∧ where . denotes the form degree. The interior product (40) is extended to higher degree graded| forms| by the rule

φ +[φ][u] u (φ σ)=(u φ) σ+( 1)| | φ (u σ). c ∧ c ∧ − ∧ c The graded exterior differential d of graded functions is introduced in accordance with the condition u df = u ( f ) for an arbitrary graded vector field u, and is extended uniquely to higher degreec graded forms by the rules

φ d(φ σ)=(dφ) σ +( 1)| |φ (dσ),dd=0. ∧ ∧ − ∧ ◦ It takes the coordinate form ∂ dφ = dzA ∂ (φ)+dca (φ), ∧ A ∧ ∂ca a where the left derivatives ∂A, ∂/∂c act on the coefficients of graded exterior forms by the rule (33), and they are graded commutative with the forms dzA, dca. The Lie derivative of a graded exterior form φ along a graded vector field u is given by the familiar formula L φ = u dφ + d(u φ). u c c Let E Z and E0 Z0 be vector bundles and Φ : E E0 a linear bundle morphism → → → over a morphism ζ : Z Z0. Then every section s∗ of the dual bundle E0∗ Z0 defines → → the pull-back section Φ∗s∗ of the dual bundle E∗ Z by the law → v Φ∗s∗(z)=Φ(v) s∗(ζ(z)), v E . z c z c ∀ z ∈ z It follows that a linear bundle morphism Φ yields a morphism

SΦ:(Z, ) (Z0, ) (41) AE → AE0 of simple graded manifolds seen as locally ringed spaces [14]. This is the pair (ζ,ζ Φ∗) of the morphism ζ of the body manifolds and the composition of the pull-back ∗ ◦f AE0 3 7→ Φ∗f E of graded functions and the direct image ζ of the sheaf E onto Z0. With ∈A A a A a ∗ A respect to local bases (z ,c )and(z0 ,c0 )for(Z, E)and(Z0, E0), the morphism (41) reads A A

a a b SΦ(z)=ζ(z),SΦ(c0 )=Φb(z)c.

Accordingly, the pull-back onto Z of graded exterior forms on Z0 is defined.

12 6 SUSY-extended Lagrangian formalism

The SUSY-extended field theory is constructed as the BRS-generalization of the vertical extension of field theory on the fibre bundle VY X in Section 4. Let us consider the vertical tangent bundle VVY→ VY of VY X and the simple → → graded manifold (VY, VVY) whose body manifold is VY and the characteristic vector bundle is VVY VY.A Its local basis is (xλ,yi,y˙i,ci,ci)where ci,ci is the fibre basis for → ˙ { } V ∗VY, dual of the holonomic fibre basis ∂i, ∂i for VVY VY. Graded vector fields and graded exterior 1-forms are introduced{ on VY} as sections→ of the vector bundles VVVY and VVY∗ , respectively. Let us complexify these bundles as C VVY and C VVY∗ .By V ⊗X V ⊗X V the BRS operator on graded functions on VY is meant the complex graded vector field 2 uQ (10). It satisfies the nilpotency rule uQ =0. The configuration space of the SUSY-extended field theory is the simple graded ma- 1 nifold (VJ Y, VVJ1Y) whose characteristic vector bundle is the vertical tangent bun- 1 A 1 1 λ i i i i i i dle VVJ Y VJ Y of VJ Y X. Its local basis is (x ,y ,yλ,c,c,cλ,cλ), where i i i i → → 1 ˙ λ ˙λ c , c ,cλ,cλ is the fibre basis for V ∗VJ Y dual of the holonomic fibre basis ∂i, ∂i,∂i ,∂y { 1 } 1 1 1 { } for VVJ Y VJ Y. The affine fibration π0 : VJ Y VY and the corresponding ver- → 1 1 → tical tangent morphism Vπ0 :VVJ Y VVY yields the associated morphism of graded 1 → manifolds (VJ Y, VVJ1Y) (VY, VVY) (41). Let us introduceA the operator→ ofA the total derivative

∂ ∂ d = ∂ + yi ∂ +˙yi∂˙+ci + ci . λ λ λ i λ i λ∂ci λ ∂ci

i i With this operator, the coordinate transformation laws of cλ and cλ read

i i i i cλ0 = dλc0 , cλ0 = dλc0 . (42)

i i i i Then one can treat cλ and cλ as the jets of c and c . Note that this is not the notion of jets of graded bundles in [19]. The transformation laws (42) show that the BRS operator uQ (10) on VY can give rise to the complex graded vector field

i λ i ∂ JuQ = uQ + cλ∂i + iy˙λ i (43) ∂cλ

1 2 on the VJ Y. It satisfies the nilpotency rule (JuQ) =0. In a similar way, the simple graded manifold with the characteristic vector bundle VVJkY VJkY can be defined. Its local basis is the collection → (xλ,yi,yi ,ci,ci,ci ,ci ), 0< Λ k. Λ Λ Λ | |≤ 13 Let us introduce the operators

∂ = ci∂ + ci ∂λ + ci ∂µ + ,∂=ci∂+ci∂λ+ci∂µ + , c i λ i λµ i ··· c i λi λµ i ··· ∂ ∂ d =∂ +yi∂ +ci + ci + . λ λ λ i λ∂ci λ ∂ci ··· It is easily verified that

dλ∂c = ∂cdλ,dλ∂c=∂cdλ. (44) As in the BRS mechanics [1-4], the main criterion of the SUSY extension of Lagran- gian formalism is its invariance under the BRS transformation (43). The BRS-invariant extension of the vertical Lagrangian LV (21) is the graded n-form

L = L + i∂ ∂ ω (45) S V c cL such that LJuQLS = 0. The corresponding Euler–Lagrange equations are defined as the kernel of the Euler–Lagrange operator δ δ =(dyiδ + dy˙iδ˙ + dci + dci ) ω. ELS i i δci δci LS ∧ They read

δ˙ = δ =0, (46a) iLS iL δ = δ + i∂ ∂ δ =0, (46b) iLS iLV c c iL δ = i∂ δ =0, (46c) δci LS − c iL δ = i∂ δ =0, (46d) δci LS c iL where the relations (44) are used. The equations (46a) are the Euler–Lagrange equations for the initial Lagrangian L, while (46b) - (46d) can be seen as the equations for a Jacobi field δyi = εci + ciε + iεεy˙i modulo terms of order > 2 in the odd parameters ε and ε.

7 SUSY-extended Hamiltonian formalism

A momentum phase space of the SUSY-extended field theory is the complexified simple graded manifold (V Π, VVΠ) whose characteristic vector bundle is VVΠ VΠ [12]. Its local basis is A →

λ i λ i λ i i λ λ (x ,y ,pi ,y˙ ,p˙i ,c,c,ci ,ci ),

14 λ λ λ λ where ci and ci have the same transformation laws as pi andp ˙i , respectively. The corresponding graded vector fields and graded 1-forms are introduced on V Π as sections of the vector bundles C VVΠ and C VV∗ Π, respectively. ⊗X V ⊗X V λ λ In accordance with the above mentioned transformation laws of ci and ci , the BRS operator uQ (10) on VY can give rise to the complex graded vector field

i ∂ λ ∂ uQ = ∂c + iy˙ i + ip˙i λ (47) ∂c ∂ci e on V Π. The BRS-invariant extension of the polysymplectic form ΩVY (25) on V Πisthe TX-valued graded form

Ω =[dp˙λ dyi + dpλ dy˙i + i(dcλ dci dci dcλ)] ω ∂ , S i ∧ i ∧ i ∧ − ∧ i ∧ ⊗ λ i λ i λ where (c , ici )and(c,ici ) are the conjugate pairs. Let γ be a Hamiltonian connection for a Hamiltonian− form H on Π. Its double vertical prolongation VVγ on VVΠ X is a linear morphism over the vertical connection Vγ on V Π X, and so defines→ the composite graded connection →

∂ ∂ ∂ ∂ (VVγ) =Vγ+dxµ [gi + gλ + gi + gλ ] S µ i µi λ µ i µi λ ⊗ ∂c ∂ci ∂c ∂ci (38) on V Π X, whose components g and g are given by the expressions → i i λ i i λ gλ = ∂c∂λ , gλi = ∂c∂i ,gλ=∂c∂λ ,gλi = ∂c∂i , i H λ i − Hi λi H − H ∂c = c ∂i + ci ∂λ,∂c=c∂i+ci∂λ.

This composite graded connection satisfies the relation

(VVγ) Ω = dH , Sc S − S and can be regarded as a Hamiltonian graded connection for the Hamiltonian graded form

H =[˙pλdyi + pλdy˙i + i(cλdci + dcicλ)]ω ω, (48) S i i i i λ −HS =(∂ +i∂ ∂ ) , HS V c c H on V Π. It is readily observed that this graded form is BRS-invariant, i.e., L H =0. uQ S Thus, it is the desired SUSY extension of the Hamiltonian form H.

The Hamiltonian graded form HS (48) defines the corresponding SUSY extensione of the Lagrangian L (16) as follows. The fibration J 1V Π V Π yields the pull-back of the H → 15 1 Hamiltonian graded form HS (48) onto J V Π. Let us consider the graded generalization of the operator h0 (12) such that h : dci ci dxµ,dcλ cλ dxµ. 0 7→ µ i7→ µi Then the graded horizontal density

L = h (H )=(L ) =L +i[(cλci + ci cλ) ∂ ∂ ]ω = L + (49) SH 0 S H S HV i λ λ i − c cH HV i[cλ(ci ∂ ∂i )+(ci ∂∂i )cλ +cλcµ∂i∂j cicj∂∂ ]ω i λ − c λH λ − c λH i i j λ µH− i jH on J 1V Π X is the SUSY extension (45) of the Lagrangian L (16). The Euler– → H Lagrange equations for LSH coincide with the Hamilton equations for HS,andread yi =∂˙i =∂i ,pλ= ∂˙ = ∂ , (50a) λ λHS λH λi − iHS − iH y˙i = ∂i =(∂ +i∂ ∂ )∂i , p˙λ = ∂ = (∂ + i∂ ∂ )∂ , (50b) λ λH V c c λH λi − iHS − V c c iH i ∂ S i λ ∂ S cλ = i Hλ = ∂c∂λ ,cλi = i Hi = ∂c∂i , (50c) ∂ci − H ∂c − H i ∂ S i λ ∂ S cλ = i Hλ = ∂c∂λ , cλi = i Hi = ∂c∂i . (50d) − ∂ci − H − ∂c − H The equations (50a) are the Hamilton equations for the initial Hamiltonian form H, while (50b) – (50d) describe the Jacobi fields

i i i i λ λ λ λ δy = εc + c ε + iεεy˙ ,δpi=εci + ci ε + iεεp˙i . Let us study the relationship between SUSY-extended Lagrangian and Hamiltonian 1 formalisms. Given a Lagrangian L on J Y , the vertical Legendre map LV (23) yields the corresponding morphism (41) of graded manifolds b 1 SL :(VJ Y, 1 ) (VΠ, ) V AVVJ Y → AVVΠ which is given by the relationsb (24) and

λ λ λ λ ci = ∂cπi , ci = ∂cπi .

Let H be a Hamiltonian form on Π. The vertical Hamiltonian map HV (29) yields the morphism of graded manifolds c 1 SH :(VΠ, ) (VJ Y, 1 ) V AVVΠ → AVVJ Y given by the relations (30)c and

ci = ∂ ∂i , ci = ∂ ∂i . λ c λH λ c λH 16 If a Hamiltonian form H is associated with L, a direct computation shows that the Hamiltonian graded form HS (48) is weakly associated with the Lagrangian LS (45), i.e.,

SL SH SL = SL , V ◦ V ◦ V V λ i λ˙i λ ∂ λ ∂ LSb SHVc=(pib∂λ+˙pib∂λ+ci λ + ci λ ) S S, ◦ ∂ci ∂ci H −H c where the second equality takes place at points of the Lagrangian constraint space L(J 1Y ).

b 8 The BRS-invariance

Now turn to the above mentioned case of fibre bundles Y X with affine transition functions. Since transition functions of the holonomic coordinates→ y ˙i on VY are indepen- i ˙ dent of y , the transformation laws of the frames ∂i and ∂i are the same, and so are the transformations laws of the coframes ci and{ }ci . Then{ } the graded vector fields (11) are globally defined on VY. The graded{ } vector{ fields} (10) and (11) constitute the above-mentioned Lie superalgebra of the supergroup ISp(2):

[uQ,uQ]=[uQ,uQ]=[uQ,uQ]=[uK,uQ]=[uK,uQ]=0,

[uK,uQ]=uQ, [uK,uQ]=uQ, [uK,uK]=uC, (51) [u ,u ]=2u , [u ,u ]= 2u . C K K C K − K Similarly to (43), let us consider the jet prolongation of the graded vector fields (11) 1 a onto VJ Y. Using the compact notation u = u ∂a, we have the formula

a λ Ju = u + dλu ∂a and, as a consequence, obtain

i λ i ∂ JuQ = uQ + cλ∂i iy˙λ i , − ∂cλ i ∂ i∂ JuK = uK + cλ i ,JuK=uK+cλi , (52) ∂cλ ∂cλ i ∂ i ∂ JuC = uC = cλ i cλ i . ∂cλ − ∂cλ

It is readily observed that the SUSY-extended Lagrangian LS (45) is invariant under the transformations (52). The graded vector fields (43) and (52) make up the Lie superalgebra (51).

17 The graded vector fields (11) can give rise to V Π by the formula

a b ∂ u = u ( 1)[y ]([pb]+[u ])∂ ubpλ . a b λ − − ∂pa We have e ∂ ∂ u = ∂ iy˙ i ip˙λ , Q c i i λ − ∂c − ∂ci ∂ ∂ ∂ ∂ ue = ci + cλ , u = ci + cλ , (53) K i i λ K i i λ ∂c ∂ci ∂c ∂ci ∂ ∂ ∂ ∂ ue = ci + cλ ci e cλ . C i i λ i i λ ∂c ∂ci − ∂c − ∂ci e A direct computation shows that the BRS-extended Hamiltonian form HS (48) is invariant under the transformations (53). Accordingly, the Lagrangian LSH (49) is invariant under the jet prolongation Ju of the graded vector fields (53). The graded vector fields (47) and (53) make up the Lie superalgebra (51). With the graded vectore fields (47) and (53), one can construct the corresponding graded currents J = u H = u H . These are the graded (n 1)-forms u c S c S − Q =(e cip˙λ y˙icλ)ω , Q=(cip˙λ y˙icλ)ω , i − i λ i − i λ K= icλciω , K = icλciω ,C=i(cci cic)ω − i λ i λ i − i λ on V Π. They form the Lie superalgebra (51) with respect to the product

[Ju,Ju0]=J[u,u0].

The following construction is similar to that in the SUSY and BRS mechanics. Given a function F on the Legendre bundle Π, let us consider the operators

βF βF βF βF F = e u e− = u β∂ F, F = e− u e = u + β∂ F, β > 0, β ◦ Q ◦ Q − c β ◦ Q ◦ Q c called the SUSYe charges, whiche act on graded functionse on V Π.e These operators are nilpotent, i.e., F F =0, F F =0. (54) β ◦ β β ◦ β By the BRS-invariant extension of a function F is meant the graded function i F = (F F + F F ). S −β β ◦ β β ◦ β

18 We have the relations

F F F F =0, F F F F =0. β ◦ S − S ◦ β β ◦ S − S ◦ β

These relations together with the relations (54) provide the operators Fβ, F β,andFS with the structure of the Lie superalgebra sl(1/1) [20]. In particular, let F be a local function in the expression (4). Then H = i(F F + F F ) HS − 1 ◦ 1 1 ◦ 1 is exactly the local function HS in the expression (48). The similar splitting of a super- Hamiltonian is the corner stone of the SUSY mechanics [21, 22].

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