The Poincaré–cartan Form in Superfield Theory

Monterde, J.; Masqué, J. Muñoz; Vallejo, José Antonio

doi:10.1142/s0219887806001375

Cited by 16 publications

(23 citation statements)

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“…If Y → X is a vector bundle, this is a particular case of graded fibre bundles in [16,27] when their base is a trivial graded manifold.…”

Section: Definitionmentioning

confidence: 99%

“…This formulation is based on the categorial equivalence of projective C ∞ (X)-modules of finite ranks and vector bundles over X in accordance with the well-known Serre-Swan theorem, generalized to an arbitrary manifold [20,28]. At the same time, different geometric models of odd variables either on graded manifolds or supermanifolds are discussed [12,13,14,26,27,34]. Both graded manifolds and supermanifolds are phrased in terms of sheaves of graded commutative algebras [4,20,31].…”

Section: Introductionmentioning

confidence: 99%

“…Obviously, the above mentioned trivial graded jet manifold (X,is a trivial graded bundle [20,34]. This definition differs from that of graded jet bundles in [16,27], but it reproduces the heuristic notion of jets of odd ghosts in field-antifield BRST theory [3,10].Furthermore, the inverse sequence of jet manifolds16) associates to them the gauge operator (74) which is a non-trivial gauge symmetry of an original Lagrangian. Its nilpotent extension is the BRST operator (77).…”

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confidence: 99%

“…is a trivial graded bundle [20,34]. This definition differs from that of graded jet bundles in [16,27], but it reproduces the heuristic notion of jets of odd ghosts in field-antifield BRST theory [3,10].…”

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SUSY Gauge Theory on Principal Graded Bundles

Sardanashvily

2016

Noether's Theorems

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Lagrangian classical field theory of even and odd fields is adequately formulated in terms of fibre bundles and graded manifolds. In particular, conventional Yang-Mills gauge theory is theory of connections on smooth principal bundles, but its BRST extension involves odd ghost fields an antifields on graded manifolds. Here, we formulate Yang-Mills theory of Grassmann-graded gauge fields associated to Lie superalgebras on principal graded bundles.A problem lies in a geometric definition of odd gauge fields. Our goal is Yang-Mills theory of graded gauge fields and its BRST extension.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, J k Y, C ∞ J k Y ). Then let us define their partners in the case of the non-trivial graded manifold (1) as follows (Section 6).Let (Y, A F ) (1) be a simple graded manifold modelled over a vector bundleWe agree to call it the graded k-order jet manifold of a graded bundle (X, Y, A F ) (Definition 7). Obviously, the above mentioned trivial graded jet manifold (X,is a trivial graded bundle [20,34]. This definition differs from that of graded jet bundles in [16,27], but it reproduces the heuristic notion of jets of odd ghosts in field-antifield BRST theory [3,10].Furthermore, the inverse sequence of jet manifolds16) associates to them the gauge operator (74) which is a non-trivial gauge symmetry of an original Lagrangian. Its nilpotent extension is the BRST operator (77). If the BRST operator exists, it provides the BRST extension of original Lagrangian field theory by Grassmann-graded ghosts and antifields [19,20]. This BRST extension is a first step towards BV quantization of classical field theory in terms of functional integrals [3,21]. As was mentioned above, classical gauge theory is formulated as Lagrangian theory of principal connections on smooth principal bundles (Section 9). Similarly, we develop SUSY gauge theory in the framework of Grassmann-graded Lagrangian formalism on principal graded bundles (Section 10). A key point is that we consider simple principal graded bundles (Definition 26) subject to an action of an even Lie group, but not the whole graded Lie one. In this case, odd gauge potentials in comparison with the even ones are linear, but not affine objects (see Remark 10). At the same time, they admit the affine gauge transformations (141) parameterised by ghosts. Our goal is Yang-Mills theory of graded gauge fields and its BRST extension (Section 11).2 Grassmann-graded algebraic calculus Throughout this work, by the Grassmann gradation is meant Z 2 -gradation, and a Grassmann graded structure simply is called the graded structure if there is no danger of confusion. Hereafter, the symbol [.] stands for the Grassmann parity. Let us summarize the relevant notions of the Grassmann-graded algebraic calculus [4,20,31]. Let K be a commutative ring. A K-module Q is called graded if it is endowed with a grading automorphism γ, γ 2 = Id . A graded module falls in...

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“…If Y → X is a vector bundle, this is a particular case of graded fibre bundles in [16,27] when their base is a trivial graded manifold.…”

Section: Definitionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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SUSY Gauge Theory on Principal Graded Bundles

Sardanashvily

2016

Noether's Theorems

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“…to which we associate a super-Lagrangian L ∈ A J 1 G (p) (see [13] for the details of this construction) proceeding by analogy with the non-graded case (which is that of harmonic functions). The coordinates in the bundle of superjets J 1 G (p) will be denoted {t, τ, x i , x −j , t i , t −j , τ i , τ −j }.…”

Section: (1|1)-supersymmetric Sigma Modelmentioning

confidence: 99%