Open Group Transformations

Batalin, I. A.; Marnelius, Robert

doi:10.1142/s0217732399001735

Cited by 22 publications

(124 citation statements)

References 10 publications

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“…[5,6,7]). Then, right-hand derivatives with respect to θ ia transform like the basis vectors of the cotangent space…”

Section: Tensor Fields Covariant Derivative and Curvature Tensor On mentioning

confidence: 99%

“…If, in addition, the tensor field ω ij obeys the property of generalized antisymmetry, 5) then, the binary operation (A.1) has the property of…”

Section: A Poisson and Symplectic Structures On Supermanifoldsmentioning

confidence: 99%

“…Modern covariant quantization methods for general gauge theories are based on the principle of BRST [2,3], or, more generally, BRST-antiBRST [4,5,8,9] invariance. The consideration of these methods in general coordinates (using appropriate supermanifolds) appears to be very important in order to reveal the geometrical meaning of the basic objects underlying these quantization schemes.…”

Section: Introductionmentioning

confidence: 99%

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Modified Triplectic Quantization in General Coordinates

Geyer

Лавров

2004

Int. J. Mod. Phys. A

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“…[5,6,7]). Then, right-hand derivatives with respect to θ ia transform like the basis vectors of the cotangent space…”

Section: Tensor Fields Covariant Derivative and Curvature Tensor On mentioning

confidence: 99%

“…If, in addition, the tensor field ω ij obeys the property of generalized antisymmetry, 5) then, the binary operation (A.1) has the property of…”

Section: A Poisson and Symplectic Structures On Supermanifoldsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Modified Triplectic Quantization in General Coordinates

Geyer

Лавров

2004

Int. J. Mod. Phys. A

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“…Triplectic quantization of general gauge theories [1,2,3] was formulated as a generalization of the Sp(2)-symmetric Lagrangian quantization [4,5], into which the ghost and antighost fields enter in a symmetric way (which is in contrast to the standard BV-formalism [7]). In the triplectic formalism, one introduces a pair of antibracket operations (in fact, a pair of odd BV ∆-operators) that are required to satisfy certain compatibility conditions.…”

Section: Introductionmentioning

confidence: 99%

“…In the triplectic formalism, one introduces a pair of antibracket operations (in fact, a pair of odd BV ∆-operators) that are required to satisfy certain compatibility conditions. In addition, one introduces two odd vector fields [5,1,6] which, again, should agree with the antibrackets in a certain sense. All these objects allow one two formulate the triplectic master-equations, whose solutions enter the corresponding path integral.…”

Section: Introductionmentioning

confidence: 99%

Canonical form of a pair of compatible antibrackets

Grigoriev

Semikhatov

1998

Physics Letters B

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In the triplectic quantization of general gauge theories, we prove a 'triplectic' analogue of the Darboux theorem: we show that the doublet of compatible antibrackets can be brought to a weakly-canonical form provided the general triplectic axioms of [2] are imposed together with some additional requirements that can be formulated in terms of marked functions of the antibrackets. The weakly-canonical antibrackets involve an obstruction to bringing them to the canonical form. We also classify the 'triplectic' odd vectors fields compatible with the weakly-canonical antibrackets and construct the Poisson bracket associated with the antibrackets and the odd vector fields. We formulate the Sp(2)-covariance requirement for the antibrackets and the vector fields; whenever the obstruction to the canonical form of the antibrackets vanishes, the Sp(2)-covariance condition implies the canonical form of the triplectic vector fields.

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A Kähler structure of the triplectic geometry

Grigoriev

Semikhatov

2000

Theor Math Phys

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We study the geometry of the triplectic quantization of gauge theories. We show that underlying the triplectic geometry is a Kähler manifold N with a pair of transversal polarizations. The antibrackets can be brought to the canonical form if and only if N admits a flat symmetric connection that is compatible with the complex structure and the polarizations. IntroductionThe Sp(2)-symmetric Lagrangian quantization [1, 2] of general gauge theories generalizes the standard BV-formalism [3] so that ghosts and antighosts enter it in a symmetric way. The triplectic quantization [4,5,6] has been formulated as the corresponding analogue of the covariant formulation of the BV scheme [7,8,9,10,11], where "covariant" refers to the space of fields. An essential point in such a formulation is to ensure that the antibracket(s) can be locally brought to the canonical ("Darboux") form, since only then the equivalence with the Hamiltonian quantization has been established.In this paper, we investigate the geometry underlying the triplectic quantization procedure. There are considerable differences from the usual BV formalism. By construction, the covariant version of the BV scheme does not differentiate between fields and antifields, which simply become non-invariant notions. In the triplectic formalism, on the other hand, the antibrackets are degenerate, therefore one can single out the marked functions (Casimir functions, or "zero modes") of the antibrackets; the marked functions then span the space of antifields. In this sense, the antifields are already encoded in the triplectic data.As we will see, the triplectic geometry is essentially concentrated on the "manifold of antifields." This turns out to be a complex manifold N , the complex structure originating from, and giving the geometric interpretation of, the e-structure entering the weakly canonical antibrackets from [12]. Further, the existence of two antibrackets induces a polarization on N , and the symmetrized Jacobi identities [1] imply then that the associated Nijenhuis tensor vanishes. Finally, the one-form F that enters the triplectic data [6,12] (the "potential" for the odd vector fields) induces a symplectic structure on N , which together with the complex structure makes it into a Kähler manifold.The properties of the "antifield" manifold N are, in particular, responsible for the possibility of bringing the triplectic antibrackets to the canonical form. The condition for the general triplectic antibrackets to allow the transformation to the canonical form reformulates as the requirement that N admit a flat symmetric connection that is compatible with the complex structure and the polarization. This solves the problem posed in [5] and addressed recently in [12].In Sec 2, we briefly recall the triplectic formulation and reformulate the structures known from [12]. In Sec. 3, we show how these translate into the language of Kähler geometry. An important fact proved in Sec. 4 is that these geometric structures distinguish different triplectic structures up to local e...

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Open Group Transformations

Abstract: Recent results on finite open group transformations are reviewed.

Cited by 22 publications

References 10 publications

Modified Triplectic Quantization in General Coordinates

Modified Triplectic Quantization in General Coordinates

Canonical form of a pair of compatible antibrackets

A Kähler structure of the triplectic geometry

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