A Kähler structure of the triplectic geometry

Abstract

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...

Modified Triplectic Quantization in General Coordinates

“…Note, that an explicit realization of the antibrackets in the form (54) for a flat symmetric connection already has been found in Ref. [7].…”

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

A Lie group structure underlying the triplectic geometry

“…This is only a half of the full triplectic geometry. The other part is concentrated on the manifold of marked functions of the antibrackets; the corresponding geometry was studied in the case of mutually commutative antibrackets in [5]. This is also interesting to generalize to the case of non-commutative antibrackets.…”

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