Triplectic quantization: A geometrically covariant description of the Sp(2)-symmetric Lagrangian formalism

Abstract

A geometric description is given for the Sp(2) covariant version of the field-antifield quantization of general constrained systems in the Lagrangian formalism. We develop differential geometry on manifolds in which a basic set of coordinates ('fields') have two superpartners ('antifields'). The quantization on such a triplectic manifold requires introducing several specific differentialgeometric objects, whose properties we study. These objects are then used to impose a set of generalized master-equations that ensure gauge-independence of the path integral. The theory thus quantized is shown to extend to a level-1 theory formulated on a manifold that includes antifields to the Lagrange multipliers. We also observe intriguing relations between triplectic and ordinary symplectic geometry.

Gauge symmetries of the master action in the Batalin–Vilkovisky formalism

“…This is given by a construction of the type of those used, with some variations, in [10,12,13,15], namely…”

Comments on the covariant Sp(2)-symmetric Lagrangian BRST formalism

“…[7,8]. In this scheme one requires the existance of a pair of ∆-operators ∆ a and odd vector fields V a (a = 1, 2) satisfying the following consistency conditions:…”

“…[2]. The main advantage of the new proposal is that it can be readily generalized to a covariant formulation, a task carried out by Batalin, Marnelius and Semikhatov [8]. Here, "covariant" refers to the supermanifold of fields (including all necessary ghosts, auxiliary fields, etc.)…”

“…[8] involves a number of new complications that makes it quite involved and which to some extent obscure its geometric meaning. It may also appear surprising that in its precise formulation, the triplectic formalism does not include the minimal solution found in ref.…”

Quantum Properties of General Gauge Theories With Composite and External Fields

“…derivation of the Ward identities, study of renormalization and gauge dependence [5], analysis of unitarity conditions [6]) or playing a key role in the interpretation of alternative quantization methods (e.g. triplectic [7], superfield [8], osp(1,2)-covariant quantization [9]). In particular, the methods [1,3] have been used to analyse the quantum structure of general gauge theories with composite fields [10,11] (in the BV and BLT formalisms, respectively); for these theories the corresponding Ward identities were derived and the related issue of gauge dependence was investigated.…”