Triplectic Quantization: A Geometrically Covariant Description of the Sp(2)-symmetric Lagrangian Formalism

TL;DR

This paper presents a geometric framework for the Sp(2)-symmetric Lagrangian quantization of constrained systems, introducing differential geometry on triplectic manifolds to ensure gauge independence and explore relations with symplectic geometry.

Contribution

It develops a geometric, covariant description of triplectic quantization, including new differential-geometric objects and master-equations, extending the formalism to include antifields for Lagrange multipliers.

Findings

Established differential geometry on triplectic manifolds.

Derived generalized master-equations ensuring gauge independence.

Connected triplectic geometry with ordinary symplectic geometry.

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 differential-geometric 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.