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The Moyal cohomology of the spinning particle

Ezra Getzler

Abstract

Felder and Kazhdan conjecture that the local cohomology in the classical Batalin-Vilkovisky formalism vanishes in sufficiently negative degrees. This hypothesis is violated by the $N=1$ spinning particle. By Barnich-Grigoriev, this cohomology is isomorphic to the cohomology of the algebra of functions on the differential graded symplectic supermanifold of the associated Batalin-Fradkin-Vilkovisky model. This cohomology is nontrivial in all negative degrees. We show in this article that replacement in this symplectic supermanifold of the Poisson bracket by the Moyal bracket eliminates these spurious cohomology classes.

The Moyal cohomology of the spinning particle

Abstract

Felder and Kazhdan conjecture that the local cohomology in the classical Batalin-Vilkovisky formalism vanishes in sufficiently negative degrees. This hypothesis is violated by the spinning particle. By Barnich-Grigoriev, this cohomology is isomorphic to the cohomology of the algebra of functions on the differential graded symplectic supermanifold of the associated Batalin-Fradkin-Vilkovisky model. This cohomology is nontrivial in all negative degrees. We show in this article that replacement in this symplectic supermanifold of the Poisson bracket by the Moyal bracket eliminates these spurious cohomology classes.

Paper Structure

This paper contains 3 sections, 7 theorems, 63 equations.

Table of Contents

  1. The symplectic supermanifold of the $N=1$ spinning particle
  2. The Moyal cohomology of the $N=1$ spinning particle: the flat case
  3. The Moyal cohomology of the $N=1$ spinning particle: the general case

Key Result

Lemma 1

$\{\pi,\{\pi,f \Theta\}\} = - \pi\{\{\pi,f\},\Theta\}$$\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (13)

  • Lemma
  • proof
  • Corollary
  • Theorem
  • proof
  • Lemma
  • proof
  • Corollary
  • proof
  • Lemma
  • ...and 3 more