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Title: Equivariant symplectic geometry of gauge fixing in Yang-Mills theory

Abstract

The Faddeev-Popov gauge fixing in Yang-Mills theory is interpreted as equivariant localization. It is shown that the Faddeev-Popov procedure amounts to a construction of a symplectic manifold with a Hamiltonian group action. The BRST cohomology is shown to be equivalent to the equivariant cohomology based on this symplectic manifold with Hamiltonian group action. The ghost operator is interpreted as a (pre)symplectic form and the gauge condition as the moment map corresponding to the Hamiltonian group action. This results in the identification of the gauge fixing action as a closed equivariant form, the sum of an equivariant symplectic form, and a certain closed equivariant 4-form, which ensures convergence. An almost complex structure compatible with the symplectic form is constructed. The equivariant localization principle is used to localize the path integrals onto the gauge slice. The Gribov problem is also discussed in the context of equivariant localization principle. As a simple illustration of the methods developed in the paper, the partition function of N=2 supersymmetric quantum mechanics is calculated by equivariant localization.

Authors:
 [1]
  1. Feza Gursey Institute, Emek Mahallesi, Rasathane Yolu No. 68, Cengelkoy, Istanbul 346 84 (Turkey)
Publication Date:
OSTI Identifier:
21100244
Resource Type:
Journal Article
Journal Name:
Journal of Mathematical Physics
Additional Journal Information:
Journal Volume: 49; Journal Issue: 3; Other Information: DOI: 10.1063/1.2897049; (c) 2008 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; CONVERGENCE; GEOMETRY; HAMILTONIANS; MAPS; PARTITION FUNCTIONS; PATH INTEGRALS; QUANTUM MECHANICS; SUPERSYMMETRY; YANG-MILLS THEORY

Citation Formats

Akant, Levent. Equivariant symplectic geometry of gauge fixing in Yang-Mills theory. United States: N. p., 2008. Web. doi:10.1063/1.2897049.
Akant, Levent. Equivariant symplectic geometry of gauge fixing in Yang-Mills theory. United States. https://doi.org/10.1063/1.2897049
Akant, Levent. Sat . "Equivariant symplectic geometry of gauge fixing in Yang-Mills theory". United States. https://doi.org/10.1063/1.2897049.
@article{osti_21100244,
title = {Equivariant symplectic geometry of gauge fixing in Yang-Mills theory},
author = {Akant, Levent},
abstractNote = {The Faddeev-Popov gauge fixing in Yang-Mills theory is interpreted as equivariant localization. It is shown that the Faddeev-Popov procedure amounts to a construction of a symplectic manifold with a Hamiltonian group action. The BRST cohomology is shown to be equivalent to the equivariant cohomology based on this symplectic manifold with Hamiltonian group action. The ghost operator is interpreted as a (pre)symplectic form and the gauge condition as the moment map corresponding to the Hamiltonian group action. This results in the identification of the gauge fixing action as a closed equivariant form, the sum of an equivariant symplectic form, and a certain closed equivariant 4-form, which ensures convergence. An almost complex structure compatible with the symplectic form is constructed. The equivariant localization principle is used to localize the path integrals onto the gauge slice. The Gribov problem is also discussed in the context of equivariant localization principle. As a simple illustration of the methods developed in the paper, the partition function of N=2 supersymmetric quantum mechanics is calculated by equivariant localization.},
doi = {10.1063/1.2897049},
url = {https://www.osti.gov/biblio/21100244}, journal = {Journal of Mathematical Physics},
issn = {0022-2488},
number = 3,
volume = 49,
place = {United States},
year = {2008},
month = {3}
}