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Title: Center vortices and the Atiyah-Singer index theorem

Abstract

The lattice index theorem is checked for classical center vortices. For non-orientable spherical vortices, lattice results differ from continuum-based expectations, possibly because of coarse discretization.

Authors:
; ;  [1];  [2]
  1. Atomic Institute, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040 Vienna (Austria)
  2. American Physical Society, One Research Road, Box 9000, Ridge, NY 11961-9000 (United States)
Publication Date:
OSTI Identifier:
21056902
Resource Type:
Journal Article
Resource Relation:
Journal Name: AIP Conference Proceedings; Journal Volume: 892; Journal Issue: 1; Conference: QCHS7: 7. conference on quark confinement and the hadron spectrum, Ponta Delgada, Acores (Portugal), 2-7 Sep 2006; Other Information: DOI: 10.1063/1.2714461; (c) 2007 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS; GAUGE INVARIANCE; INDEXES; LATTICE FIELD THEORY; QUANTUM CHROMODYNAMICS; SPHERICAL CONFIGURATION; VORTICES

Citation Formats

Jordan, Gerald, Pullirsch, Rainer, Faber, Manfried, and Heller, Urs. Center vortices and the Atiyah-Singer index theorem. United States: N. p., 2007. Web. doi:10.1063/1.2714461.
Jordan, Gerald, Pullirsch, Rainer, Faber, Manfried, & Heller, Urs. Center vortices and the Atiyah-Singer index theorem. United States. doi:10.1063/1.2714461.
Jordan, Gerald, Pullirsch, Rainer, Faber, Manfried, and Heller, Urs. Tue . "Center vortices and the Atiyah-Singer index theorem". United States. doi:10.1063/1.2714461.
@article{osti_21056902,
title = {Center vortices and the Atiyah-Singer index theorem},
author = {Jordan, Gerald and Pullirsch, Rainer and Faber, Manfried and Heller, Urs},
abstractNote = {The lattice index theorem is checked for classical center vortices. For non-orientable spherical vortices, lattice results differ from continuum-based expectations, possibly because of coarse discretization.},
doi = {10.1063/1.2714461},
journal = {AIP Conference Proceedings},
number = 1,
volume = 892,
place = {United States},
year = {Tue Feb 27 00:00:00 EST 2007},
month = {Tue Feb 27 00:00:00 EST 2007}
}
  • The Atiyah-Singer index theorem has recently been used to determine the number of parameters entering the general self-dual Euclidena Yang-Mills configuration with a given topological charge. We give a simple derivation of the form which the index theorem takes in this application.
  • With the chiral representation, the Friedel sum rule for chiral fermions in odd space dimensions is given and the Levinson theorem which determines the number of discrete zero modes is obtained. Furthermore, the connections among the Friedel sum, Levinson theorem, and the Atiyah-Singer index are discussed, thereby proving that the Euclidean Atiyah-Singer index for the background field that describes an arbitrary short-range central force is zero.
  • We calculate explicitly the eta invariant for the Euclidean Dirac operator in the background field of an instanton on a four-dimensional manifold with a three-sphere boundary. The total chiral anomaly is found by invoking the Atiyah-Patodi-Singer theorem.
  • In the fundamental paper [APS], Atiyah, Patoki and Singer showed a formula for the index of certain first order elliptic differential operators P:C[infinity](E) [yields] C[infinity](F), where E and F are Hermitian C[infinity] vector bundles over a compact n-dimensional C[infinity] manifold X with boundary Y, under the assumption that the operator as well as the manifold and bundles have a product structure near the boundary. In the present paper we treat the general non-product case, deriving a formula with a new boundary term; and we establish asymptotic expansions of the heat operators associated with the problem, that have not been treatedmore » individually before.« less
  • We consider the problem of determining the zero modes for the massless Dirac operator in a background Abelian gauge field F on a four-dimensional Euclidean manifold M with a boundary partialM. We solve exactly the case when F is constant self-dual or anti-self-dual and M is B/sup 4/, a four-dimensional ball, so that partialM is S/sup 3/, a three-dimensional sphere. The analysis employs the Atiyah-Patodi-Singer index theorem for manifolds with boundaries.