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Title: Note on Z{sub 2} symmetries of the Knizhnik-Zamolodchikov equation

Abstract

We continue the study of hidden Z{sub 2} symmetries of the four-point sl(2){sub k} Knizhnik-Zamolodchikov equation initiated by Giribet [Phys. Lett. B 628, 148 (2005)]. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector {omega}=1 is involved. We give a formula that shows how this observable can be expressed in terms of the four-point function of non spectral flowed states. This means that the formula holding for the winding violating four-string scattering processes in AdS{sub 3} has a simple expression in terms of the one for the conservative case, generalizing what is known for the case of three-point functions, where the violating and the nonviolating structure constants turn out to be connected one to each other in a similar way. What makes this connection particularly simple is the fact that, unlike what one would naively expect, it is not necessary to explicitly solve the five-point function containing a single spectral flow operator to this end. Instead, nondiagonal functional relations between different solutions of the Knizhnik-Zamolodchikov equation turn out to be the key point for this short path to exist. Considering such functional relation is necessary but it ismore » not sufficient; besides, the formula also follows from the relation existing between correlators in both Wess-Zumino-Novikov-Witten (WZNW) and Liouville conformal theories.« less

Authors:
 [1]
  1. Departamento de Fisica, Universidad de Buenos Aires, FCEN UBA, Ciudad Universitaria, Pabellon I, 1428 Buenos Aires (Argentina)
Publication Date:
OSTI Identifier:
20929615
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 48; Journal Issue: 1; Other Information: DOI: 10.1063/1.2424789; (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; ANTI DE SITTER SPACE; CONFORMAL INVARIANCE; CORRELATION FUNCTIONS; EQUATIONS; MATHEMATICAL SOLUTIONS; QUANTUM FIELD THEORY; STRING MODELS; SYMMETRY

Citation Formats

Giribet, Gaston E. Note on Z{sub 2} symmetries of the Knizhnik-Zamolodchikov equation. United States: N. p., 2007. Web. doi:10.1063/1.2424789.
Giribet, Gaston E. Note on Z{sub 2} symmetries of the Knizhnik-Zamolodchikov equation. United States. doi:10.1063/1.2424789.
Giribet, Gaston E. Mon . "Note on Z{sub 2} symmetries of the Knizhnik-Zamolodchikov equation". United States. doi:10.1063/1.2424789.
@article{osti_20929615,
title = {Note on Z{sub 2} symmetries of the Knizhnik-Zamolodchikov equation},
author = {Giribet, Gaston E.},
abstractNote = {We continue the study of hidden Z{sub 2} symmetries of the four-point sl(2){sub k} Knizhnik-Zamolodchikov equation initiated by Giribet [Phys. Lett. B 628, 148 (2005)]. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector {omega}=1 is involved. We give a formula that shows how this observable can be expressed in terms of the four-point function of non spectral flowed states. This means that the formula holding for the winding violating four-string scattering processes in AdS{sub 3} has a simple expression in terms of the one for the conservative case, generalizing what is known for the case of three-point functions, where the violating and the nonviolating structure constants turn out to be connected one to each other in a similar way. What makes this connection particularly simple is the fact that, unlike what one would naively expect, it is not necessary to explicitly solve the five-point function containing a single spectral flow operator to this end. Instead, nondiagonal functional relations between different solutions of the Knizhnik-Zamolodchikov equation turn out to be the key point for this short path to exist. Considering such functional relation is necessary but it is not sufficient; besides, the formula also follows from the relation existing between correlators in both Wess-Zumino-Novikov-Witten (WZNW) and Liouville conformal theories.},
doi = {10.1063/1.2424789},
journal = {Journal of Mathematical Physics},
number = 1,
volume = 48,
place = {United States},
year = {Mon Jan 15 00:00:00 EST 2007},
month = {Mon Jan 15 00:00:00 EST 2007}
}
  • The crystal structure, electronic structure, and photoluminescence properties of Eu{sub x}Si{sub 6-z}Al{sub z-x}O{sub z+x}N{sub 8-z-x} (x=0-0.1, 0<z<1) and Eu{sub x}M{sub y}Si{sub 6-z}Al{sub z-x-y}O{sub z+x+y}N{sub 8-z-x-y} (M=2Li, Mg, Ca, Sr, Ba) have been studied. Single-phase Eu{sub x}Si{sub 6-z}Al{sub z-x}O{sub z+x}N{sub 8-z-x} can be obtained in very narrow ranges of x{<=}0.06 (z=0.15) and z<0.5 (x=0.3), indicating that limited Eu{sup 2+} ions can be incorporated into nitrogen-rich Si{sub 6-z}Al{sub z}O{sub z}N{sub 8-z}. The Eu{sup 2+} ion is found to occupy the 2b site in a hexagonal unit cell (P6{sub 3}/m) and directly connected by six adjacent nitrogen/oxygen atoms ranging 2.4850-2.5089 A. The calculatedmore » host band gaps by the relativistic DV-X{alpha} method are about 5.55 and 5.45 eV (without Eu{sup 2+} 4f5d levels) for x=0 and 0.013 in Eu{sub x}Si{sub 6-z}Al{sub z-x}O{sub z+x}N{sub 8-z-x} (z=0.15), in which the top of the 5d orbitals overlap with the Si-3s3p and N-2p orbitals within the bottom of the conduction band of the host. Eu{sub x}Si{sub 6-z}Al{sub z-x}O{sub z+x}N{sub 8-z-x} shows a strong green emission with a broad Eu{sup 2+} band centered at about 530 nm under UV to near-UV excitation range. The excitation and emission spectra are hardly modified by Eu concentration and dual-doping ions of Li and other alkaline-earth ions with Eu. Higher Eu concentrations can significantly quench the luminescence of Eu{sup 2+} and decrease the thermal quenching temperature. In addition, the emission spectrum can only be slightly tuned to the longer wavelengths ({approx}529-545 nm) by increasing z within the solid solution range of z<0.5. Furthermore, the luminescence intensity of Eu{sub x}Si{sub 6-z}Al{sub z-x}O{sub z+x}N{sub 8-z-x} can be improved by increasing z and the dual-doping of Li and Ba. - Graphical abstract: Excitation and emission spectra of Eu{sub x}Si{sub 6-z}Al{sub z-x}O{sub z+x}N{sub 8-z-x} with the project of a 2x2x2 supercell crystal structure viewed along (001), in which red spheres are the Eu atoms.« less
  • Unitarity can be proven from the usual Z/sub 2/ grading of gauge and ghost fields, or from a Z/sub 2/ x Z/sub 2/ grading, geometrically derived by Ne'eman and Thierry-Mieg, or from a Z/sub 2/ x Z/sub 2/ x Z/sub 2/ grading derived here. The claim that only the Z/sub 2/ x Z/sub 2/ grading leads to unitarity is incorrect. The opposite is shown to hold: signs due to different gradings are physically unobservable. We show how the Z/sub 2/ x Z/sub 2/ grading follows from the Z/sub 2/ grading by taking a product space.
  • The detection of granular superconductivity in Pr222Nb-10 leads to the conclusion that R{sub 2-z}Ce{sub z}CuO{sub 4} (R21-4) compounds and R{sub 2-z}Ce{sub z}Sr{sub 2}Cu{sub 2}NbO{sub 10} (R222Nb-10) compounds both superconduct for R=Pr. The fact that Gd21-4 does not superconduct, but Gd222Nb-10 does, indicates that the superconducting layers are different in the two classes of compounds. Magnetic resonance studies of Gd222Ru-10 and Gd21-4 indicate either that ordered magnetism and superconductivity co-exist in the cuprate-planes or else that the cuprate-planes are not the primary superconducting layers in Gd222Ru-10. (c) 1999 American Institute of Physics.
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