6j Symbols for the Modular Double, Quantum Hyperbolic Geometry, and Supersymmetric Gauge Theories
We revisit the definition of the 6 j symbols from the modular double of U q ( sl ( 2 , R ) ) , referred to as b-6 j symbols. Our new results are (1) the identification of particularly natural normalization conditions, and (2) new integral representations for this object. This is used to briefly disc...
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| Published in | Letters in mathematical physics Vol. 104; no. 5; pp. 527 - 551 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Dordrecht
Springer Netherlands
01.05.2014
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| Subjects | |
| Online Access | Get full text |
| ISSN | 0377-9017 1573-0530 |
| DOI | 10.1007/s11005-014-0684-3 |
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| Abstract | We revisit the definition of the 6
j
symbols from the modular double of
U
q
(
sl
(
2
,
R
)
)
, referred to as b-6
j
symbols. Our new results are (1) the identification of particularly natural normalization conditions, and (2) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6
j
symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. Our new integral representations finally indicate a possible interpretation of the b-6
j
symbols as partition functions of non-abelian three-dimensional
N
=
2
supersymmetric gauge theories. |
|---|---|
| AbstractList | We revisit the definition of the 6
j
symbols from the modular double of
U
q
(
sl
(
2
,
R
)
)
, referred to as b-6
j
symbols. Our new results are (1) the identification of particularly natural normalization conditions, and (2) new integral representations for this object. This is used to briefly discuss possible applications to quantum hyperbolic geometry, and to the study of certain supersymmetric gauge theories. We show, in particular, that the b-6
j
symbol has leading semiclassical asymptotics given by the volume of a non-ideal tetrahedron. Our new integral representations finally indicate a possible interpretation of the b-6
j
symbols as partition functions of non-abelian three-dimensional
N
=
2
supersymmetric gauge theories. |
| Author | Vartanov, Grigory Teschner, Jörg |
| Author_xml | – sequence: 1 givenname: Jörg surname: Teschner fullname: Teschner, Jörg email: teschner@mail.desy.de organization: DESY Theory – sequence: 2 givenname: Grigory surname: Vartanov fullname: Vartanov, Grigory organization: DESY Theory |
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| Cites_doi | 10.1016/S0550-3213(01)00464-3 10.1007/s11139-005-4846-x 10.1007/s00220-013-1863-2 10.1088/0264-9381/18/23/201 10.1007/JHEP03(2010)089 10.1007/s00029-012-0112-4 10.1103/PhysRevLett.68.1795 10.1007/JHEP12(2010)079 10.1007/s00220-012-1485-0 10.1142/S0217751X0100444X 10.1007/JHEP08(2011)135 10.1142/S0129055X00000344 10.1016/j.jnt.2009.03.005 10.1007/s00220-005-1342-5 10.1007/PL00005590 10.1007/s00220-013-1861-4 10.4310/CAG.2005.v13.n2.a5 10.1023/A:1007364912784 10.1016/j.geomphys.2007.03.008 10.1007/JHEP03(2011)127 10.1016/0003-4916(72)90335-1 10.1007/s11005-010-0369-5 10.1016/0040-9383(92)90015-A 10.1142/S0217732394000447 10.1016/j.physletb.2011.09.007 10.1007/JHEP05(2011)014 10.1016/j.nuclphysb.2007.05.013 10.1007/s00220-003-0894-5 10.1007/JHEP06(2011)025 10.4310/CNTP.2009.v3.n2.a4 10.1088/0305-4470/39/41/S11 10.1007/BF01872779 10.1098/rsta.1901.0006 10.1016/j.nuclphysb.2009.08.022 10.1063/1.531809 10.1007/JHEP05(2012)159 10.1007/978-90-481-2810-5_46 10.1070/RM2008v063n03ABEH004533 10.1007/JHEP05(2013)109 10.1007/978-94-010-0670-5_13 10.1070/RM2001v056n01ABEH000374 |
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| References | Turaev, Viro (CR47) 1992; 31 Pestun (CR32) 2012; 313 CR39 Mizoguchi, Tada (CR30) 1992; 68 CR38 CR37 Hikami (CR21) 2001; 16 CR34 CR33 Bazhanov, Mangazeev, Sergeev (CR3) 2007; B 784 Faddeev (CR15) 1995; 34 Ip (CR25) 2013; 19 Murakami, Yano (CR31) 2005; 13 Teschner (CR45) 2004; 437 CR2 Dolan, Spiridonov, Vartanov (CR13) 2011; 704 Volkov (CR48) 2005; 258 Hikami (CR23) 2007; 57 Spiridonov, Vartanov (CR41) 2014; 325 CR46 Gaiotto, Witten (CR18) 2009; 13 Drukker, Gaiotto, Gomis (CR14) 2011; 1106 Hama, Hosomichi, Lee (CR20) 2011; 1105 Alday, Gaiotto, Tachikawa (CR1) 2010; 91 Bytsko, Teschner (CR7) 2003; 240 Ruijsenaars (CR36) 1997; 38 Kashaev (CR28) 1997; 39 Bytsko, Teschner (CR8) 2006; 39 Dimofte, Gukov, Lenells, Zagier (CR10) 2009; 3 CR17 CR11 Barnes (CR5) 1901; 196 Bazhanov, Mangazeev, Sergeev (CR4) 2008; 0807 Baxter (CR6) 1972; 70 Faddeev, Kashaev (CR16) 1994; 9 Woronowicz (CR49) 2000; 12 Kapustin, Willett, Yaakov (CR27) 2010; 1003 Hama, Hosomichi, Lee (CR19) 2011; 1103 Ponsot, Teschner (CR35) 2001; 224 Hikami (CR22) 2001; 616 CR29 Terashima, Yamazaki (CR43) 2011; 1108 CR26 Spiridonov, Vartanov (CR40) 2010; 824 Spreafico (CR42) 2009; 129 van Diejen, Spiridonov (CR9) 2005; 10 Teschner (CR44) 2001; 18 Hosomichi, Lee, Park (CR24) 2010; 1012 Dimofte, Gaiotto, Gukov (CR12) 2014; 325 R.M. Kashaev (684_CR28) 1997; 39 R.J. Baxter (684_CR6) 1972; 70 A. Kapustin (684_CR27) 2010; 1003 N. Hama (684_CR19) 2011; 1103 E.W. Barnes (684_CR5) 1901; 196 684_CR33 B. Ponsot (684_CR35) 2001; 224 K. Hikami (684_CR22) 2001; 616 684_CR34 684_CR29 684_CR26 V.P. Spiridonov (684_CR40) 2010; 824 K. Hikami (684_CR21) 2001; 16 T. Dimofte (684_CR10) 2009; 3 V.V. Bazhanov (684_CR4) 2008; 0807 L.D. Faddeev (684_CR16) 1994; 9 D. Gaiotto (684_CR18) 2009; 13 S.N.M. Ruijsenaars (684_CR36) 1997; 38 L.F. Alday (684_CR1) 2010; 91 T. Dimofte (684_CR12) 2014; 325 N. Drukker (684_CR14) 2011; 1106 M. Spreafico (684_CR42) 2009; 129 684_CR17 Y. Terashima (684_CR43) 2011; 1108 J.F. van Diejen (684_CR9) 2005; 10 A.Yu. Volkov (684_CR48) 2005; 258 J. Murakami (684_CR31) 2005; 13 684_CR2 684_CR11 A.G. Bytsko (684_CR7) 2003; 240 V.G. Turaev (684_CR47) 1992; 31 S.L. Woronowicz (684_CR49) 2000; 12 S. Mizoguchi (684_CR30) 1992; 68 I.C.H. Ip (684_CR25) 2013; 19 J. Teschner (684_CR44) 2001; 18 A.G. Bytsko (684_CR8) 2006; 39 684_CR46 F.A.H. Dolan (684_CR13) 2011; 704 L.D. Faddeev (684_CR15) 1995; 34 V.P. Spiridonov (684_CR41) 2014; 325 684_CR45 684_CR39 V.V. Bazhanov (684_CR3) 2007; B 784 684_CR37 N. Hama (684_CR20) 2011; 1105 684_CR38 K. Hosomichi (684_CR24) 2010; 1012 V. Pestun (684_CR32) 2012; 313 K. Hikami (684_CR23) 2007; 57 |
| References_xml | – volume: 616 start-page: 537 year: 2001 end-page: 548 ident: CR22 article-title: Hyperbolicity of partition function and quantum gravity publication-title: Nucl. Phys. B doi: 10.1016/S0550-3213(01)00464-3 – volume: 10 start-page: 187 year: 2005 end-page: 204 ident: CR9 article-title: Unit circle elliptic beta integrals publication-title: Ramanujan J. doi: 10.1007/s11139-005-4846-x – volume: 325 start-page: 367 year: 2014 end-page: 419 ident: CR12 article-title: Gauge theories labelled by three-manifolds publication-title: Commun. Math. Phys. doi: 10.1007/s00220-013-1863-2 – volume: 18 start-page: R153 year: 2001 end-page: R222 ident: CR44 article-title: Liouville theory revisited publication-title: Class. Quant. Grav. doi: 10.1088/0264-9381/18/23/201 – volume: 1003 start-page: 089 year: 2010 ident: CR27 article-title: Exact results for Wilson loops in superconformal Chern–Simons theories with matter publication-title: J. High Energy Phys. doi: 10.1007/JHEP03(2010)089 – ident: CR39 – volume: 19 start-page: 987 year: 2013 end-page: 1082 ident: CR25 article-title: Representation of the quantum plane, its quantum double, and harmonic analysis on publication-title: Selecta Math. doi: 10.1007/s00029-012-0112-4 – volume: 437 start-page: 459 year: 2004 end-page: 477 ident: CR45 article-title: On the relation between quantum Liouville theory and the quantized Teichmüller spaces publication-title: Int. J. Mod. Phys. – ident: CR29 – volume: 68 start-page: 1795 year: 1992 end-page: 1798 ident: CR30 article-title: Three-dimensional gravity from the Turaev–Viro invariant publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.68.1795 – volume: 1012 start-page: 079 year: 2010 ident: CR24 article-title: AGT on the -duality Wall publication-title: J. High Energy Phys. doi: 10.1007/JHEP12(2010)079 – volume: 313 start-page: 71 year: 2012 end-page: 129 ident: CR32 article-title: Localization of gauge theory on a four-sphere and supersymmetric Wilson loops publication-title: Commun. Math. Phys. doi: 10.1007/s00220-012-1485-0 – volume: 16 start-page: 3309 year: 2001 end-page: 3333 ident: CR21 article-title: Hyperbolic structure arising from a knot invariant publication-title: Int. J. Mod. Phys. A doi: 10.1142/S0217751X0100444X – volume: 1108 start-page: 135 year: 2011 ident: CR43 article-title: Chern–Simons, Liouville, and gauge theory on duality walls publication-title: J. High Energy Phys. doi: 10.1007/JHEP08(2011)135 – ident: CR46 – volume: 12 start-page: 873 year: 2000 end-page: 920 ident: CR49 article-title: Quantum exponential function publication-title: Rev. Math. Phys. doi: 10.1142/S0129055X00000344 – volume: 129 start-page: 2035 year: 2009 end-page: 2063 ident: CR42 article-title: On the Barnes double zeta and Gamma functions publication-title: J. Number Theory doi: 10.1016/j.jnt.2009.03.005 – volume: 258 start-page: 257 year: 2005 end-page: 273 ident: CR48 article-title: Noncommutative hypergeometry publication-title: Commun. Math. Phys. doi: 10.1007/s00220-005-1342-5 – ident: CR11 – volume: 224 start-page: 613 year: 2001 end-page: 655 ident: CR35 article-title: Clebsch–Gordan and Racah–Wigner coefficients for a continuous series of representations of publication-title: Commun. Math. Phys. doi: 10.1007/PL00005590 – volume: 325 start-page: 421 year: 2014 end-page: 486 ident: CR41 article-title: Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices publication-title: Commun. Math. Phys. doi: 10.1007/s00220-013-1861-4 – volume: 13 start-page: 379 issue: 2 year: 2005 end-page: 400 ident: CR31 article-title: On the volume of a hyperbolic and spherical tetrahedron publication-title: Commun. Anal. Geom. doi: 10.4310/CAG.2005.v13.n2.a5 – volume: 39 start-page: 269 year: 1997 end-page: 275 ident: CR28 article-title: The hyperbolic volume of knots from the quantum dilogarithm publication-title: Lett. Math. Phys. doi: 10.1023/A:1007364912784 – ident: CR26 – volume: 57 start-page: 1895 year: 2007 end-page: 1940 ident: CR23 article-title: Generalized volume conjecture and the -polynomials – the Neumann–Zagier potential function as a classical limit of quantum invariant publication-title: J. Geom. Phys. doi: 10.1016/j.geomphys.2007.03.008 – volume: 1103 start-page: 127 year: 2011 ident: CR19 article-title: Notes on SUSY gauge theories on three-sphere publication-title: J. High Energy Phys. doi: 10.1007/JHEP03(2011)127 – volume: 70 start-page: 193 year: 1972 end-page: 228 ident: CR6 article-title: Partition function of the eight-vertex lattice model publication-title: Ann. Phys. (NY) doi: 10.1016/0003-4916(72)90335-1 – volume: 13 start-page: 721 year: 2009 end-page: 896 ident: CR18 article-title: -duality of boundary conditions in super Yang–Mills theory publication-title: ATMP – volume: 91 start-page: 167 year: 2010 end-page: 197 ident: CR1 article-title: Liouville correlation functions from four-dimensional gauge theories publication-title: Lett. Math. Phys. doi: 10.1007/s11005-010-0369-5 – volume: 31 start-page: 865 year: 1992 end-page: 902 ident: CR47 article-title: State sum invariants of 3-manifolds and quantum 6 -symbols publication-title: Topology doi: 10.1016/0040-9383(92)90015-A – ident: CR2 – volume: 9 start-page: 427 year: 1994 end-page: 434 ident: CR16 article-title: Quantum dilogarithm publication-title: Mod. Phys. Lett. A doi: 10.1142/S0217732394000447 – ident: CR37 – ident: CR33 – volume: 704 start-page: 234 year: 2011 end-page: 241 ident: CR13 article-title: From 4 superconformal indices to 3 partition functions publication-title: Phys. Lett. B doi: 10.1016/j.physletb.2011.09.007 – volume: 1105 start-page: 014 year: 2011 ident: CR20 article-title: SUSY gauge theories on squashed three-spheres publication-title: J. High Energy Phys. doi: 10.1007/JHEP05(2011)014 – volume: B 784 start-page: 234 year: 2007 end-page: 258 ident: CR3 article-title: Faddeev–Volkov solution of the Yang–Baxter equation and discrete conformal symmetry publication-title: Nucl. Phys. doi: 10.1016/j.nuclphysb.2007.05.013 – volume: 240 start-page: 171 year: 2003 end-page: 196 ident: CR7 article-title: -operator, co-product and Haar-measure for the modular double of publication-title: Commun. Math. Phys. doi: 10.1007/s00220-003-0894-5 – volume: 1106 start-page: 025 year: 2011 ident: CR14 article-title: The virtue of defects in 4 gauge theories and 2 CFTs publication-title: J. High Energy Phys. doi: 10.1007/JHEP06(2011)025 – volume: 3 start-page: 363 year: 2009 end-page: 443 ident: CR10 article-title: Exact results for perturbative Chern–Simons theory with complex gauge group publication-title: Commun. Number Theor. Phys. doi: 10.4310/CNTP.2009.v3.n2.a4 – ident: CR38 – ident: CR17 – volume: 39 start-page: 12927 year: 2006 end-page: 12981 ident: CR8 article-title: Quantization of models with non-compact quantum group symmetry: Modular magnet and lattice sinh-Gordon model publication-title: J. Phys. A doi: 10.1088/0305-4470/39/41/S11 – volume: 34 start-page: 249 year: 1995 end-page: 254 ident: CR15 article-title: Discrete Heisenberg–Weyl group and modular group publication-title: Lett. Math. Phys. doi: 10.1007/BF01872779 – volume: 196 start-page: 265 year: 1901 end-page: 388 ident: CR5 article-title: Theory of the double gamma function publication-title: Philos. Trans. R. Soc. A doi: 10.1098/rsta.1901.0006 – ident: CR34 – volume: 0807 start-page: P07004 year: 2008 ident: CR4 article-title: Quantum geometry of 3-dimensional lattices publication-title: J. Stat. Mech. – volume: 824 start-page: 192 year: 2010 end-page: 216 ident: CR40 article-title: Superconformal indices for theories with multiple duals publication-title: Nucl. Phys. B doi: 10.1016/j.nuclphysb.2009.08.022 – volume: 38 start-page: 1069 year: 1997 end-page: 1146 ident: CR36 article-title: First order analytic difference equations and integrable quantum systems publication-title: J. Math. Phys. doi: 10.1063/1.531809 – volume: 616 start-page: 537 year: 2001 ident: 684_CR22 publication-title: Nucl. Phys. B doi: 10.1016/S0550-3213(01)00464-3 – volume: 10 start-page: 187 year: 2005 ident: 684_CR9 publication-title: Ramanujan J. doi: 10.1007/s11139-005-4846-x – volume: 13 start-page: 721 year: 2009 ident: 684_CR18 publication-title: ATMP – volume: 240 start-page: 171 year: 2003 ident: 684_CR7 publication-title: Commun. Math. Phys. doi: 10.1007/s00220-003-0894-5 – volume: 31 start-page: 865 year: 1992 ident: 684_CR47 publication-title: Topology doi: 10.1016/0040-9383(92)90015-A – volume: 258 start-page: 257 year: 2005 ident: 684_CR48 publication-title: Commun. Math. Phys. doi: 10.1007/s00220-005-1342-5 – ident: 684_CR33 – volume: 39 start-page: 269 year: 1997 ident: 684_CR28 publication-title: Lett. Math. Phys. doi: 10.1023/A:1007364912784 – volume: 0807 start-page: P07004 year: 2008 ident: 684_CR4 publication-title: J. Stat. Mech. – volume: 1003 start-page: 089 year: 2010 ident: 684_CR27 publication-title: J. High Energy Phys. doi: 10.1007/JHEP03(2010)089 – volume: 129 start-page: 2035 year: 2009 ident: 684_CR42 publication-title: J. Number Theory doi: 10.1016/j.jnt.2009.03.005 – volume: 68 start-page: 1795 year: 1992 ident: 684_CR30 publication-title: Phys. Rev. Lett. doi: 10.1103/PhysRevLett.68.1795 – volume: B 784 start-page: 234 year: 2007 ident: 684_CR3 publication-title: Nucl. Phys. doi: 10.1016/j.nuclphysb.2007.05.013 – volume: 325 start-page: 367 year: 2014 ident: 684_CR12 publication-title: Commun. Math. Phys. doi: 10.1007/s00220-013-1863-2 – ident: 684_CR17 – ident: 684_CR38 – ident: 684_CR26 doi: 10.1007/JHEP05(2012)159 – ident: 684_CR46 doi: 10.1007/978-90-481-2810-5_46 – volume: 1106 start-page: 025 year: 2011 ident: 684_CR14 publication-title: J. High Energy Phys. doi: 10.1007/JHEP06(2011)025 – ident: 684_CR34 – volume: 824 start-page: 192 year: 2010 ident: 684_CR40 publication-title: Nucl. Phys. B doi: 10.1016/j.nuclphysb.2009.08.022 – volume: 1108 start-page: 135 year: 2011 ident: 684_CR43 publication-title: J. High Energy Phys. doi: 10.1007/JHEP08(2011)135 – volume: 313 start-page: 71 year: 2012 ident: 684_CR32 publication-title: Commun. Math. Phys. doi: 10.1007/s00220-012-1485-0 – volume: 9 start-page: 427 year: 1994 ident: 684_CR16 publication-title: Mod. Phys. Lett. A doi: 10.1142/S0217732394000447 – ident: 684_CR39 doi: 10.1070/RM2008v063n03ABEH004533 – volume: 18 start-page: R153 year: 2001 ident: 684_CR44 publication-title: Class. Quant. Grav. doi: 10.1088/0264-9381/18/23/201 – volume: 325 start-page: 421 year: 2014 ident: 684_CR41 publication-title: Commun. Math. Phys. doi: 10.1007/s00220-013-1861-4 – ident: 684_CR45 – volume: 196 start-page: 265 year: 1901 ident: 684_CR5 publication-title: Philos. Trans. R. Soc. A doi: 10.1098/rsta.1901.0006 – volume: 34 start-page: 249 year: 1995 ident: 684_CR15 publication-title: Lett. Math. Phys. doi: 10.1007/BF01872779 – ident: 684_CR11 doi: 10.1007/JHEP05(2013)109 – ident: 684_CR29 doi: 10.1007/978-94-010-0670-5_13 – volume: 38 start-page: 1069 year: 1997 ident: 684_CR36 publication-title: J. Math. Phys. doi: 10.1063/1.531809 – ident: 684_CR37 doi: 10.1070/RM2001v056n01ABEH000374 – volume: 16 start-page: 3309 year: 2001 ident: 684_CR21 publication-title: Int. J. Mod. Phys. A doi: 10.1142/S0217751X0100444X – volume: 19 start-page: 987 year: 2013 ident: 684_CR25 publication-title: Selecta Math. doi: 10.1007/s00029-012-0112-4 – volume: 1105 start-page: 014 year: 2011 ident: 684_CR20 publication-title: J. High Energy Phys. doi: 10.1007/JHEP05(2011)014 – volume: 13 start-page: 379 issue: 2 year: 2005 ident: 684_CR31 publication-title: Commun. Anal. Geom. doi: 10.4310/CAG.2005.v13.n2.a5 – ident: 684_CR2 – volume: 1012 start-page: 079 year: 2010 ident: 684_CR24 publication-title: J. High Energy Phys. doi: 10.1007/JHEP12(2010)079 – volume: 1103 start-page: 127 year: 2011 ident: 684_CR19 publication-title: J. High Energy Phys. doi: 10.1007/JHEP03(2011)127 – volume: 704 start-page: 234 year: 2011 ident: 684_CR13 publication-title: Phys. Lett. B doi: 10.1016/j.physletb.2011.09.007 – volume: 57 start-page: 1895 year: 2007 ident: 684_CR23 publication-title: J. Geom. Phys. doi: 10.1016/j.geomphys.2007.03.008 – volume: 224 start-page: 613 year: 2001 ident: 684_CR35 publication-title: Commun. Math. Phys. doi: 10.1007/PL00005590 – volume: 39 start-page: 12927 year: 2006 ident: 684_CR8 publication-title: J. Phys. A doi: 10.1088/0305-4470/39/41/S11 – volume: 3 start-page: 363 year: 2009 ident: 684_CR10 publication-title: Commun. Number Theor. Phys. doi: 10.4310/CNTP.2009.v3.n2.a4 – volume: 12 start-page: 873 year: 2000 ident: 684_CR49 publication-title: Rev. Math. Phys. doi: 10.1142/S0129055X00000344 – volume: 91 start-page: 167 year: 2010 ident: 684_CR1 publication-title: Lett. Math. Phys. doi: 10.1007/s11005-010-0369-5 – volume: 70 start-page: 193 year: 1972 ident: 684_CR6 publication-title: Ann. Phys. (NY) doi: 10.1016/0003-4916(72)90335-1 |
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| Snippet | We revisit the definition of the 6
j
symbols from the modular double of
U
q
(
sl
(
2
,
R
)
)
, referred to as b-6
j
symbols. Our new results are (1) the... |
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| SubjectTerms | Complex Systems Geometry Group Theory and Generalizations Mathematical and Computational Physics Physics Physics and Astronomy Theoretical |
| Title | 6j Symbols for the Modular Double, Quantum Hyperbolic Geometry, and Supersymmetric Gauge Theories |
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