메뉴 건너뛰기




Volumn 38, Issue 2, 2001, Pages 81-139

Statistical mechanics of 2+1 gravity from Riemann zeta function and Alexander polynomial: Exact results

Author keywords

83C45; Dynamical systems; Gravity; Quantum gravity; Surface automorphisms

Indexed keywords


EID: 0009442787     PISSN: 03930440     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0393-0440(00)00049-8     Document Type: Article
Times cited : (9)

References (111)
  • 4
    • 0032647538 scopus 로고    scopus 로고
    • The Riemann zeros and eigenvalue asymptotics
    • Berry M., Keating J. The Riemann zeros and eigenvalue asymptotics. SIAM Rev. 41:1999;236-266.
    • (1999) SIAM Rev. , vol.41 , pp. 236-266
    • Berry, M.1    Keating, J.2
  • 5
    • 53149152987 scopus 로고    scopus 로고
    • Trace formula in noncommutative geometry and zeros of the Riemann zeta function
    • Connes A. Trace formula in noncommutative geometry and zeros of the Riemann zeta function. Sel. Math. New Ser. 5:1999;23-106.
    • (1999) Sel. Math. New Ser. , vol.5 , pp. 23-106
    • Connes, A.1
  • 6
    • 36149023450 scopus 로고
    • Statistical theory of equations of state and phase transitions
    • Yang C., Lee T. Statistical theory of equations of state and phase transitions. Phys. Rev. 87:1952;404-419.
    • (1952) Phys. Rev. , vol.87 , pp. 404-419
    • Yang, C.1    Lee, T.2
  • 7
    • 21144479691 scopus 로고
    • On a ferromagnetic spin chain
    • Knauf A. On a ferromagnetic spin chain. Commun. Math. Phys. 153:1993;77-115.
    • (1993) Commun. Math. Phys. , vol.153 , pp. 77-115
    • Knauf, A.1
  • 8
    • 0031512851 scopus 로고    scopus 로고
    • The phase transition of the number-theoretic spin chain
    • Kontucchi A., Knauf A. The phase transition of the number-theoretic spin chain. Forum Mathematicum. 9:1997;547-567.
    • (1997) Forum Mathematicum , vol.9 , pp. 547-567
    • Kontucchi, A.1    Knauf, A.2
  • 9
    • 21344481086 scopus 로고
    • Phases of the number-theoretic spin chain
    • Knauf A. Phases of the number-theoretic spin chain. J. Statist. Phys. 73:1993;423-431.
    • (1993) J. Statist. Phys. , vol.73 , pp. 423-431
    • Knauf, A.1
  • 10
    • 0037583140 scopus 로고
    • Scattering theory and automorphic functions
    • Pavlov B., Faddeev L. Scattering theory and automorphic functions. J. Sov. Math. 3:1975;522-548.
    • (1975) J. Sov. Math. , vol.3 , pp. 522-548
    • Pavlov, B.1    Faddeev, L.2
  • 12
    • 0000291634 scopus 로고
    • Stochastic behavior in quantum scattering
    • Gutzwiller M. Stochastic behavior in quantum scattering. Physica D. 7:1983;341-355.
    • (1983) Physica D , vol.7 , pp. 341-355
    • Gutzwiller, M.1
  • 13
    • 0009450293 scopus 로고
    • Phase transitions on strange irrational sets
    • Artuso R., Cvitanovic P., Kenny B. Phase transitions on strange irrational sets. Phys. Rev. A. 39:1989;268-281.
    • (1989) Phys. Rev. a , vol.39 , pp. 268-281
    • Artuso, R.1    Cvitanovic, P.2    Kenny, B.3
  • 14
    • 0007140841 scopus 로고
    • Circle maps: Irrationally winding
    • in: M. Waldschmidt, P. Moussa, J.-M. Luck, C. Itzykson (Eds.), Springer, Berlin
    • P. Cvitanovic, Circle maps: irrationally winding, in: M. Waldschmidt, P. Moussa, J.-M. Luck, C. Itzykson (Eds.), From Number Theory to Physics, Springer, Berlin, 1992.
    • (1992) From Number Theory to Physics
    • Cvitanovic, P.1
  • 15
    • 0034148604 scopus 로고    scopus 로고
    • Use of meanders and train tracks for description of defects and textures in liquid crystals and 2+1 gravity
    • Kholodenko A. Use of meanders and train tracks for description of defects and textures in liquid crystals and. 2+1 gravity J. Geom. Phys. 33:2000;23-58.
    • (2000) J. Geom. Phys. , vol.33 , pp. 23-58
    • Kholodenko, A.1
  • 16
    • 0034148318 scopus 로고    scopus 로고
    • Use of quadratic differentials for description of defects and textures in liquid crystals and 2+1 gravity
    • Kholodenko A. Use of quadratic differentials for description of defects and textures in liquid crystals and. 2+1 gravity J. Geom. Phys. 33:2000;59-102.
    • (2000) J. Geom. Phys. , vol.33 , pp. 59-102
    • Kholodenko, A.1
  • 17
  • 22
    • 33646668033 scopus 로고
    • Metastability and phase transitions in two-dimensional systems
    • Kosterlitz J., Thouless D. Metastability and phase transitions in two-dimensional systems. J. Phys. C. 6:1973;1181-1203.
    • (1973) J. Phys. C , vol.6 , pp. 1181-1203
    • Kosterlitz, J.1    Thouless, D.2
  • 23
    • 0002692928 scopus 로고
    • Trace maps
    • in: F. Axel, D. Gratias (Eds.), Springer, Berlin
    • J. Peyriere, Trace maps, in: F. Axel, D. Gratias (Eds.), Beyond Quasicrystals, Springer, Berlin, 1995, pp. 465-480.
    • (1995) Beyond Quasicrystals , pp. 465-480
    • Peyriere, J.1
  • 24
    • 0009378748 scopus 로고    scopus 로고
    • Boundary conformal field theories, limit sets of Kleinian groups and holography
    • A. Kholodenko, Boundary conformal field theories, limit sets of Kleinian groups and holography, J. Geom. Phys. 35 (2000) 193-238.
    • (2000) J. Geom. Phys. , vol.35 , pp. 193-238
    • Kholodenko, A.1
  • 25
    • 0001155283 scopus 로고
    • The arithmetic and geometry of some hyperbolic 3-manifolds
    • Sarnak P. The arithmetic and geometry of some hyperbolic 3-manifolds. Acta Math. 151:1983;253-295.
    • (1983) Acta Math. , vol.151 , pp. 253-295
    • Sarnak, P.1
  • 26
    • 84968469826 scopus 로고
    • The determinant of the Eisenstein matrix and Hilbert class fields
    • Efrat I., Sarnak P. The determinant of the Eisenstein matrix and Hilbert class fields. AMS Trans. 290:1985;815-824.
    • (1985) AMS Trans. , vol.290 , pp. 815-824
    • Efrat, I.1    Sarnak, P.2
  • 27
    • 0000762274 scopus 로고
    • Sui gruppi de sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginari
    • Bianchi L. Sui gruppi de sostituzioni lineari con coefficienti appartenenti a corpi quadratici immaginari. Math. Ann. 40:1892;332-412.
    • (1892) Math. Ann. , vol.40 , pp. 332-412
    • Bianchi, L.1
  • 28
    • 0039549424 scopus 로고
    • An elliptical path from parabolic representations to hyperbolic structures
    • Riley R. An elliptical path from parabolic representations to hyperbolic structures. Lect. Notes Math. 722:1979;99-133.
    • (1979) Lect. Notes Math. , vol.722 , pp. 99-133
    • Riley, R.1
  • 29
    • 0040757095 scopus 로고
    • Arithmetic properties of discrete groups
    • Margulis G. Arithmetic properties of discrete groups. Russ. Math. Surv. 29:1974;49-98.
    • (1974) Russ. Math. Surv. , vol.29 , pp. 49-98
    • Margulis, G.1
  • 32
    • 84971137955 scopus 로고
    • Pseudo-Anosov homeomorphisms on a sphere with four punctures have all periods
    • Libre J., Mackey R. Pseudo-Anosov homeomorphisms on a sphere with four punctures have all periods. Math. Proc. Camb. Phil. Soc. 112:1992;539-549.
    • (1992) Math. Proc. Camb. Phil. Soc. , vol.112 , pp. 539-549
    • Libre, J.1    Mackey, R.2
  • 33
    • 0346624074 scopus 로고
    • An introduction to train tracks
    • Penner R. An introduction to train tracks. LMS Lect. Notes Ser. 112:1986;77-90.
    • (1986) LMS Lect. Notes Ser. , vol.112 , pp. 77-90
    • Penner, R.1
  • 34
    • 0012755709 scopus 로고    scopus 로고
    • Markov triples and quasi-Fuchsian groups
    • Bowdich B. Markov triples and quasi-Fuchsian groups. Proc. London Math. Soc. 77:1998;697-736.
    • (1998) Proc. London Math. Soc. , vol.77 , pp. 697-736
    • Bowdich, B.1
  • 36
    • 0031094096 scopus 로고    scopus 로고
    • A variation of McShane's identity for once punctures torus bundles
    • Bowdich B. A variation of McShane's identity for once punctures torus bundles. Topology. 36:1997;325-334.
    • (1997) Topology , vol.36 , pp. 325-334
    • Bowdich, B.1
  • 38
    • 0002554739 scopus 로고
    • Approach to Markoff's minimal forms through modular functions
    • Cohn H. Approach to Markoff's minimal forms through modular functions. Ann. Math. 61:1955;1-12.
    • (1955) Ann. Math. , vol.61 , pp. 1-12
    • Cohn, H.1
  • 39
    • 0030164711 scopus 로고    scopus 로고
    • Escaping orbits and trace maps
    • Roberts J. Escaping orbits and trace maps. Physica A. 228:1996;295-325.
    • (1996) Physica A , vol.228 , pp. 295-325
    • Roberts, J.1
  • 41
    • 84968517859 scopus 로고
    • Bounds on the least dilatations
    • Penner R. Bounds on the least dilatations. AMS Proc. 113:1991;443-450.
    • (1991) AMS Proc. , vol.113 , pp. 443-450
    • Penner, R.1
  • 42
    • 0002594183 scopus 로고
    • Diophantine approximation on hyperbolic Riemann surfaces
    • Haas A. Diophantine approximation on hyperbolic Riemann surfaces. Acta Math. 156:1986;33-82.
    • (1986) Acta Math. , vol.156 , pp. 33-82
    • Haas, A.1
  • 43
    • 0009362511 scopus 로고    scopus 로고
    • The boundary of the Giseking tree in hyperbolic three space
    • Alperin R., Dicks W., Porti J. The boundary of the Giseking tree in hyperbolic three space. Topol. Appl. 93:1999;219-259.
    • (1999) Topol. Appl. , vol.93 , pp. 219-259
    • Alperin, R.1    Dicks, W.2    Porti, J.3
  • 45
    • 0002189261 scopus 로고
    • The modular surface and continued fractions
    • Series C. The modular surface and continued fractions. J. London Math. Soc. 31:1985;69-80.
    • (1985) J. London Math. Soc. , vol.31 , pp. 69-80
    • Series, C.1
  • 46
    • 0033468108 scopus 로고    scopus 로고
    • The classification of punctured torus groups
    • Minsky Y. The classification of punctured torus groups. Ann. Math. 149:1999;559-626.
    • (1999) Ann. Math. , vol.149 , pp. 559-626
    • Minsky, Y.1
  • 48
    • 0347884486 scopus 로고    scopus 로고
    • Mathematical microcosm of geodesics, free groups and Markoff forms
    • Cohn H. Mathematical microcosm of geodesics, free groups and Markoff forms. LNPAM. 149:1999;559-626.
    • (1999) LNPAM , vol.149 , pp. 559-626
    • Cohn, H.1
  • 54
    • 33744722968 scopus 로고
    • 2+1-dimensional gravity as exactly soluble problem
    • Witten E. 2+1 -dimensional gravity as exactly soluble problem Nucl. Phys. B. 311:1988;46-78.
    • (1988) Nucl. Phys. B , vol.311 , pp. 46-78
    • Witten, E.1
  • 56
    • 0000286629 scopus 로고    scopus 로고
    • Some geometrical and topological problems in polymer physics
    • Kholodenko A., Vilgis Th. Some geometrical and topological problems in polymer physics. Phys. Rep. 298:1998;251-370.
    • (1998) Phys. Rep. , vol.298 , pp. 251-370
    • Kholodenko, A.1    Vilgis, Th.2
  • 60
    • 0041140805 scopus 로고    scopus 로고
    • Le theoreme d'hyperbolization pour les varietes fibres de dimension 3
    • Otal J.-P. Le theoreme d'hyperbolization pour les varietes fibres de dimension 3. Asterisque. 235:1996;1-159.
    • (1996) Asterisque , vol.235 , pp. 1-159
    • Otal, J.-P.1
  • 61
    • 0345992936 scopus 로고
    • Fibered knots with given Alexander polynomial
    • Morton H. Fibered knots with given Alexander polynomial. Enseignment Math. 31:1983;205-222.
    • (1983) Enseignment Math. , vol.31 , pp. 205-222
    • Morton, H.1
  • 62
    • 0347243300 scopus 로고    scopus 로고
    • On the relation between 2+1 Einstein gravity and Chern-Simons theory
    • Matschul H. On the relation between. 2+1 Einstein gravity and Chern-Simons theory Class. Quant. Gravity. 16:1999;2599-2609.
    • (1999) Class. Quant. Gravity , vol.16 , pp. 2599-2609
    • Matschul, H.1
  • 64
    • 0000581872 scopus 로고
    • Seifert fibered spaces in 3-manifolds
    • Jaco W., Shalen P. Seifert fibered spaces in 3-manifolds. AMS Memoirs. 21:1979;1-292.
    • (1979) AMS Memoirs , vol.21 , pp. 1-292
    • Jaco, W.1    Shalen, P.2
  • 65
    • 0003348894 scopus 로고
    • Seifert Manifolds
    • Orlik P. Seifert Manifolds. Lect. Notes Math. 291:1972;1-155.
    • (1972) Lect. Notes Math. , vol.291 , pp. 1-155
    • Orlik, P.1
  • 66
    • 0041842002 scopus 로고
    • Deformation spaces for Seifert manifolds
    • Kulkarni R., Lee K., Raymond F. Deformation spaces for Seifert manifolds. Lect. Notes Math. 1167:1985;180-216.
    • (1985) Lect. Notes Math. , vol.1167 , pp. 180-216
    • Kulkarni, R.1    Lee, K.2    Raymond, F.3
  • 69
    • 34250141655 scopus 로고
    • Incompressible branched surfaces
    • Oertel U. Incompressible branched surfaces. Invent. Math. 76:1984;385-410.
    • (1984) Invent. Math. , vol.76 , pp. 385-410
    • Oertel, U.1
  • 72
    • 0001721424 scopus 로고
    • Incompressible surfaces in once punctured torus bundles
    • Culler M., Jaco W., Rubinstein H. Incompressible surfaces in once punctured torus bundles. Proc. London Math. Soc. 45:1982;385-419.
    • (1982) Proc. London Math. Soc. , vol.45 , pp. 385-419
    • Culler, M.1    Jaco, W.2    Rubinstein, H.3
  • 73
    • 49049130340 scopus 로고
    • Incompressible surfaces in punctured torus bundles
    • Floyd W., Hatcher A. Incompressible surfaces in punctured torus bundles. Topol. Appl. 13:1982;263-282.
    • (1982) Topol. Appl. , vol.13 , pp. 263-282
    • Floyd, W.1    Hatcher, A.2
  • 75
    • 0347884487 scopus 로고    scopus 로고
    • Notes on geometry and 3-manifolds in Low-dimensional Topology
    • Budapest
    • W. Neumann, Notes on geometry and 3-manifolds in Low-dimensional Topology, Journal of Bolyai Mathematical Society, Budapest, 1999.
    • (1999) Journal of Bolyai Mathematical Society
    • Neumann, W.1
  • 76
    • 84972500142 scopus 로고
    • On the boundary curves of incompressible surfaces
    • Hatcher A. On the boundary curves of incompressible surfaces. Pacific J. Math. 99:1982;373-377.
    • (1982) Pacific J. Math. , vol.99 , pp. 373-377
    • Hatcher, A.1
  • 78
    • 0008951344 scopus 로고
    • Fermi-Bose transmutation: From semiflexible polymers to superstrings
    • Kholodenko A. Fermi-Bose transmutation: from semiflexible polymers to superstrings. Ann. Phys. 202:1990;186-225.
    • (1990) Ann. Phys. , vol.202 , pp. 186-225
    • Kholodenko, A.1
  • 80
    • 0040947147 scopus 로고
    • Incompressibility of surfaces after Dehn surgery
    • Przytycki J. Incompressibility of surfaces after Dehn surgery. Michigan Math. J. 30:1983;289-308.
    • (1983) Michigan Math. J. , vol.30 , pp. 289-308
    • Przytycki, J.1
  • 82
    • 0033242960 scopus 로고    scopus 로고
    • A Farey fraction spin chain
    • Kleban P., Qzluk A. A Farey fraction spin chain. Commun. Math. Phys. 203:1999;635-647.
    • (1999) Commun. Math. Phys. , vol.203 , pp. 635-647
    • Kleban, P.1    Qzluk, A.2
  • 83
    • 0009364733 scopus 로고    scopus 로고
    • The universal Ptolemy group and its completions
    • Penner R. The universal Ptolemy group and its completions. London Math. Soc. Lect. Notes Ser. 243:1997;293-312.
    • (1997) London Math. Soc. Lect. Notes Ser. , vol.243 , pp. 293-312
    • Penner, R.1
  • 84
    • 0029681224 scopus 로고    scopus 로고
    • A proof of McShanes identity via Markoff triples
    • Bowditch B. A proof of McShanes identity via Markoff triples. Bull. London Math. Soc. 28:1996;73-78.
    • (1996) Bull. London Math. Soc. , vol.28 , pp. 73-78
    • Bowditch, B.1
  • 86
    • 0004117363 scopus 로고
    • The University of Chicago Press, Chicago
    • S. Katok, Fuchsian Groups, The University of Chicago Press, Chicago, 1992.
    • (1992) Fuchsian Groups
    • Katok, S.1
  • 87
    • 34250224475 scopus 로고
    • Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics
    • Sullivan D. Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics. Acta Math. 149:1982;215-237.
    • (1982) Acta Math. , vol.149 , pp. 215-237
    • Sullivan, D.1
  • 88
    • 0033276293 scopus 로고    scopus 로고
    • Equidistribution of parabolic fixed points in the limit set of Kleinian groups
    • Cosentino S. Equidistribution of parabolic fixed points in the limit set of Kleinian groups. Ergod. Theory Dyn. Syst. 19:1999;1437-1484.
    • (1999) Ergod. Theory Dyn. Syst. , vol.19 , pp. 1437-1484
    • Cosentino, S.1
  • 89
    • 0000144097 scopus 로고
    • Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups
    • Sullivan D. Entropy, Hausdorff measures old and new, and limit sets of geometrically finite Kleinian groups. Acta Math. 153:1984;259-277.
    • (1984) Acta Math. , vol.153 , pp. 259-277
    • Sullivan, D.1
  • 92
    • 0001907888 scopus 로고
    • Spectral theory of automorphic functions, Selberg's Zeta function and some problems of analytical number theory and mathematical physics
    • Venkov A. Spectral theory of automorphic functions, Selberg's Zeta function and some problems of analytical number theory and mathematical physics. Russ. Math. Surv. 34:1979;69-135.
    • (1979) Russ. Math. Surv. , vol.34 , pp. 69-135
    • Venkov, A.1
  • 93
    • 0000459324 scopus 로고
    • Volumes of hyperbolic three manifolds
    • Neumann W., Zagier D. Volumes of hyperbolic three manifolds. Topology. 24:1985;307-332.
    • (1985) Topology , vol.24 , pp. 307-332
    • Neumann, W.1    Zagier, D.2
  • 94
    • 0001772981 scopus 로고
    • Arithmetic hyperbolic surface bundles
    • Bowditch B., McLachlan C., Reid A. Arithmetic hyperbolic surface bundles. Math. Ann. 302:1995;31-60.
    • (1995) Math. Ann. , vol.302 , pp. 31-60
    • Bowditch, B.1    McLachlan, C.2    Reid, A.3
  • 95
    • 0039114135 scopus 로고
    • Link complements arising from arithmetic group actions
    • Grunewald F., Hirsch U. Link complements arising from arithmetic group actions. Int. J. Math. 6:1995;337-370.
    • (1995) Int. J. Math. , vol.6 , pp. 337-370
    • Grunewald, F.1    Hirsch, U.2
  • 96
    • 0001564601 scopus 로고
    • Arithmeticity of knot complements
    • Reid A. Arithmeticity of knot complements. J. London Math. Soc. 43:1991;171-184.
    • (1991) J. London Math. Soc. , vol.43 , pp. 171-184
    • Reid, A.1
  • 103
    • 0346624072 scopus 로고
    • Princeton University Press, Princeton, NJ
    • H. Weil, Algebraic Theory of Numbers, Princeton University Press, Princeton, NJ, 1940.
    • (1940) Algebraic Theory of Numbers
    • Weil, H.1
  • 104
    • 0004257005 scopus 로고    scopus 로고
    • Cambridge University Press, Cambridge
    • H. McKean, V. Moll, Elliptic Curves, Cambridge University Press, Cambridge, 1999.
    • (1999) Elliptic Curves
    • McKean, H.1    Moll, V.2
  • 108
    • 21344485562 scopus 로고
    • What have we learned from two-dimensional models of quantum black holes?
    • Teo E. What have we learned from two-dimensional models of quantum black holes? Gen. Relativity and Gravitation. 26:1994;13-19.
    • (1994) Gen. Relativity and Gravitation , vol.26 , pp. 13-19
    • Teo, E.1
  • 109
    • 84975966715 scopus 로고
    • A quadratic parabolic group
    • Riley A. A quadratic parabolic group. Math. Soc. Camb. Phil. Soc. 77:1975;281-288.
    • (1975) Math. Soc. Camb. Phil. Soc. , vol.77 , pp. 281-288
    • Riley, A.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.