Abstract
We study simplicial quantum gravity in three dimensions. Motivated by Boulatov’s model which generates a sum over simplicial complexes weighted with the Turaev-Viro invariant, we introduce boundary operators in the simplicial gravity associated to compact orientable surfaces. An amplitude of the boundary operator is given by a sum over triangulations in the interior of the boundary surface. It turns out that the amplitude solves the Schwinger-Dyson equation even if we restrict the topology in the interior of the surface, so long as the surface is non-degenerate. We propose a set of factorization conditions on the amplitudes which singles out a solution associated to triangulations of S 3.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. Regge, Nuovo Cimento 19 (1961) 558.
For a review on recent numerical results in this approach, see H. Hamber, Nucl. Phys. (Proc. Suppl.) 25A (1992) 150.
W.T. Tutte, Can. J. Math. 14 (1962) 21.
E. Brézin, C. Itzykson, G. Parisi and J.-B. Zuber, Commun. Math. Phys. 59 (1978) 35.
T. Eguchi and H. Kawai, Phys. Lett. 114B (1982) 247.
F. David, Nucl. Phys. B257 (1985) 45.
V.A. Kazakov, Phys. Lett. 150B (1985) 282.
D. Bessis, C. Itzykson and J.-B. Zuber, Adv. Appl. Math. 1 (1980) 109.
I. Rostov and M.L. Metha, Phys. Lett. 189B (1987) 118.
V.G. Knizhnik, A.M. Polyakov and A.B. Zamolodchikov, Mod. Phys. Lett. A3 (1988) 819.
F. David, Mod. Phys. Lett. A3 (1988) 1651.
J. Distler and H. Kawai, Nucl. Phys B321 (1989) 509.
M. Douglas and S. Shenker, Nucl. Phys. B335 (1990) 635.
D.J. Gross and A.A. Migdal, Phys. Rev. Lett. 64 (1990) 127.
E. Brezin and V.A. Kazakov, Phys. Lett. B236 (1990) 144.
E. Witten, Nucl. Phys. B340 (1990) 281.
R. Dijkgraaf and E. Witten, Nucl. Phys. B342 (1990) 342.
R. Dijkgraaf, E. Verlinde and H. Verlinde, Nucl. Phys. B348 (1991) 435.
E. Verlinde and H. Verlinde, Nucl. Phys. B348 (1991) 457.
M. Fukuma, H. Kawai and R. Nakayama, Int. J. Mod. Phys. A6 (1991) 1385.
G. Moore, N. Seiberg and M. Staudacher, Nucl. Phys. B362 (1991) 665.
G. Moore and N. Seiberg, Int. J. Mod. Phys. A7 (1991) 190.
J. Ambjørn and S. Varsted, Phys. Lett. 266B (1991) 285.
M.E. Agishtein and A.A. Migdal, Mod. Phys. Lett. A6 (1991) 1863.
D.V. Boulatov and A. Krywicki Mod. Phys. Lett. A6 (1991) 3005.
J. Ambjørn and S. Varsted, Nucl. Phys. B373 (1992) 557.
J. Ambjørn, D.V. Boulatov, A. Krywicki and S. Varsted, Phys. Lett. 276B (1992) 432.
J. Ambjørn, B. Durhuus and T. Jonsson, Mod. Phys. Lett. A6 (1991) 1133.
N. Sasakura, Mod. Phys. Lett. A6 (1991) 2613.
M. Godfrey and M. Gross, Phys. Rev. D43 (1992) R1749.
D.V. Boulatov, Mod. Phys. Lett. A7 (1992) 1629.
V.G. Turaev and O.Y. Viro, Toplogy 31 (1992) 865.
S.R. Wadia, Phys. Rev. D24 (1981) 970.
A.A. Migdal, Phys. Rep. 102 (1983) 199.
A.N. Kirillov and N.Yu. Reshetikhin, in Infinite Dimensional Lie Algebras and Groups, edited by V.G. Kac (World Scientific, 1989).
V.G. Turaev, C. R. Acad. Sci. Paris, t. 313, Série I (1991) 395; “Topology of Shadow”, preprint (1991).
K. Walker, “On Witten’s 3-Manifold Invariants”, preprint (1990).
E. Witten, Commun. Math. Phys. 121 (1989) 3
N.Y. Reshetikhin and V.G. Turaev, Invent. Math. 103 (1991) 547.
H. Ooguri and N. Sasakura, Mod. Phys. Lett. A6 (1991) 3591.
H. Ooguri, Nucl. Phys. B382 (1992) 276.
F. Archer and R. Williams, Phys. Lett. B273 (1991) 438.
S. Mizoguchi and T. Tada, Phys. Rev. Lett. 68 (1992) 1795.
T. Kohno, Topology 31 (1992) 203.
F. Waldhausen, Topology 7 (1968) 195.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Science+Business Media New York
About this chapter
Cite this chapter
Ooguri, H. (1993). Schwinger-Dyson Equation in Three-Dimensional Simplicial Quantum Gravity. In: Osborn, H. (eds) Low-Dimensional Topology and Quantum Field Theory. NATO ASI Series, vol 315. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1612-9_4
Download citation
DOI: https://doi.org/10.1007/978-1-4899-1612-9_4
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4899-1614-3
Online ISBN: 978-1-4899-1612-9
eBook Packages: Springer Book Archive