×

Geometric topology and field theory on 3-manifolds. (English) Zbl 1221.57023

Banagl, Markus (ed.) et al., The mathematics of knots. Theory and application. Berlin: Springer (ISBN 978-3-642-15636-6/hbk; 978-3-642-15637-3/ebook). Contributions in Mathematical and Computational Sciences 1, 199-256 (2011).
The author describes the interaction between geometric topology and classical and quantum field theories concentrating on the low dimensional topology of 3-manifolds. The article on geometric topology and field theory on 3-manifolds provides a survey of this area without proofs. Closely related to the topics here are the articles [K. Marathe, Topological quantum field theory as topological quantum gravity. Fauser, Bertfried (ed.) et al., Quantum gravity. Mathematical models and experimental bounds. Papers based on the presentations at the workshop ‘Mathematical and physical aspects of quantum gravity’, Blaubeuren, Germany, July 28th–August 1st, 2005. Basel: Birkhäuser. 221–235 (2007; Zbl 1120.83023)] and the textbook [K. Marathe, Topics in physical mathematics. London: Springer. (2007; Zbl 1291.58001)]. A wide variety of topics are covered, beginning with Gauss’s formula for the linking number as a first example of topological field theory and ending with a discussion of Perelman’s proof of Thurston’s Geometrization conjecture. Other topics covered are Chern Simons theory, knot polynomials and their categorification by Khovanov, Whitten-Reshetikhin-Turaev invariants, Fukaya-Floer homology and topological quantum field theory.
For the entire collection see [Zbl 1205.57002].

MSC:

57M27 Invariants of knots and \(3\)-manifolds (MSC2010)
57R56 Topological quantum field theories (aspects of differential topology)
58J28 Eta-invariants, Chern-Simons invariants
81T13 Yang-Mills and other gauge theories in quantum field theory
81T45 Topological field theories in quantum mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aganagic, M.; Mari no, M.; Vafa, C., All loop topological string amplitudes from Chern-Simons theory, Commun. Math. Phys., 247, 467-512 (2004) · Zbl 1055.81055 · doi:10.1007/s00220-004-1067-x
[2] Atiyah, M. F., Topological quantum field theories, Publ. Math. Inst. Hautes Etud. Sci., 68, 175-186 (1989) · Zbl 0692.53053 · doi:10.1007/BF02698547
[3] Atiyah, M. F.; Bott, R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. R. Soc. Lond. A, 308, 523-615 (1982) · Zbl 0509.14014 · doi:10.1098/rsta.1983.0017
[4] Axelrod, S.; Singer, I., Chern-Simons perturbation theory, J. Differ. Geom., 39, 787-902 (1994) · Zbl 0889.53053
[5] Bar-Natan, D.: Perturbative aspects of the Chern-Simons topological quantum field theory. PhD thesis, Princeton University (1991) · Zbl 0738.53041
[6] Bar-Natan, D., On Khovanov’s categorification of the Jones polynomial, Algebraic. & Geom. Topol., 2, 337-370 (2002) · Zbl 0998.57016 · doi:10.2140/agt.2002.2.337
[7] Besse, A., Einstein Manifolds (1986), Berlin: Springer, Berlin
[8] Blanchet, C., Hecke algebras, modular categories and 3-manifolds quantum invariants, Topology, 39, 193-223 (2000) · Zbl 0938.57009 · doi:10.1016/S0040-9383(98)00066-4
[9] Borcherds, R. E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math., 109, 405-444 (1992) · Zbl 0799.17014 · doi:10.1007/BF01232032
[10] Bott, R.; Cattaneo, A. S., Integral invariants of 3-manifolds, J. Differ. Geom., 48, 357-361 (1998)
[11] Bott, R.; Cattaneo, A. S., Integral invariants of 3-manifolds. II, J. Differ. Geom., 53, 1-13 (1999) · Zbl 1036.57500
[12] Bott, R.; Taubes, C., On the self-linking of knots, J. Math. Phys., 35, 5247-5287 (1994) · Zbl 0863.57004 · doi:10.1063/1.530750
[13] Canarutto, D., Marathe’s generalized gravitational fields and singularities, Nuovo Cimento, 75B, 134-144 (1983)
[14] Cao, H.-D.; Zhu, X.-P., A complete proof of the Poincaré and geometrization conjectures—application of the Hamilton-Perelman theory of the Ricci flow, Asian J. Math., 10, 2, 165-492 (2006) · Zbl 1200.53057
[15] Chang, K. C., Infinite Dimensional Morse Theory and Multiple Solution Problems (1993), Boston: Birkhäuser, Boston · Zbl 0779.58005
[16] Crane, L., 2-d physics and 3-d topology, Commun. Math. Phys., 135, 615-640 (1991) · Zbl 0717.57007 · doi:10.1007/BF02104124
[17] Dijkgraaf, R., Fuji, H.: The volume conjecture and topological strings. arXiv:0903.2084 [hep-th] (2009) · Zbl 1210.81081
[18] Dijkgraaf, R.; Witten, E., Topological gauge theories and group cohomology, Commun. Math. Phys., 129, 393-429 (1990) · Zbl 0703.58011 · doi:10.1007/BF02096988
[19] Dimofte, T., Gukov, S., Lenells, J., Zagier, D.: Exact results for perturbative Chern-Simons theory with complex gauge group. arXiv:0903.2472v1 (2009) · Zbl 1214.81151
[20] Dunfield, N.M., Gukov, S., Rasmussen, J.: The superpolynomial for knot homologies. arXiv:math/0505662v2 [math.GT] (2005) · Zbl 1118.57012
[21] Eilenberg, S.; Steenrod, N., Foundations on Algebraic Topology (1952), Princeton: Princeton University Press, Princeton · Zbl 0047.41402
[22] Fintushel, R.; Stern, R., Instanton homology of Seifert fibered homology three spheres, Proc. Lond. Math. Soc., 61, 109-137 (1990) · Zbl 0705.57009 · doi:10.1112/plms/s3-61.1.109
[23] Floer, A., An instanton-invariant for 3-manifolds, Commun. Math. Phys., 118, 215-240 (1988) · Zbl 0684.53027 · doi:10.1007/BF01218578
[24] Floer, A., Witten’s complex and infinite dimensional Morse theory, J. Differ. Geom., 30, 207-221 (1989) · Zbl 0678.58012
[25] Frenkel, I.; Lepowsky, J.; Meurman, A., Vertex Operator Algebras and the Monster (1988), New York: Academic Press, New York · Zbl 0674.17001
[26] Gopakumar, R., Vafa, C.: M-theory and topological strings—I. arXiv:hep-th/9809187v1 (1998) · Zbl 0922.32015
[27] Gopakumar, R., Vafa, C.: M-theory and topological strings—II. arXiv:hep-th/9812127v1 (1998) · Zbl 0922.32015
[28] Gopakumar, R.; Vafa, C., On the gauge theory/geometry correspondence, Adv. Theor. Math. Phys., 3, 1415-1443 (1999) · Zbl 0972.81135
[29] Guillemin, V. W.; Sternberg, S., Supersymmetry and Equivariant de Rham Theory (1999), Berlin: Springer, Berlin · Zbl 0934.55007
[30] Gukov, S., Witten, E.: Branes and quantization. arXiv:0809.0305v2 [hep-th] (2008) · Zbl 1247.81378
[31] Hamilton, R. S., Three manifolds with positive Ricci curvature, J. Differ. Geom., 17, 255-306 (1982) · Zbl 0504.53034
[32] Hamilton, R. S., Four manifolds with positive curvature operator, J. Differ. Geom., 24, 153-179 (1986) · Zbl 0628.53042
[33] Harer, J.; Zagier, D., The Euler characteristic of the moduli space of curves, Invent. Math., 85, 457-485 (1986) · Zbl 0616.14017 · doi:10.1007/BF01390325
[34] Hickerson, D., A proof of the mock theta conjectures, Invent. Math., 94, 639-660 (1988) · Zbl 0661.10059 · doi:10.1007/BF01394279
[35] Kauffman, L. H., State models and the Jones polynomial, Topology, 26, 395-407 (1987) · Zbl 0622.57004 · doi:10.1016/0040-9383(87)90009-7
[36] Kauffman, L. H., Statistical mechanics and the Jones polynomial, Braids Contemp. Math. Pub. 78, 263-297 (1988), Providence: Am. Math. Soc., Providence · Zbl 0664.57002
[37] Kauffman, L. H., Knots and Physics (1991), Singapore: World Scientific, Singapore · Zbl 0733.57004 · doi:10.1142/9789812796226
[38] Khovanov, M., A categorification of the Jones polynomial, Duke Math. J., 101, 359-426 (2000) · Zbl 0960.57005 · doi:10.1215/S0012-7094-00-10131-7
[39] Kirby, R.; Melvin, P.; Donaldson, S. K.; Thomas, C. B., Evaluation of the 3-manifold invariants of Witten and Reshetikhin-Turaev, Geometry of Low-dimensional Manifolds, 101-114 (1990), London: Lond. Math. Soc., London · Zbl 0756.57006
[40] Kirby, R.; Melvin, P., The 3-manifold invariants of Witten and Reshetikhin-Turaev for sl(2,ℂ), Inven. Math., 105, 473-545 (1991) · Zbl 0745.57006 · doi:10.1007/BF01232277
[41] Kirillov, A. N.; Reshetikhin, N. Y.; Kac, V. G., Representations of the algebra \(u_q(sl(2, \c{)})\), q-orthogonal polynomials and invariants of links, Infinite Dimensional Lie Algebras and Groups, 285-339 (1988), Singapore: World Scientific, Singapore
[42] Knizhnik, V. G.; Zamolodchikov, A. B., Current algebra and Wess-Zumino models in two dimensions, Nucl. Phys. B, 247, 83-103 (1984) · Zbl 0661.17020 · doi:10.1016/0550-3213(84)90374-2
[43] Kock, J., Frobenius Algebras and 2d Topological Quantum Field Theories (2004), Cambridge: Cambridge University Press, Cambridge · Zbl 1046.57001
[44] Kodiyalam, V., Sunder, V.S.: Topological quantum field theories from subfactors. Res. Not. Math. 423 (2001) · Zbl 1050.46038
[45] Kohno, T., Topological invariants for three manifolds using representations of the mapping class groups I, Topology, 31, 203-230 (1992) · Zbl 0762.57011 · doi:10.1016/0040-9383(92)90016-B
[46] Kohno, T., Topological Invariants for Three Manifolds Using Representations of the Mapping Class Groups Ii: Estimating Tunnel Number of Knots, Mathematical Aspects of Conformal and Topological Field Theories and Quantum Groups, 193-217 (1994), Providence: Am. Math. Soc., Providence · Zbl 0823.57004
[47] Kohno, T., Conformal Field Theory and Topology (2002), Providence: Am. Math. Soc., Providence · Zbl 1024.81001
[48] Kontsevich, M., Feynman diagrams and low-dimensional topology, First European Cong. Math., 97-121 (1994), Berlin: Birkhäuser, Berlin · Zbl 0872.57001
[49] Lawrence, R., An introduction to topological field theory, The Interface of Knots and Physics, 89-128 (1996), Providence: Am. Math. Soc., Providence · Zbl 0844.57021
[50] Lawrence, R.; Zagier, D., Modular forms and quantum invariants of 3-manifolds, Asian J. Math., 3, 93-108 (1999) · Zbl 1024.11028
[51] Li, J.; Liu, K.; Zhou, J., Topological string partition functions as equivariant indices, Asian J. Math., 10, 1, 81-114 (2006) · Zbl 1129.14024
[52] Manturov, V., Knot Theory (2004), London: Chapman & Hall/CRC, London · Zbl 1052.57001 · doi:10.1201/9780203402849
[53] Marathe, K.B.: Structure of relativistic spaces. PhD thesis, University of Rochester (1971)
[54] Marathe, K. B., Generalized field equations of gravitation, Rend. Mat. (Roma), 6, 439-446 (1972) · Zbl 0257.53051
[55] Marathe, K.; Engquist, B.; Schmidt, W., A chapter in physical mathematics: theory of knots in the sciences, Mathematics Unlimited—2001 and Beyond, 873-888 (2001), Berlin: Springer, Berlin · Zbl 1038.00005
[56] Marathe, K., Chern-Simons and string theory, J. Geom. Symm. Phys., 5, 36-47 (2006) · Zbl 1104.81078
[57] Marathe, K.: Topological Quantum Field Theory as Topological Gravity. Mathematical and Physical Aspects of Quantum Gravity, pp. 189-205. Birkhäuser, Berlin (2006)
[58] Marathe, K., The review of Symmetry and the Monster by Marc Ronan (Oxford), Math. Intell., 31, 76-78 (2009) · doi:10.1007/s00283-008-9007-9
[59] Marathe, K., Topics in Physical Mathematics (2010), London: Springer, London · Zbl 1291.58001 · doi:10.1007/978-1-84882-939-8
[60] Marathe, K. B.; Martucci, G., The geometry of gauge fields, J. Geom. Phys., 6, 1-106 (1989) · Zbl 0679.53023 · doi:10.1016/0393-0440(89)90002-8
[61] Marathe, K. B.; Martucci, G., The Mathematical Foundations of Gauge Theories (1992), Amsterdam: North-Holland, Amsterdam · Zbl 0920.58079
[62] Marathe, K. B.; Martucci, G.; Francaviglia, M., Gauge theory, geometry and topology, Semin. Mat. Univ. Bari, 262, 1-90 (1995) · Zbl 1204.57036
[63] Milnor, J., Morse Theory (1973), Princeton: Princeton University Press, Princeton
[64] Modugno, M., Sur quelques propriétés de la double 2-forme gravitationnelle W, Ann. Inst. Henri Poincaré, XVIII, 251-262 (1973)
[65] Murakami, H., Quantum SU(2)-invariants dominate Casson’s SU(2)-invariant, Math. Proc. Camb. Philos. Soc., 115, 253-281 (1993) · Zbl 0832.57005 · doi:10.1017/S0305004100072078
[66] Ocneanu, A., Quantized Groups, String Algebras and Galois Theory for Algebras, Operator Algebras and Applications, 119-172 (1988), Cambridge: Cambridge University Press, Cambridge · Zbl 0696.46048
[67] Ozsváth, P.; Szabó, Z., On knot Floer homology and the four-ball genus, Geom. Topol., 7, 225-254 (2003) · Zbl 1130.57303 · doi:10.2140/gt.2003.7.225
[68] Pandharipande, R., Hodge integrals and degenerate contributions, Commun. Math. Phys., 208, 489-506 (1999) · Zbl 0953.14036 · doi:10.1007/s002200050766
[69] Pandharipande, R.: Three Questions in Gromov-Witten Theory. In: Proc. ICM 2002 vol. II, pp. 503-512. Beijing (2002) · Zbl 1047.14043
[70] Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math/0211159v1 [math.DG] (2002) · Zbl 1130.53001
[71] Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math/0303109v1 [math.DG] (2003) · Zbl 1130.53002
[72] Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math/0307245v1 [math.DG] (2003) · Zbl 1130.53003
[73] Petrov, A. Z., Einstein Spaces (1969), New York: Pergamon, New York · Zbl 0174.28305
[74] Piunikhin, S., Reshetikhin-Turaev and Kontsevich-Kohno-Crane 3-manifold invariants coincide, J. Knot Theory, 2, 65-95 (1993) · Zbl 0791.57012 · doi:10.1142/S0218216593000052
[75] Przytycki, J. H., Fundamentals of Kauffman bracket skein modules, Kobe J. Math., 16, 45-66 (1999) · Zbl 0947.57017
[76] Reshetikhin, N.; Turaev, V. G., Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math., 103, 547-597 (1991) · Zbl 0725.57007 · doi:10.1007/BF01239527
[77] Sachs, R. K.; Wu, H., General Relativity for Mathematicians (1977), Berlin: Springer, Berlin · Zbl 0373.53001
[78] Salamon, D., Morse theory, the Conley index and Floer homology, Bull. Lond. Math. Soc., 22, 113-140 (1990) · Zbl 0709.58011 · doi:10.1112/blms/22.2.113
[79] Saveliev, N., Lectures on the Topology of 3-manifolds (1999), Berlin: de Gruyter, Berlin · Zbl 0932.57001
[80] Sawin, S., Links, quantum groups and TQFTs, Bull. Am. Math. Soc., 33, 413-445 (1996) · Zbl 0872.57002 · doi:10.1090/S0273-0979-96-00690-8
[81] Schwarz, M.: Morse homology. Prog. Math. 11 (1993) · Zbl 0806.57020
[82] Segal, G., Two-dimensional conformal field theories and modular functors, Proc. IXth Int. Cong. on Mathematical Physics, 22-37 (1989), Bristol: Hilger, Bristol
[83] Smith, I.; Thomas, R. P.; Yau, S.-T., Symplectic conifold transitions, J. Differ. Geom., 62, 209-242 (2002) · Zbl 1071.53541
[84] Stern, R. J., Gauge Theories as a Tool for Low-dimensional Topologists, Perspectives in Mathematics, 495-507 (1984), Basel: Birkhauser, Basel · Zbl 0557.57011
[85] Taubes, C. H., Casson’s invariant and gauge theory, J. Differ. Geom., 31, 547-599 (1990) · Zbl 0702.53017
[86] Turaev, V. G., The Yang-Baxter equation and invariants of link, Invent. Math., 92, 527-553 (1988) · Zbl 0648.57003 · doi:10.1007/BF01393746
[87] Turaev, V. G., Quantum Invariants of Knots and 3-manifolds (1994), Amsterdam: de Gruyter, Amsterdam · Zbl 0812.57003
[88] Turaev, V. G.; Viro, O. Y., State sum invariants of 3-manifolds and quantum 6j-symbols, Topology, 31, 865-895 (1992) · Zbl 0779.57009 · doi:10.1016/0040-9383(92)90015-A
[89] Turaev, V. G.; Wenzl, H., Quantum invariants of 3-manifolds associated with classical simple Lie algebras, Int. J. Math., 4, 323-358 (1993) · Zbl 0784.57007 · doi:10.1142/S0129167X93000170
[90] Verlinde, E., Fusion rules and modular transformations in 2d conformal field theory, Nucl. Phys. B, 300, 360-376 (1988) · Zbl 1180.81120 · doi:10.1016/0550-3213(88)90603-7
[91] Wenzl, H., Braids and invariants of 3-manifolds, Invent. Math., 114, 235-275 (1993) · Zbl 0804.57007 · doi:10.1007/BF01232670
[92] Witten, E., Supersymmetry and Morse theory, J. Differ. Geom., 17, 661-692 (1982) · Zbl 0499.53056
[93] Witten, E., Topological quantum field theory, Commun. Math. Phys., 117, 353-386 (1988) · Zbl 0656.53078 · doi:10.1007/BF01223371
[94] Witten, E., Quantum field theory and the Jones polynomial, Commun. Math. Phys., 121, 359-399 (1989) · Zbl 0667.57005 · doi:10.1007/BF01217730
[95] Witten, E., Quantization of Chern-Simons gauge theory with complex gauge group, Commun. Math. Phys., 137, 29-66 (1991) · Zbl 0717.53074 · doi:10.1007/BF02099116
[96] Witten, E., Chern-Simons gauge theory as string theory, Prog. Math., 133, 637-678 (1995) · Zbl 0844.58018
[97] Wu, T. T.; Yang, C. N., Concept of nonintegrable phase factors and global formulation of gauge fields, Phys. Rev. D, 12, 3845-3857 (1975) · doi:10.1103/PhysRevD.12.3845
[98] Yetter, D. N., Functorial Knot Theory (2001), Singapore: World Scientific, Singapore · Zbl 0977.57005
[99] Yokota, Y., Skeins and quantum SU(N) invariants of 3-manifolds, Math. Ann., 307, 109-138 (1997) · Zbl 0953.57009 · doi:10.1007/s002080050025
[100] Zwegers, S. P., Mock θ-functions and real analytic modular forms, q-Series with Applications to Combinatorics, Number Theory, and Physics, 269-277 (2001), Providence: Am. Math. Soc., Providence · Zbl 1044.11029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.