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Quantum field theory and the Jones polynomial

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Abstract

It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the Jones polynomial of knot theory in three dimensional terms. In this version, the Jones polynomial can be generalized fromS 3 to arbitrary three manifolds, giving invariants of three manifolds that are computable from a surgery presentation. These results shed a surprising new light on conformal field theory in 1+1 dimensions.

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References

  1. Atiyah, M. F.: New invariants of three and four dimensional manifolds. In: The mathematical heritage of Hermann Weyl. Proc. Symp. Pure Math., vol.48. Wells R. (ed.). Providence, RI: American Mathematical Society 1988, pp. 285–299

    Google Scholar 

  2. Donaldson, S.: An application of gauge theory to the topology of four manifolds. J. Diff. Geom.18, 269 (1983), Polynomial invariants for smooth four-manifolds. Oxford preprint

    Google Scholar 

  3. Floer, A.: An instanton invariant for three manifolds. Courant Institute preprint (1987). Morse theory for fixed points of symplectic diffeomorphisms. Bull. AMS16, 279 (1987)

    Google Scholar 

  4. Witten, E.: Topological quantum field theory. Commun. Math. Phys.117, 353 (1988)

    Google Scholar 

  5. Jones, V. F. R.: Index for subfactors. Invent. Math.72, 1 (1983). A polynomial invariant for links via von Neumann algebras. Bull. AMS12, 103 (1985), Hecke algebra representations of braid groups and link polynomials. Ann. Math.126, 335 (1987)

    Google Scholar 

  6. Freyd, P., Yetter, D., Hoste, J., Lickorish, W. B. R., Millett, K., Ocneanu, A.: A new polynomial invariant of knots and links. Bull. AMS12, 239 (1985)

    Google Scholar 

  7. Kauffman, L.: State models and the Jones polynomial. Topology26, 395 (1987). Statistical mechaniscs and the Jones polynomial, to appear in the Proceedings of the July, 1986, conference on Artin's braid group, Santa Cruz, California; An invariant of regular isotopy preprint

    Google Scholar 

  8. Turaev, V. G.: The Yang-Baxter equation and invariants of links. LOMI preprint E-3-87, Inv. Math.92, 527 (1988)

    Google Scholar 

  9. Przytycki, J. H., Traczyk, P.: Invariants of links of conway type. Kobe J. Math.,4, 115 (1988)

    Google Scholar 

  10. Birman, J.: On the Jones polynomial of closed 3-braids. Invent. Math.81, 287 (1985).

    Google Scholar 

  11. Birman, J., Wenzl, H. Link polynomials and a new algebra, preprint

  12. Tsuchiya, A., Kanie, Y.: In: Conformal field theory and solvable lattice models. Adv. Stud. Pure math.16, 297 (1988); Lett. Math. Phys.13, 303 (1987)

    Google Scholar 

  13. Verlinde, E.: Fusion rules and modular transformations in 2d conformal field theory. Nucl. Phys.B 300, 360 (1988)

    Google Scholar 

  14. Moore, G., Seiberg, N.: Polynomial equations for rational conformal field theories. To appear in Phys. Lett. B, Naturality in conformal field theory. To appear in Nucl. Phys. B, Classical and quantum conformal field theory. IAS preprint HEP-88/35

  15. Schroer, B.: Nucl. Phys.295, 4 (1988) K.-H. Rehren, Schroer, B.: Einstein causality and Artin braids. FU preprint (1987)

    Google Scholar 

  16. Fröhlich, J.: Statistics of fields, the Yang-Baxter equation, and the theory of links and knots. 1987 Cargese lectures, to appear In Nonperturbative quantum field theory. New York: Plenum Press

    Google Scholar 

  17. Segal, G.: Conformal field theory. Oxford preprint; and lecture at the IAMP Congress, Swansea, July, 1988

  18. Gromov, M.: Pseudo-holomorphic curves in symplectic manifolds. Invent. Math.82, 307 (1985)

    Google Scholar 

  19. Schwarz, A.: The partition function of degenerate quadratic functional and Ray-Singer invariants. Lett. Math. Phys.2, 247 (1978)

    Google Scholar 

  20. Schonfeld, J.: A mass term for three dimensional gauge fields. Nucl. Phys.B 185, 157 (1981)

    Google Scholar 

  21. Jackiw, R., Templeton, S.: How superrenormalizable theories cure their infrared divergences. Phys. Rev.D 23, 2291 (1981)

    Google Scholar 

  22. Deser, S., Jackiw, R., Templeton, S.: Three dimensional massive gauge theories. Phys. Rev. Lett.48, 975 (1983). Deser, S., Jackiw, R., Templeton, S. Topologically massive gauge theory. Ann. Phys. NY140, 372 (1984)

    Google Scholar 

  23. Zuckerman, G.: Action principles and global geometry. In: The proceedings of the 1986 San Diego Summer Workshop, Yau, S.-T. (ed.)

  24. Polyakov, A. M.: Fermi-bose transmutations induced by gauge fields. Mod. Phys. Lett.A 3, 325 (1988)

    Google Scholar 

  25. Hagen, C. R.: Ann. Phys.157, 342 (1984)

    Google Scholar 

  26. Arovas, D., Schrieffer, R., Wilczek, F., Zee, A.: Statistical mechanics of anyons. Nucl. Phys.B 251 [FS 13], 117 (1985)

    Google Scholar 

  27. Witten, E.: Non-Abelian bosonization in two dimensions. Commun. Math. Phys.92, 455 (1984)

    Google Scholar 

  28. Ray, D., Singer, I.: Adv. Math.7, 145 (1971), Ann. Math.98 154 (1973)

    Google Scholar 

  29. Deser, S., Jackiw, R., Templeton, S.: In [21] ; Affleck, I., Harvey, J., Witten, E.: Nucl. Phys.B 206, 413 (1982)

    Google Scholar 

  30. Redlich, A. N.: Gauge invariance and parity conservation of three-dimensional fermions. Phys. Rev. Lett.52, 18 (1984)

    Google Scholar 

  31. Alvarez-Gaumé, L., Witten, E.: Gravitational anomalies. Nucl. Phys.B 234, 269 (1983)

    Google Scholar 

  32. Alvarez-Gaumé, Della Pietra, S., Moore, G.: Anomalies and odd dimensions. Ann. Phys. (NY)163, 288 (1985)

    Google Scholar 

  33. Atiyah, M.F.: A note on the eta invariant (unpublished)

  34. Singer, I.M.: Families of Dirac operators with applications to Physics. Astérisque,1985, p. 323

  35. Atiyah, M. F., Patodi, V., Singer, I.: Math. Proc. Camb. Phil. Soc.77, 43 (1975),78, 405 (1975),79, 71 (1976)

    Google Scholar 

  36. Wilczek, F., Zee, A.: Linking numbers, spin, and statistics of solitons. Phys. Rev. Lett.51, 2250 (1983)

    Google Scholar 

  37. Friedan, D., Shenker, S.: Nucl. Phys.B 281, 509 (1987)

    Google Scholar 

  38. Belavin, A., Polyakov, A. M., Zamolodchikov, A.: Nucl. Phys.B (1984)

  39. Atiyah, M. F., Bott, R.: The Yang-Mills equations over Riemann surfaces. Phil. Trans. R. Soc. Lond.A 308, 523 (1982)

    Google Scholar 

  40. Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, 31 (1986)

    Google Scholar 

  41. Drinfeld, V.: Quantum groups. In: The Proceedings of the International Congress of Mathematicians, Berkeley 1986, Vol. 1, pp. 798–820

  42. Gepner, D., Witten, E.: String theory on group manifolds. Nucl. Phys. B278, 493 (1986)

    Google Scholar 

  43. Kac, V. G.: Infinite dimensional Lie algebras. Cambridge: Cambridge University Press (1985)

    Google Scholar 

  44. Kac, V. G., Peterson, D. H.: Adv. Math.53, 125 (1984)

    Google Scholar 

  45. Kac, V. G., Wakimoto, M.: Adv. Math.70, 156 (1988)

    Google Scholar 

  46. Witten, E.: 2+1 dimensional gravity as an exactly soluble system. IAS preprint HEP-88/32

  47. Segal, G.: Unitary representation of some infinite dimensional groups. Commun. Math. Phys.80, 301 (1981)

    Google Scholar 

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Authors and Affiliations

  1. School of Natural Sciences, Institute for Advanced Study, Olden Lane, 08540, Princeton, NJ, USA

    Edward Witten

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  1. Edward Witten
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Additional information

Communicated by A. Jaffe

An expanded version of a lecture at the IAMP Congress, Swansea, July, 1988

Research supported in part by NSF Grant No. 86-20266, and NSF Waterman Grant 88–17521

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Witten, E. Quantum field theory and the Jones polynomial. Commun.Math. Phys. 121, 351–399 (1989). https://doi.org/10.1007/BF01217730

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  • DOI: https://doi.org/10.1007/BF01217730

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