A TQFT for Wormhole cobordisms over the field of rational functions

Patrick Gilmer

Banach Center Publications (1998)

  • Volume: 42, Issue: 1, page 119-127
  • ISSN: 0137-6934

Abstract

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We consider a cobordism category whose morphisms are punctured connected sums of ’s (wormhole spaces) with embedded admissibly colored banded trivalent graphs. We define a TQFT on this cobordism category over the field of rational functions in an indeterminant A. For r large, we recover, by specializing A to a primitive 4rth root of unity, the Witten-Reshetikhin-Turaev TQFT restricted to links in wormhole spaces. Thus, for r large, the rth Witten-Reshetikhin-Turaev invariant of a link in some wormhole space, properly normalized, is the value of a certain rational function at . We relate our work to Hoste and Przytycki’s calculation of the Kauffman bracket skein module of .

How to cite

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Gilmer, Patrick. "A TQFT for Wormhole cobordisms over the field of rational functions." Banach Center Publications 42.1 (1998): 119-127. <http://eudml.org/doc/208799>.

@article{Gilmer1998,
abstract = {We consider a cobordism category whose morphisms are punctured connected sums of $S^1 × S^2$’s (wormhole spaces) with embedded admissibly colored banded trivalent graphs. We define a TQFT on this cobordism category over the field of rational functions in an indeterminant A. For r large, we recover, by specializing A to a primitive 4rth root of unity, the Witten-Reshetikhin-Turaev TQFT restricted to links in wormhole spaces. Thus, for r large, the rth Witten-Reshetikhin-Turaev invariant of a link in some wormhole space, properly normalized, is the value of a certain rational function at $e^\{(πi)/(2r)\}$. We relate our work to Hoste and Przytycki’s calculation of the Kauffman bracket skein module of $S^1 × S^2$.},
author = {Gilmer, Patrick},
journal = {Banach Center Publications},
keywords = {Kauffman bracket; fusion rules},
language = {eng},
number = {1},
pages = {119-127},
title = {A TQFT for Wormhole cobordisms over the field of rational functions},
url = {http://eudml.org/doc/208799},
volume = {42},
year = {1998},
}

TY - JOUR
AU - Gilmer, Patrick
TI - A TQFT for Wormhole cobordisms over the field of rational functions
JO - Banach Center Publications
PY - 1998
VL - 42
IS - 1
SP - 119
EP - 127
AB - We consider a cobordism category whose morphisms are punctured connected sums of $S^1 × S^2$’s (wormhole spaces) with embedded admissibly colored banded trivalent graphs. We define a TQFT on this cobordism category over the field of rational functions in an indeterminant A. For r large, we recover, by specializing A to a primitive 4rth root of unity, the Witten-Reshetikhin-Turaev TQFT restricted to links in wormhole spaces. Thus, for r large, the rth Witten-Reshetikhin-Turaev invariant of a link in some wormhole space, properly normalized, is the value of a certain rational function at $e^{(πi)/(2r)}$. We relate our work to Hoste and Przytycki’s calculation of the Kauffman bracket skein module of $S^1 × S^2$.
LA - eng
KW - Kauffman bracket; fusion rules
UR - http://eudml.org/doc/208799
ER -

References

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  1. [A] M. F. Atiyah, Topological quantum field theories, Inst. Hautes Études Sci. Publ. Math. 68 (1988), 175-186. Zbl0692.53053
  2. [BHMV] C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Topological Quantum Field Theories Derived from the Kauffman Bracket, Topology 34 (1995), 883-927. Zbl0887.57009
  3. [G] P. Gilmer, Invariants for 1-dimensional cohomology classes arising from TQFT, Top. and its Appl. 75 (1997), 217-259. Zbl0870.57015
  4. [HP] J. Hoste, J. Przytycki, The Kauffman Bracket Skein Module of , Math. Zeit. 220 (1995), 65-73. Zbl0826.57007
  5. [J] V. F. R. Jones, A polynomial invariant for knots via von Neumann algebras, Bulletin AMS 12 (1985), 103-111. Zbl0564.57006
  6. [K] L. H. Kauffman, State models and the Jones polynomial, Topology 26 (1987), 395-407. Zbl0622.57004
  7. [KL] L. H. Kauffman, S. Lins, Temperley Lieb recoupling theory and invariants of -manifolds, Annals of Math Studies 91994), Princeton Univ. Press, Princeton N.J. 
  8. [Ki] R. Kirby, The Topology of 4-manifolds, Springer Lecture Notes in Math. 1374. 
  9. [La] R. Lawrence, Asymptotic expansions of Witten-Reshetikhin-Turaev invariants for some simple 3-manifolds, J. Math. Phys. 36 (1995), 6106-6129. Zbl0877.57008
  10. [L] W. B. R. Lickorish, The skein Method for three manifolds, J. of Knot Th. and its Ramif. 2 (1993), 171-194. Zbl0793.57003
  11. [MV] G. Masbaum, P. Vogel, Verlinde Formulae for surfaces with spin structure, Geometric Topology, Joint U.S. Israel Workshop on Geometric Topology June 10-16, 1992 Technion, Haifa, Israel Contemporary Mathematics 164 (Gordon, Moriah, Waynryb, ed.), American Math. Soc., 1994, 119-137. Zbl0824.57012
  12. [RT] N. Reshetikhin, V. Turaev, Invariants of 3-manifolds via link-polynomials and quantum groups, Invent. Math. 103 (1991), 547-597. Zbl0725.57007
  13. [R] J. Roberts, Skein Theory and Turaev-Viro invariants, Topology 34 (1995), 771-787. Zbl0866.57014
  14. [W] E. Witten, Quantum field theory and the Jones polynomial, Commun. Math. Phys. 121 (1989), 351-399. Zbl0667.57005

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