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A Simple Model of 4d-TQFT

Part of the MATRIX Book Series book series (MXBS,volume 1)

Abstract

We show that, associated with any complex root of unity ω, there exists a particularly simple 4d-TQFT model defined on the cobordism category of ordered triangulations of oriented 4-manifolds.

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References

  1. Atiyah, M.: Topological quantum field theories. Inst. Hautes Études Sci. Publ. Math. 68, 175–186 (1988/1989). http://www.numdam.org/item?id=PMIHES_1988__68__175_0

    MathSciNet  CrossRef  Google Scholar 

  2. Bazhanov, V.V., Baxter, R.J.: New solvable lattice models in three dimensions. J. Stat. Phys. 69(3–4), 453–485 (1992). http://dx.doi.org/10.1007/BF01050423

    MathSciNet  CrossRef  Google Scholar 

  3. Biedenharn, L.C.: An identity by the Racah coefficients. J. Math. Phys. 31, 287–293 (1953)

    MathSciNet  CrossRef  Google Scholar 

  4. Carter, J.S., Kauffman, L.H., Saito, M.: Structures and diagrammatics of four-dimensional topological lattice field theories. Adv. Math. 146(1), 39–100 (1999). http://dx.doi.org/10.1006/aima.1998.1822

    MathSciNet  CrossRef  Google Scholar 

  5. Crane, L., Frenkel, I.B.: Four-dimensional topological quantum field theory, Hopf categories, and the canonical bases. J. Math. Phys. 35(10), 5136–5154 (1994). http://dx.doi.org/10.1063/1.530746. Topology and physics

    MathSciNet  CrossRef  Google Scholar 

  6. Crane, L., Yetter, D.: A categorical construction of 4D topological quantum field theories. In: Quantum Topology. Knots Everything, vol. 3, pp. 120–130. World Scientific Publications, River Edge, NJ (1993). http://dx.doi.org/10.1142/9789812796387_0005

    Google Scholar 

  7. Elliott, J.P.: Theoretical studies in nuclear structure. V. The matrix elements of non-central forces with an application to the 2p-shell. Proc. R. Soc. Lond. Ser. A 218, 345–370 (1953). http://dx.doi.org/10.1098/rspa.1953.0109

    Google Scholar 

  8. Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  9. Kashaev, R.: A simple model of 4d-TQFT (2014). arXiv:1405.5763

    Google Scholar 

  10. Kashaev, R.M.: On realizations of Pachner moves in 4d. J. Knot Theory Ramif. 24(13), 1541002, 13 (2015). http://dx.doi.org/10.1142/S0218216515410023

    MathSciNet  CrossRef  Google Scholar 

  11. Korepanov, I.G.: Euclidean 4-simplices and invariants of four-dimensional manifolds. I. Surgeries 3 → 3. Teor. Mat. Fiz. 131(3), 377–388 (2002). http://dx.doi.org/10.1023/A:1015971322591

  12. Korepanov, I.G., Sadykov, N.M.: Parameterizing the simplest Grassmann-Gaussian relations for Pachner move 3–3. Symmetry Integr. Geom. Methods Appl. 9, Paper 053, 19 (2013)

    Google Scholar 

  13. Lickorish, W.B.R.: Simplicial moves on complexes and manifolds. In: Proceedings of the Kirbyfest (Berkeley, CA, 1998). Geometry & Topology Monographs, vol. 2, pp. 299–320. Geometry & Topology Publications, Coventry (1999) (electronic). http://dx.doi.org/10.2140/gtm.1999.2.299

  14. Oeckl, R.: Discrete Gauge Theory. Imperial College Press, London (2005). http://dx.doi.org/10.1142/9781860947377. From lattices to TQFT

  15. Pachner, U.: PL homeomorphic manifolds are equivalent by elementary shellings. Eur. J. Comb. 12(2), 129–145 (1991). http://dx.doi.org/10.1016/S0195-6698(13)80080-7

  16. Ponzano, G., Regge, T.: Semiclassical limit of Racah coefficients. In: Spectroscopic and group theoretical methods in physics, pp. 1–58. North-Holland, Amsterdam (1968)

    Google Scholar 

  17. Turaev, V.G.: Quantum Invariants of Knots and 3-Manifolds. de Gruyter Studies in Mathematics, vol. 18. Walter de Gruyter, Berlin (1994)

    Google Scholar 

  18. Turaev, V.G., Viro, O.Y.: State sum invariants of 3-manifolds and quantum 6j-symbols. Topology 31(4), 865–902 (1992). http://dx.doi.org/10.1016/0040-9383(92)90015-A

    MathSciNet  CrossRef  Google Scholar 

  19. Walker, K.: On Witten’s 3-manifold invariants (1991). Preprint

    Google Scholar 

  20. Witten, E.: Topological quantum field theory. Commun. Math. Phys. 117(3), 353–386 (1988). http://projecteuclid.org/getRecord?id=euclid.cmp/1104161738

    MathSciNet  CrossRef  Google Scholar 

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Acknowledgements

This work is supported in part by the Swiss National Science Foundation.

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Correspondence to Rinat Kashaev .

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Kashaev, R. (2018). A Simple Model of 4d-TQFT. In: de Gier, J., Praeger, C., Tao, T. (eds) 2016 MATRIX Annals. MATRIX Book Series, vol 1. Springer, Cham. https://doi.org/10.1007/978-3-319-72299-3_14

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