These notes are the outcome of a mini-course on TQFTs held at the edition of Winter Braids in Pau in February 2015. We define the notion of TQFT and provide the first basic examples obtained via the universal construction and via Frobenius algebras. After recalling some basic notions on the mapping class groups of surfaces, we concentrate on the Reshetikhin-Turaev construction via the skein theoretical approach: we first define the skein module of a -manifold and the RT invariants; then we apply the universal construction to get the RT -TQFTs. We conclude with an overview of the main results on these TQFTs and on some recent developments. An appendix summarizes the basic notions and facts in category theory used here.
@article{WBLN_2015__2__A1_0, author = {Francesco Costantino}, title = {Notes on {Topological} {Quantum} {Field} {Theories}}, journal = {Winter Braids Lecture Notes}, note = {talk:1}, pages = {1--45}, publisher = {Winter Braids School}, volume = {2}, year = {2015}, doi = {10.5802/wbln.7}, mrnumber = {3705873}, zbl = {1423.81169}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/wbln.7/} }
TY - JOUR AU - Francesco Costantino TI - Notes on Topological Quantum Field Theories JO - Winter Braids Lecture Notes N1 - talk:1 PY - 2015 SP - 1 EP - 45 VL - 2 PB - Winter Braids School UR - https://proceedings.centre-mersenne.org/articles/10.5802/wbln.7/ DO - 10.5802/wbln.7 LA - en ID - WBLN_2015__2__A1_0 ER -
Francesco Costantino. Notes on Topological Quantum Field Theories. Winter Braids Lecture Notes, Volume 2 (2015), Talk no. 1, 45 p. doi : 10.5802/wbln.7. https://proceedings.centre-mersenne.org/articles/10.5802/wbln.7/
[1] L. Abrams, Two-dimensional topological quantum field theories and Frobenius algebras, J. Knot Theory and its Ramifications 5 (1996), 569–587. | DOI | MR | Zbl
[2] J.E. Andersen, Asymptotic faithfulness of the quantum SU(n) representations of the mapping class groups, Ann. of Math. (2), 163(1):347–368, 2006. | DOI | MR | Zbl
[3] J.E. Andersen and J. Fjelstad, Reducibility of quantum representations of mapping class groups, Lett. Math. Phys., 91(3):215–239, 2010. | DOI | MR | Zbl
[4] J.E. Andersen, G. Masbaum and K. Ueno, Topological quantum field theory and the Nielsen-Thurston classification of Math. Proc. Cambridge Philos. Soc., 141(3) :477–488, 2006. | DOI | MR | Zbl
[5] M. Atiyah, Topological quantum field theories, Inst. Hautes Etudes Sci. Publ. Math., (68):175–186 (1989), 1988. | DOI | Zbl
[6] C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Three manifold invariants derived from the Kauffman bracket, Topology 31 no 4 (1992), 685–699. | DOI | MR | Zbl
[7] C. Blanchet, N. Habegger, G. Masbaum, P. Vogel, Topological quantum field theories derived from the Kauffman bracket, Topology 34 (1995), 883–927. | DOI | Zbl
[8] C. Blanchet, F. Costantino, N. Geer, B. Patureau, Non semi-simple TQFTs, Reidemeister torsion and Kashaev’s invariants, arXiv:1404.7289, to appear in Advances in Mathematics. | DOI | MR | Zbl
[9] B. Berndt, R. Evans, S. Williams, Gauss and Jacobi sums, Wiley (1988). | DOI | Zbl
[10] K. Brown, Cohomology of groups, Graduate Texts in Mathematics, No. 87, Springer Verlag. | DOI
[11] D. Bullock, Rings of -characters and the Kauffman bracket skein module, Comment. Math. Helv., 72(4):521–542, 1997. | DOI | MR | Zbl
[12] L. Charles, J. Marché, Multicurves and regular functions on the representation variety of a surface in SU(2), Commentarii Mathematici Helvetici, Vol 87 (2012), p. 409–431 arXiv:0901.3064. | DOI | MR | Zbl
[13] F. Costantino, N. Geer, B. Patureau, Quantum invariants of -manifolds via link surgery presentations and non semi-simple categories, Journal of Topology, vol 7. Issue 1, pp. 1-49 (2014). | DOI | MR | Zbl
[14] F. Costantino, B. Martelli, An analytic family of representations for the mapping class group of punctured surfaces, Geometry & Topology, vol. 3, no 18 (2014), 1485-1538. | DOI | MR | Zbl
[15] R. Dijkgraaf, A geometric approach to two dimensional conformal field theory, PhD thesis, University of Utrecht, 1989.
[16] D. B. A. Epstein, Curves on 2-manifolds and isotopies, Acta Math., 115:83–107, 1966. | DOI | MR | Zbl
[17] A. Fathi, F. Laudenbach, V. Poenaru, Travaux de Thurston sur les surfaces, Astérisque 66, Société Mathématique de France, Paris (1979) Séminaire Orsay. | Numdam | Zbl
[18] B. Farb, D. Margalit, A primer on mapping class groups, Princeton Mathematical Series, 2011. | DOI | Zbl
[19] M. H. Freedman, K. Walker, and Z. Wang, Quantum SU(2) faithfully detects mapping class groups modulo center, Geom. Topol., 6:523–539 (electronic), 2002. | DOI | Zbl
[20] L. Funar, On the TQFT representations of the mapping class groups, Pacific J. Math., 188(2):251–274, 1999. | DOI | MR | Zbl
[21] P. Gilmer, On the Witten-Reshetikhin-Turaev Representations of mapping class groups, Proceedings of the American Mathematical Society Volume 127, Number 8, Pages 2483–2488 (1999). | Zbl
[22] P. Gilmer, G. Masbaum, Maslov index, Lagrangians, Mapping Class Groups and TQFT, Forum Mathematicum, Volume 25, Issue 5 (2011) 1067-1106 arXiv:0912.4706. | MR | Zbl
[23] P. Gilmer, G. Masbaum, Integral lattices in TQFT, Ann. Sci. Ecole Norm. Sup. (4), 40(5):815–844, 2007. | DOI | Numdam | MR | Zbl
[24] P. Gilmer, X. Wang, Extra structure and the universal construction for the witten-reshetikhin-turaev TQFT, arXiv 1201.1921v2 (2012). | DOI | MR | Zbl
[25] R. Gompf, A. Stipsicz, -manifolds and Kirby calculus, Graduate Studies in Mathematics 20, American Mathematical Society (1999) | DOI | Zbl
[26] M. Handel, W. Thurston, New proofs of some results of Nielsen Adv. in Mathematics 56, 173-191 (1985). | DOI | MR | Zbl
[27] C. Kassel, Quantum Groups, Graduate Texts in Mathematics, 155. Springer-Verlag, New York, 1995. | DOI
[28] J. Kock, Frobenius Algebras and 2D Topological Quantum Field Theories Cambridge University Press. 2003. | DOI | Zbl
[29] J. Korinman, Decomposition f some Reshitekhin-Turaev representations into irreducible factors, ArXiv e-prints, June 2014. | DOI | Zbl
[30] S. Kwasik, R. Schultz, Pseudo isotopies of -manifolds, Topology 35 no 2, pp 363-376, 1996. | DOI | MR | Zbl
[31] L. Kauffman, S. Lins, Temperley-Lieb recoupling theory and invariants of 3- manifolds, Ann. of Math. Studies 143, Princeton University Press, 1994. | DOI | Zbl
[32] W. B. R. Lickorish, Three manifolds and the Temperley-Lieb algebra, Math. Annalen 290 (1991), 657–670. | DOI | MR | Zbl
[33] W. B. R. Lickorish, Skeins and handlebodies, Pacif. Journal of Math. 159 no. 2 (1993), 337–349. | DOI | MR | Zbl
[34] W. B. R. Lickorish, An Introduction to Knot Theory, Graduate Texts in Mathematics 175, Springer, 1997. | DOI | Zbl
[35] Y. Manin, Topics in Noncommutative Geometry Princeton University Press (1991). | DOI | Zbl
[36] J. Marché and M. Narimannejad, Some asymptotics of topological quantum field theory via skein theory, Duke Math. J., 141(3), 573–587, 2008. | DOI | MR | Zbl
[37] G. Masbaum, P. Vogel, Three-valent graphs and the Kauffman bracket, Pac. J. Math. Math. 164, 361–381 (1994). | DOI | MR | Zbl
[38] J. H Przytycki, Fundamentals of Kauffman bracket skein modules, Kobe J. Math. 16 (1999) 45–66 | Zbl
[39] V. A. Rokhlin, A -dimensional manifold is the boundary of a -dimensional manifold, Dokl. Akad. Nauk. SSSR 81 (1951).
[40] J. Roberts, Irreducibility of some quantum representations of mapping class groups, J. Knot Theory Ramifications, 10(5):763–767, 2001. Knots in Hellas 98, Vol. 3 (Delphi). | DOI | MR | Zbl
[41] J. Roberts, Kirby calculus in manifolds with boundary, Turkish J. Math. 21 (1997), 111–117. | Zbl
[42] C. Rourke, A new proof that is zero, London Math. Soc. (2) 31 (1985), no. 2, 373–376. | DOI | Zbl
[43] N. Reshetikhin, V. Turaev, Invariants of 3-manifolds via link polynomials and quantum groups, Invent. Math., 103(3),547–597 (1991). | DOI | MR | Zbl
[44] R. Santharoubane, Limits of the quantum SO(3) representations for the one-holed torus , Journal of Knot Theory and its Ramifications, 21(11), (2012). | DOI | MR | Zbl
[45] A. Schwarz, The partition function of degenerate quadratic functional and Ray-Singer invariants, Letters in Math. Phys. vol 2 Issue 3 (1978) pp. 247-252. | DOI | MR | Zbl
[46] V. G. Turaev, Quantum invariants of knots and three-manifolds, W. de Gruyter, New York (1994). | DOI | Zbl
[47] V. G. Turaev, Skein quantization of Poisson algebras of loops on surfaces, Ann. Sci. Sc. Norm. Sup. (4) 24 (1991), no. 6, 635–704. | DOI | Numdam | MR | Zbl
[48] E. Witten, Quantum field theory and Jones polynomial, Comm. Math. Phys. 121 (1989), 351–399. | DOI | MR | Zbl
[49] E. Witten, Topological quantum field theories, Comm. Math. Phys. 117 (3) (1988), 353–386. | DOI | MR | Zbl
[50] D. Zagier, Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula, Israel Mathematical Conference Proceedings 9 (1996) 445-462. | Zbl
Cited by Sources: