nLab perturbative quantization of Chern-Simons theory -- references

Perturbative quantum Chern-Simons theory

Discussion of perturbative quantization of Chern-Simons theory (via Kontsevich integrals/knot graph cohomology on Jacobi diagrams, regarding Feynman amplitudes as differential forms on configuration spaces of points and yielding universalVassiliev invariants):

• Dror Bar-Natan, Perturbative aspects of the Chern-Simons topological quantum field theory, thesis 1991 (spire:323500, proquest:303979053, BarNatanPerturbativeCS91.pdf)

• Scott Axelrod, Isadore Singer, Chern-Simons Perturbation Theory, in S. Catto, A. Rocha (eds.) Proc. XXthe DGM Conf. World Scientific Singapore, 1992, 3-45; (arXiv:hep-th/9110056)

• Scott Axelrod, Isadore Singer, Chern–Simons Perturbation Theory II, J. Diff. Geom. 39 (1994) 173-213 (arXiv:hep-th/9304087)

• Maxim Kontsevich, Vassiliev’s knot invariants, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 (pdf)

• Maxim Kontsevich, Feynman diagrams and low-dimensional topology, in First European Congress of Mathematics, Vol. II (Paris, 1992), volume 120 of Progr. Math., pages 97–121, Birkhäuser, Basel, 1994. (pdf)

• Dror Bar-Natan, Perturbative Chern-Simons theory, Journal of Knot Theory and Its RamificationsVol. 04, No. 04, pp. 503-547 (1995) (doi:10.1142/S0218216595000247)

• Daniel Altschuler, Laurent Freidel, Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, Commun. Math. Phys. 187 (1997) 261-287 $[$arXiv:q-alg/9603010, doi:10.1007/s002200050136$]$

• Pascal Lambrechts, Ismar Volić, sections 6 and 7 of Formality of the little N-disks operad, Memoirs of the American Mathematical Society ; no. 1079, 2014 (arXiv:0808.0457, doi:10.1090/memo/1079)

Review:

• Robbert Dijkgraaf, Perturbative topological field theory, In: Trieste 1993, Proceedings, String theory, gauge theory and quantum gravity ‘93 189-227 (spire:399223, pdf)

• Sergei Chmutov, Sergei Duzhin, Jacob Mostovoy, Section 4 of: Introduction to Vassiliev knot invariants, Cambridge University Press, 2012 (arxiv/1103.5628, doi:10.1017/CBO9781139107846)

See also at correlator as differential form on configuration space of points and see at graph complex as a model for the spaces of knots.

The “Wheels theorem”, saying that the perturbative Chern-Simons Wilson loop observable of the unknot is, as a universal Vassiliev invariant, a series of wheel-shaped Jacobi diagrams with coefficients the modified Bernoulli numbers, is due to

• Dror Bar-Natan, Thang T Q Le, Dylan Thurston, Two applications of elementary knot theory to Lie algebras and Vassiliev invariants, Geom. Topol. Volume 7, Number 1 (2003), 1-31 (euclid.gt/1513883092)

following

• Dror Bar-Natan, Stavros Garoufalidis, Lev Rozansky, Dylan Thurston, Wheels, wheeling, and the Kontsevich integral of the unknot (q-alg/9703025)