nLab Chern-Simons propagator

Idea

The Chern-Simons propagator is the Feynman propagator for Chern-Simons theory regarded as a Euclidean quantum field theory. Since the Feynman propagator only depends on the free field theory-equations of motion (the staring point of perturbative QFT), which here is $d A = 0$, its form is independent of the gauge group and in fact applies also to all higher Chern-Simons theories such as notably the AKSZ sigma-models.

In the perturbative quantization of 3d Chern-Simons theory the CS propagator was first studied in Axelrod-Singer 91.

As a smooth differential form on compactified configuration spaces

The Chern-Simons propagator was re-formulated as a smooth differential form on compactified configuration space in Axelrod-Singer 93, p. 5-6. In this specific form its mathematical nature was amplified in Bott-Cattaneo 97, remark 3.6 and Cattaneo-Mnev 10, Remark 11. It was suggested that this serves to exhibit its Feynman amplitudes as exhibiting a graph complex-model for the de Rham cohomology of configuration spaces of points in Kontsevich 93, Kontsevich 94, which was proven in Lambrechts-Volic 14. This role of the Chern-Simons propagator is advertized also in Campos-Idrissi-Lambrechts-Willwacher 18, p. 66.

References

Detailed review is in

• Konstantin Wernli, section 3 of Perturbative Quantization of Split Chern-Simons Theory on Handlebodies and Lens Spaces by the BV-BFV formalism, 2018

Original articles include the following

• 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)

• Raoul Bott, Alberto Cattaneo, Remark 3.6 in Integral invariants of 3-manifolds, J. Diff. Geom., 48 (1998) 91-133 (arXiv:dg-ga/9710001)

• Alberto Cattaneo, Pavel Mnev, Remark 11 in Remarks on Chern-Simons invariants, Commun.Math.Phys.293:803-836,2010 (arXiv:0811.2045)

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

• Maxim Kontsevich, pages 11-12 of Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics 120, Birkhäuser (1994), 97–121 (pdf)

• 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)

• Alberto Cattaneo, Pavel Mnev, Nicolai Reshetikhin, appendix B of Perturbative quantum gauge theories on manifolds with boundary, Communications in Mathematical Physics, January 2018, Volume 357, Issue 2, pp 631–730 (arXiv:1507.01221, doi:10.1007/s00220-017-3031-6)

• Ricardo Campos, Najib Idrissi, Pascal Lambrechts, Thomas Willwacher, Configuration Spaces of Manifolds with Boundary (arXiv:1802.00716)