nLab correlator as differential form on configuration space of points

Idea

In Euclidean field theory, an alternative to regarding propagators/correlators as distributions of several variables with singularities on the fat diagonal, is to pull-back these distributions to smooth functions/differential forms on (Fulton-MacPherson compactifications of) configuration spaces of points and regard them in this incarnation, in particular discuss their renormalization from this perspective.

Analogous to the perspective of wavefront sets for distributions, this perspective amounts to recording around each potentially singular point an (d-1)-sphere worth of extra directional information carried by the correlator/Feynman amplitude in the vicinity of the point.

This approach goes back to Axelrod-Singer 93 in the discussion of perturbative quantization of Chern-Simons theory. Here the graph complex of Kontsevich 94 (full details due to Lambrechts-Volić 14) shows that the de Rham algebra of the configuration space of points is actually quasi-isomorphic to all possible Feynman amplitudes for free Chern-Simons/AKSZ theory.

A general and systematic discussion of perturbative quantum field theory and its renormalization from this perspective is offered in Berghoff 14a, Berghoff 14b (albeit presently only for Euclidean quantum field theory, not for relativistic quantum field theory).

Examples

Higher Chern-Simons theory

This approach to pQFT was originally considered specifically for the Chern-Simons propagator in quantization of 3d Chern-Simons theory in Axelrod-Singer 93, see also Bott-Cattaneo 97, Remark 3.6 and Cattaneo-Mnev 10, Remark 11. The analysis applies verbatim to higher Chern-Simons theory such as notably the AKSZ sigma-models, too, since the Feynman propagator depends only on the free field theory-equations of motion, which is $d A = 0$ in all these cases.

Here the Chern-Simons propagator regarded as a non-singular differential form on the compactification of the configuration space of points serves to exhibit its Feynman amplitudes as providing graph complex-models for the de Rham cohomology of these compactified configuration spaces of points, a point due to Kontsevich 94, Kontsevich 93, 5:

(1)$$\underset{ \color{blue} \array{ \phantom{A} \\ \text{graph complex} \\ \text{of n-point Feynman diagrams} \\ \text{for Chern-Simons theory} \\ \text{on} \; \Sigma } }{ Graphs_n(\Sigma) } \underoverset{ \simeq_{\mathrlap{qi}} } { \color{blue} \array{ \text{assign Feynman amplitudes} \\ \text{of Chern-Simons theory} \\ \phantom{A} } } { \longrightarrow } \underset{ \color{blue} \array{ \phantom{A} \\ \text{de Rham algebra} \\ \text{of semi-algebraic differential forms} \\ \text{on the FM-compactification} \\ \text{of the configuration space of n points} \\ \text{in}\; \Sigma } }{ \Omega^\bullet_{PA} \big( Conf_n\big( \Sigma \big) \big) } \,.$$

Related concepts

• quantization of 3d Chern-Simons theory

References

The approach was originally considered specifically for Chern-Simons theory in

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

which was re-amplified in

• Daniel Altschuler, Laurent Freidel, Section 3 of: Vassiliev knot invariants and Chern-Simons perturbation theory to all orders, Commun. Math. Phys. 187 (1997) 261-287 (arxiv:q-alg/9603010)

• 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)

• 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)

and highlighted as a means to obtain graph complex-models for the de Rham cohomology of configuration spaces of points/knot spaces in

• 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)

with full details and proofs in

• 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)

see also

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

A systematic development of Euclidean perturbative quantum field theory with n-point functions considered as smooth functions on Fulton-MacPherson compactifications/wonderful compactifications of configuration spaces of points and more generally of subspace arrangements is due to

• Christoph Bergbauer, Romeo Brunetti, Dirk Kreimer, Renormalization and resolution of singularities, ESI preprint 2010 (arXiv:0908.0633, ESI:2244)

• Christoph Bergbauer, Renormalization and resolution of singularities, talks as IHES and Boston, 2009 (pdf)

• Marko Berghoff, Wonderful renormalization, 2014 (pdf, doi:10.18452/17160)

• Marko Berghoff, Wonderful compactifications in quantum field theory, Communications in Number Theory and Physics Volume 9 (2015) Number 3 (arXiv:1411.5583)

Analogous discussion for configuration spaces of points regarded as Hilbert schemes of points and with their ordinary cohomology replaced by K-theory:

• Jian Zhou, K-Theory of Hilbert Schemes as a Formal Quantum Field Theory (arXiv:1803.06080)

Discussion specifically in topological quantum field theory with an eye towards supersymmetric field theory, in terms of the ordinary homology of configuration spaces of points:

• Christopher Beem, David Ben-Zvi, Mathew Bullimore, Tudor Dimofte, Andrew Neitzke, Secondary products in supersymmetric field theory (arXiv:1809.00009)