# nLab correlator as differential form on configuration space of points

Contents

### Context

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

singular cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

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

## References

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

which was re-amplified in

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