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correlator as differential form on configuration space of points

Contents

Context

Algebraic Quantum Field Theory

algebraic quantum field theory (perturbative, on curved spacetimes, homotopical)

Introduction

Concepts

field theory:

Lagrangian field theory

quantization

quantum mechanical system, quantum probability

free field quantization

gauge theories

interacting field quantization

renormalization

Theorems

States and observables

Operator algebra

Local QFT

Perturbative QFT

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \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 dA=0d 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)Graphs n(Σ)A graph complex of n-point Feynman diagrams for Chern-Simons theory onΣ qiassign Feynman amplitudes of Chern-Simons theory AΩ PA (Conf n(Σ))A de Rham algebra of semi-algebraic differential forms on the FM-compactification of the configuration space of n points inΣ. \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 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

see also

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

Last revised on October 2, 2019 at 06:17:25. See the history of this page for a list of all contributions to it.