Feynman Diagrams and Low-Dimensional Topology

  title={Feynman Diagrams and Low-Dimensional Topology},
  author={M. Kontsevich},
We shall describe a program here relating Feynman diagrams, topology of manifolds, homotopical algebra, non-commutative geometry and several kinds of “topological physics.” 
Weyl n-Algebras
We introduce Weyl n-algebras and show how their factorization complex may be used to define invariants of manifolds. In the appendix, we heuristically explain why these invariants must beExpand
Perturbative Topological Field Theory
We give a review of the application of perturbative techniques to topologi-cal quantum eld theories, in particular three-dimensional Chern-Simons-Witten theory and its various generalizations. ToExpand
Weyl n-algebras
We introduce Weyl n-algebras and show how their factorization homology may be used to define invariants of manifolds. In the appendix we heuristically explain why these invariants must beExpand
Combinatorics and algebra of tensor calculus
In this paper, motivated by the theory of operads and PROPs we reveal the combinatorial nature of tensor calculus for strict tensor categories and show that there exists a monad which is described byExpand
Homological algebra related to surfaces with boundary
In this article we describe an algebraic framework which can be used in three related but different contexts: string topology, symplectic field theory, and Lagrangian Floer theory of higher genus. ItExpand
Topological Field Theories and Harrison Homology
Tools and arguments developed by Kevin Costello are adapted to families of “Outer Spaces” or spaces of graphs. This allows us to prove a version of Deligne’s conjecture: the Harrison homologyExpand
Homotopy Gerstenhaber algebras and topological field theory
We prove that the BRST complex of a topological conformal field theory is a homotopy Gerstenhaber algebra, as conjectured by Lian and Zuckerman in 1992. We also suggest a refinement of the originalExpand
Graph cohomology classes in the Batalin-Vilkovisky formalism
This paper gives a conceptual formulation of Kontsevich’s ‘dual construction’ producing graph cohomology classes from a differential graded Frobenius algebra with an odd scalar product. OurExpand
On operad structures of moduli spaces and string theory
We construct a real compactification of the moduli space of punctured rational algebraic curves and show how its geometry yields operads governing homotopy Lie algebras, gravity algebras andExpand
Abstract Hodge Decomposition and Minimal Models for Cyclic Algebras
We show that an algebra over a cyclic operad supplied with an additional linear algebra datum called Hodge decomposition admits a minimal model whose structure maps are given in terms of summationExpand


Some time ago B. Feigin, V. Retakh and I had tried to understand a remark of J. Stasheff [S1] on open string theory and higher associative algebras [S2]. Then I found a strange construction ofExpand
Intersection theory on the moduli space of curves and the matrix airy function
We show that two natural approaches to quantum gravity coincide. This identity is nontrivial and relies on the equivalence of each approach to KdV equations. We also investigate related mathematicalExpand
Quantum field theory techniques in graphical enumeration
We present a method for counting closed graphs on a compact Riemannian surface, based on techniques suggested by quantum field theory.
Topological quantum field theory
A twisted version of four dimensional supersymmetric gauge theory is formulated. The model, which refines a nonrelativistic treatment by Atiyah, appears to underlie many recent developments inExpand
Ribbon graphs and their invaraints derived from quantum groups
The generalization of Jones polynomial of links to the case of graphs inR3 is presented. It is constructed as the functor from the category of graphs to the category of representations of the quantumExpand
Perturbative aspects of the Chern-Simons field theory
Abstract The quantization of the non-abelian Chern-Simons theory in three dimensions is performed in the framework of the BRS formalism. General covariance is preserved on the physical subspace. TheExpand
Perturbative series and the moduli space of Riemann surfaces
On utilise des techniques de la theorie quantique des champs pour calculer des quantites reliees aux groupes de symetrie de paires (F, G), ou F est une surface et G une epine de F
Computer tests of Witten's Chern-Simons theory against the theory of three-manifolds.
  • Freed, Gompf
  • Physics, Medicine
  • Physical review letters
  • 1991
Witten's (2+1)-dimensional Chern-Simons theory with gauge group SU(2) is studied and the perturbation theory is checked against the exact solution of the path integral. Expand
Quantum field theory and the Jones polynomial
It is shown that 2+1 dimensional quantum Yang-Mills theory, with an action consisting purely of the Chern-Simons term, is exactly soluble and gives a natural framework for understanding the JonesExpand
Chern-Simons-Witten invariants of lens spaces and torus bundles, and the semiclassical approximation
We derive explicit formulas for the Chern-Simons-Witten invariants of lens spaces and torus bundles overS1, for arbitrary values of the levelk. Most of our results are for the groupG=SU(2), thoughExpand