Feynman diagrams and low-dimensional topology. (English) Zbl 0872.57001

Kontsevich, Maxim

Joseph, A. (ed.) et al., First European congress of mathematics (ECM), Paris, France, July 6-10, 1992. Volume II: Invited lectures (Part 2). Basel: Birkhäuser. Prog. Math. 120, 97-121 (1994).

This sweeping paper outlines a program relating Feynman diagrams, topology of manifolds, homotopical algebra, noncommutative geometry and “topological physics”. The first section gives two constructions relating associative algebras with moduli spaces of algebraic curves. For \(g\geq 0\), \(n\geq 1\) and \(2g+ n>2\) let \(M_{g,n}\) be the coarse moduli space of smooth complex algebraic curves of genus \(g\) with \(n\) unlabelled punctures. The first construction associates with certain differential algebras \(V\) cohomology classes in \(H^{\text{even}} (M_{g,n},C)\), the second homology classes in \(H_{\text{even}} (M_{g,n},C)\). The first construction is outlined and the second indicated. In the second a chain is defined which can be identified with the decomposition over Feynman diagrams of a finite dimensional integral.
The second section, independent of the first, describes perturbative Chern-Simons theory. If \(M\) is a closed oriented 3-manifold and \(G\) a compact Lie subgroup of unitary matrices \(U(N)\), let \(A\) be a 1-form on \(M\) with values in the Lie algebra of the group \(G\). There is an extensive study of the numbers \(Z_k\), defined by \(\int (e^{\text{CS} (A)})^k\) \(DA\) over the quotient space of the space of all connections in \(G\)-bundles over \(M\) modulo gauge transformations \(-\text{ CS} (A)\) is the Chern-Simons functional of \(A\). For a family of loops \(L_i\) in a 3-manifold and \(\rho_i\) finite-dimensional unitary representations of the group \(G\), \(Z_k ((L_1,\rho_1) \dots (L_n, \rho_n))\) is defined as a Feynman integral along the lines of \(\mathbb Z_k\). This second section ends with an exploration of the connections between the perturbative Chern-Simons theory for knots and the Vassiliev invariants.
The third and final section develops a non-commutative symplectic geometry which gives the common algebraic background of the first two parts. Three infinite dimensional Lie algebras are defined and the goal is the computation of their stable homology. The stable homology is a free polynomial algebra generated by the subspace of primitive elements. The main result gives an explicit computation of the primitive elements for these three Lie algebras.
For the entire collection see [Zbl 0807.00008].

Reviewer: George E. Lang jun. (Fairfield)

MSC:

57R57	Applications of global analysis to structures on manifolds
57M25	Knots and links in the \(3\)-sphere (MSC2010)
81T18	Feynman diagrams
14H15	Families, moduli of curves (analytic)
32G15	Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)

Keywords:

Feynman diagrams; manifolds; homotopical algebra; noncommutative geometry; moduli spaces; algebraic curves; Chern-Simons theory; 3-manifold; Vassiliev invariants