Matrix Integrals and Feynman Diagrams in the Kontsevich Model

Feynman diagrams and the KdV hierarchy

The generating series of the intersection numbers of the stable cohomology classes on moduli spaces of curves satisfies the string equation and a KdV hierarchy. Kontsevich's original proof of this result uses a matrix model and the matrix Airy equation. Witten then recasted Kontsevich's results in terms of Virasoro algebras, by means of an ingenious mixture of Feynman diagrams techniques and integrations by parts with some "rather formidable choice" of the integrands. In this note we show how Witten's formidable choices can be bartered for standard Feynman diagram manipulations, as soon as one…

Matroids, motives and conjecture of Kontsevich

Let G be a finite connected graph. The Kirchhoff polynomial of G is a certain homogeneous polynomial whose degree is equal to the first betti number of G. These polynomials appear in the study of electrical circuits and in the evaluation of Feynman amplitudes. Motivated by work of D. Kreimer and D. J. Broadhurst associating multiple zeta values to certain Feynman integrals, Kontsevich conjectured that the number of zeros of a Kirchhoff polynomial over the field with q elements is always a polynomial function of q. We show that this conjecture is false by relating the schemes defined by Kirchho…

Multiple zeta values in deformation quantization

Kontsevich's 1997 formula for the deformation quantization of Poisson brackets is a Feynman expansion involving volume integrals over moduli spaces of marked disks. We develop a systematic theory of integration on these moduli spaces via suitable algebras of polylogarithms, and use it to prove that Kontsevich's integrals can be expressed as integer-linear combinations of multiple zeta values. Our proof gives a concrete algorithm for calculating the integrals, which we have used to produce the first software package for the symbolic calculation of Kontsevich's formula.

广义Kontsevich图与r-自旋相交数的拓扑递归

Topological recursion for generalised Kontsevich graphs and r-spin intersection numbers

本文定义四类广义Kontsevich图并证明主要类型ciliated maps满足Tutte递归，从而可由拓扑递归计算，进而建立其与r-自旋相交数的恒等式，为组合理解Witten类及探索拓扑递归的辛不变性提供新工具。

Periods of Feynman Diagrams and GKZ D-Modules

We study differential equations for Feynman amplitudes and we show that the corresponding D-module is isomorphic to a GKZ D-modules. We show that the sheaf of solutions to the D-module is isomorphic to a certain relative homology and the amplitudes are periods of a relative motive. Using these ideas, we develop a method of regularization which specializes to dimensional regularization and analytic regularization.

Wick rotations, Eichler integrals, and multi-loop Feynman diagrams

Using contour deformations and integrations over modular forms, we compute certain Bessel moments arising from diagrammatic expansions in two-dimensional quantum field theory. We evaluate these Feynman integrals as either explicit constants or critical values of modular $L$-series, and verify several recent conjectures of Broadhurst.