The Feymann integral and completely integrable systems

Abstract

We review recent attempts to relate the concept of Feynman integral with integrable systems. We then present a framework which is rooted in the hypothetical relationship between the heuristic concept of Feynman integral in theoretical physics and the rigorous mathematical results derived from the theory of (physically signi?cant) completely integrable systems. This idea originates primarily from Witten’s conjecture and Kontsevich’s model which conjecturally able to formulate this remarkable connection. Essentially this link refers to a generator function of intersection numbers on moduli space for stable curves (or r-spin curves) and the tau-function of Korteweg-de Vries (or Gelfand-Dikii) hierarchy. Based on Witten-Kontsevich’s result, we also discuss the idea of recasting the tau function in terms of the Feynman diagrams.