Holonomy Structure of Landau Singularities and Feynman Integrals

Since the pioneering work of Landau and Nakanishi on general Feynman integrals, it has been known that the singularities of the S-matrix, causal Green's functions and related functions are described by the so called Landau equations. These Landau singularities were physically interpreted as the macroscopic causality by the theoretical physicists working in 5-matrix theory, and the notion of essential support was obtained (Chandler-Stapp [3], lagolnitzer-Stapp [5]). In the branch of mathematics, on the other side, the theory of microfunction has evolved and has been applied powerfully to the general theory of partial differential equations (Sato-Kawai-Kashiwara [11]). It contained the essential support theory as the singularity spectrum of a function (i.e. its support viewed as a microfunction), and was far-reaching because of its close connection with the theory of differential equations. Namely the method of microlocal analysis, based on the theory of holonomic systems, has provided a systematic way of handling functions with "natural background" and found most effective applications to various problems of mathematics, such as the theory of ^-functions and Fourier transformations (Sato [10], Kashiwara [6]). It was then recognized that the Landau equations give holonomic varieties, which led one to the holonomicity postulate of ^-matrix and related quantities (Sato [10]). In the present paper we shall study the holonomy structure of Landau singularities and Feynman integrals from this standpoint. First we review the notion of Landau varieties. In contrast with