Ebook: Planar Ising Correlations

Understanding certain exactly solvable models in statistical mechanics and quantum field theory from a mathematical physics perspective is a very important and active area of research. These models include the scaling limits of the 2-D Ising (lattice) model, and more generally, a class of 2-D quantum fields known as holonomic fields. Steady progress has been made in understanding the special mathematical features of these models. Over the years, new results have made it possible to obtain a detailed nonperturbative analysis of the many spin correlations. This book is principally devoted to an analysis showing that the scaling functions are tau functions associated with monodromy-preserving deformations of the Dirac equation. While charting a fairly direct route to this analysis via new results of Palmer and others, as well as previous research of the Kyoto School---Sato, Miwa, and Jimbo---are the primary focus of this book, other interesting mathematical insights occur all along the way. For example, the Ising model has been a source of rich mathematics from Szego limit theorems to Wick type theorems for infinite spin groups. Also, some aspects of the solution of the Ising model are elegantly expressed in terms of Pfaffian and determinant bundles over infinite dimensional Grassmannians. These construct generalize the more familar objects in finite dimensional algebraic geometry and have appeared only recently in the mathematics literature. Exploring the Ising model as a microcosm of the confluence of interesting ideas in mathematics and physics, this work will appeal to graduate students, mathematicians, and physicists interested in the mathematics of statistical mechanics and quantum field theory.