This book addresses the need among theoretical physicists and mathematicians for a modern, intuitive and moderately comprehensive introduction to the subject of integrable systems in two dimensions, one plus one or two plus zero. The requisite background for reading this book profitably amounts to elementary quantum field theory and statistical mechanics, in addition to basic group theory. We have tried to present all the material, both standard and new, in modern language and consistent notation.

It is perhaps still premature to evaluate the real physical impact of string theory, but it is certainly true that the current renaissance of two-dimensional physics owes much to the string wave. Traditionally, physics in two dimensions was considered a theoretical laboratory, the realm of toy models. Only after the recent work on string theory did two-dimensional quantum field theories graduate from pedagogical simplifications to serious candidates for the understanding of nature: physics in the purest aristotelian sense.

Independently of how much truth lies within string theory or elsewhere, a beautiful feature of physics in two dimensions is of course its mathematical richness. Astonishingly, almost any branch of mathematics becomes relevant in the study of two-dimensional field theories. The main physical reason for such mathematical inflation is the existence of non-trivial completely integrable two-dimensional field theories. More technically, the wonders of two dimensions have their origin in the powerful artillery of complex analysis. It is remarkable that so much of what we now understand in great generality was already contained in Onsager's solution (1944) of the two-dimensional Ising model.