In mathematical physics, one of the fascinating issues is the study of
integrable systems. In particular, non-perturbative techniques that have
been developed have triggered significant insight for real physics.
There are basically two notions of integrability: classical
integrability and quantum integrability. In this book, the focus is on
the former, classical integrability. When the system has a finite number
of degrees of freedom, it has been well captured by the Arnold-Liouville
theorem. However, when the number of degrees of freedom is infinite, as
in classical field theories, the integrable structure is enriched
profoundly. In fact, the study of classically integrable field theories
has a long history and various kinds of techniques, including the
classical inverse scattering method, which have been developed so far.
In previously published books, these techniques have been collected and
well described and are easy to find in traditional, standard
textbooks.
One of the intriguing subjects in classically integrable systems is the
investigation of deformations preserving integrability. Usually, it is
not considered systematic to perform such a deformation, and one must
study systems case by case and show the integrability of the deformed
systems by constructing the associated Lax pair or action-angle
variables.
Recently, a new, systematic method to perform integrable deformations of
2D non-linear sigma models was developed. It was invented by C. Klimcik
in 2002, and the integrability of the deformed sigma models was shown in
2008. The original work was done for 2D principal chiral models, but it
has been generalized in various directions nowadays. In this book, the
recent progress on this Yang-Baxter deformation is described in a
pedagogical manner, including some simple examples. Applications of
Yang-Baxter deformation to string theory are also described briefly.
