# A primer on topological solitons in scalar field theories

As the title suggests I want to learn more about topological solitons in scalar field theories. I am searching for a resource which is self-contained, in the sense that it also explains the mathematics needed, at least to some extent. It should be a text for physicists. My main goal is to learn about skyrmions, so it would be nice if skyrmions are covered. But this is optional, so if there is a good primer on topological solitons not covering skyrmions this would be fine too.

To be more precise about the "self-containedness" of the source I should maybe give some information about my maths background. I do know linear algebra, calculus and some "group and representation theory" (in the way it is used by physicists, so no in-depth knowledge here). I have very little prior knowledge of topology and almost no formal knowledge about differential geometry (Here also superficial knowledge from physics lectures.).

I do already know the book "Topological and Non-Topological Solitons in Scalar Field Theories" by Y.M. Shnir. This book gives a good overview, but I miss some mathematical details.

Two standard references are "Topological solitons" by Nic Manton and Paul Sutcliffe [1], and "Solitons and instantons" by R. Rajaraman [2].

In both cases they are fully fledged textbooks rather than short primers, but selective reading of a few chapters should suffice as a primer, with later chapters devoted to more specialised topics.

Chapters 2 to 4 of Manton and Sutcliffe cover the basics of the Lagrangian formulation of classical field theory (including spontaneous symmetry breaking and the Higgs mechanism), topology in field theory, and solitons, respectively. The introductory chapter on solitons is, unsurprisingly, focused on mathematical aspects related to group theory and topology. If you are interested in Skyrmions you should probably read chapter 9 at some point - at least one of the authors is an expert in the field.

The second chapter of Rajaraman can also be viewed as a simple primer to solitons, starting from a simpler definition of them as special solutions to non-linear wave equations. This is likely to be more mathematically accessible if you are more familiar with differential equations than with notions from group theory and topology. Chapter 5 has a section on solitons in scalar field theories, in the guise of the quantisation of static solutions to such theories, including the $$\phi^4$$ theory (it will take some reading to follow the argument).

As a primer I think reading chapter 2 of Rajaraman, followed by chapters 2 to 4 of Manton and Sutcliffe should suffice. Based on a quick scan of the contents of "Topological and Non-Topological Solitons in Scalar Field Theories" you mentioned, this should provide the missing mathematical details you are after.

References

[1] Manton, Nicholas, and Paul Sutcliffe. Topological solitons. Cambridge University Press, 2004.

[2] Rajaraman, Ramamurti. "Solitons and instantons." (1982).