Because this is not a textbook, it is not designed around a traditional course or a body of specialized research results. Instead, it is a reference to help place such courses and research programs in a broader context. To make clear the somewhat unusual viewpoint taken here, we give an explicit enumeration of attributes. This book will focus on:

• Defining and visualizing concepts and relationships between them
• Stating related results and introducing related concepts for further study
• Explaining the jargon and alternative treatments found in the literature

The book will avoid:

• Historical motivations and attributions
• Proofs of theorems and derivations of results
• Tools for practical calculations

The idea is to take advantage of the reader’s intuitive ability to grasp a concept through in most cases mathematically rigorous definitions, descriptive pictures, and related results. Such an approach allows us to concisely cover a large breadth of material, hopefully providing a cross-subject synthesis while at the same time serving as a useful reference.

Of course there is no doubt that a certain level of insight only comes through studying actual derivations or proofs; in particular, this is required to obtain a clear understanding of why many of the results stated here follow from the relevant definitions. Furthermore, any significant work using or extending these ideas will require tools for calculation and exercises to build facility with these tools. Thus the present book is meant as a companion to textbooks that provide these attributes.

In this book we also go out of our way to avoid messy details, special cases, and pathological exceptions. Our goal is to provide a unified presentation of core concepts so as to illuminate the assumptions and shared structures used in our models of nature. To realize this goal, the presentation strives to be consistent and intuitive. Looking up a concept here should wherever possible not only give a definition, but also place this definition in context with related concepts.

It should also be mentioned that this book in no respect attempts to be a comprehensive reference. In particular, a thin slice of the vast world of mathematics has been selected, and definitions are not necessarily the most general, the goal being coverage of structures and usage as commonly found in theoretical physics. The path chosen through theoretical physics itself reflects an end destination of theoretical particle physics, and thus omits many other branches of study.