Wilson Lines in Quantum Field Theory / Igor Olegovich Cherednikov, Tom Mertens, Frederik F. Van der Veken.

Includes bibliographical references (pages 249-251) and index.

Preface; 1 Introduction: What are Wilson lines?; 2 Prolegomena to the mathematical theory of Wilson lines; 2.1 Shuffle algebra and the idea of algebraic paths; 2.1.1 Shuffle algebra: Definition and properties; 2.1.2 Chen's algebraic paths; 2.1.3 Chen iterated integrals; 2.2 Gauge fields as connections on a principal bundle; 2.2.1 Principal fiber bundle, sections and associated vector bundle; 2.2.2 Gauge field as a connection; 2.2.3 Horizontal lift and parallel transport; 2.3 Solving matrix differential equations: Chen iterated integrals; 2.3.1 Derivatives of a matrix function.

880-01 3.4 The group of generalized loops3.5 Generalized loops and the Ambrose-Singer theorem; 3.6 The Lie algebra of the group of the generalized loops; 4 Shape variations in the loop space; 4.1 Path derivatives; 4.2 Area derivative; 4.3 Variational calculus; 4.4 Fréchet derivative in a generalized loop space; 5 Wilson lines in high-energy QCD; 5.1 Eikonal approximation; 5.1.1 Wilson line on a linear path; 5.1.2 Wilson line as an eikonal line; 5.2 Deep inelastic scattering; 5.2.1 Kinematics; 5.2.2 Invitation: the free parton model; 5.2.3 A more formal approach; 5.2.4 Parton distribution functions.

5.2.5 Operator definition for PDFs5.2.6 Gauge invariant operator definition; 5.2.7 Collinear factorization and evolution of PDFs; 5.3 Semi-inclusive deep inelastic scattering; 5.3.1 Conventions and kinematics; 5.3.2 Structure functions; 5.3.3 Transverse momentum dependent PDFs; 5.3.4 Gauge-invariant definition for TMDs; A Mathematical vocabulary; A.1 General topology; A.2 Topology and basis; A.3 Continuity; A.4 Connectedness; A.5 Local connectedness and local path-connectedness; A.6 Compactness; A.7 Countability axioms and Baire theorem; A.8 Convergence; A.9 Separation properties.

A.10 Local compactness and compactificationA. 11 Quotient topology; A.12 Fundamental group; A.13 Manifolds; A.14 Differential calculus; A.15 Stokes' theorem; A.16 Algebra: Rings and modules; A.17 Algebra: Ideals; A.18 Algebras; A.19 Hopf algebra; A.20 Topological, C*-, and Banach algebras; A.21 Nuclear multiplicative convex Hausdorff algebras and the Gel'fand spectrum; B Notations and conventions in quantum field theory; B.1 Vectors and tensors; B.2 Spinors and gamma matrices; B.3 Light-cone coordinates; B.4 Fourier transforms and distributions; B.5 Feynman rules for QCD; C Color algebra.

The objective of this book is to get the reader acquainted with theoretical and mathematical foundations of the concept of Wilson loops in the context of modern quantum field theory. Itteaches how to perform independently with some elementary calculationson Wilson lines, and shows the recent development of the subject in different important areas of research.

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