Simple Lie Groups and Grand Unified Theories

Lecture series in the Ecole Doctorale de Physique by I. Schienbein, A. Wingerter

• Part I: Simple Lie Groups
  10.4., 12.4., 17.4., 19.4., 24.4., 26.4.


• Part II: Grand Unified Theories
  5.6., 7.6., 12.6., 14.6., 19.6., 21.6.

• Knowledge of basic group theory
• Knowledge of the groups SU(2), SO(3), SU(3) helpful.
  Will be repeated in the first two lectures.

• Content of Part I
• Knowledge of the Standard Model of Particle Physics

• Lecture 01: SU(2)
  Introduction to the simplest, non-trivial Lie algebra SU(2); definition of the algebra;
  generators; connection between group and algebra; notion of representation;
  reducible and irreducible representations; spin algebra; rotations and SO(3)
• Lecture 02: SU(3)
  Next-to-simplest Lie algebra; generators (Gell-Mann matrices); SU(2) subalgebras;
  commutation relations and abstract definition of the algebra; adjoint representation;
  Cartan subalgebra; root vectors; color SU(3) and flavor SU(3); classification of mesons (the eightfold way)
• Lecture 03: The General Structure of Simple Lie Algebras
  Killing form; simple and semi-simple Lie algebras; roots and weights; simple roots;
  Cartan matrix; Dynkin diagram of SU(2) and SU(3)
• Lecture 04: Classification of Lie Algebras
  Dynkin diagrams; ADE classification of simple Lie algebras; classical and exceptional Lie algebras
• Lecture 05: Dynkin's Approach to Representation Theory
  Roots and weights again; finding all representations of a given algebra; finding all weights
  of a given representation; determining the subalgebras; regular and irregular embedding
• Lecture 06: Computational Aspects of Representation Theory
  Dimension of representations; Freudenthal's formula; Weyl's dimension formula; tensor products;
  breaking to subalgebras and reducing representations; Casimir operator(s)

• Lecture 07: Motivation and Background for Grand Unification
  Theoretical motivation for beyond-the-Standard-Model physics; gauge coupling unification;
  big desert hypothesis; grand unification scale;
  SU(5), SO(10), SU(4)xSU(2)xSU(2), E_6 as groups proposed for the unification of particle interactions
• Lecture 08: The Simplest Grand Unified Theory: SU(5)
  Details of SU(5); representations; choice of fermion and Higgs representations;
  doublet-triplet splitting problem; spontaneous symmetry breaking of SU(5)
• Lecture 09: Phenomenology of SU(5)
  Lagrangian; fermion masses; extra gauge bosons; interactions; Feynman diagrams; low-energy effective theory
• Lecture 10: Predictions of SU(5)
  Running of the gauge couplings; beta-functions; grand unification scale M_GUT and
  Weinberg angle sin^2 theta_W; proton decay; limits from experiment
• Lecture 11: SO(10) and Neutrino Masses
  Sketch of SO(10) properties; representations; choice of fermion and Higgs representations;
  seesaw mechanism and neutrino masses; B-L symmetry
• Lecture 12: Summary & Outlook
  Supersymmetry; extra dimensions; orbifold GUTs; string theory; ultraviolet completion;
  open problems and critique

• R. N. Cahn, Semisimple Lie Algebras and Their Representations.
  Benjamin/Cummings, Menlo Park, USA, 1984
• G. G. Ross, Grand Unified Theories.
  Benjamin/Cummings, Menlo Park, USA, 1984