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Course, academic year 2024/2025
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Geometrical Methods of Theoretical Physics II - NTMF060
Title: Geometrické metody teoretické fyziky II
Guaranteed by: Institute of Theoretical Physics (32-UTF)
Faculty: Faculty of Mathematics and Physics
Actual: from 2023
Semester: summer
E-Credits: 4
Hours per week, examination: summer s.:3/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Additional information: http://utf.mff.cuni.cz/vyuka/NTMF060
Guarantor: prof. RNDr. Pavel Krtouš, Ph.D.
Comes under: Doporučené přednášky 1/2
Annotation -
Riemann geometry in terms of forms, Hodge theory, topological methods. Lie groups and algebras. Fibre bundles, geometry of gauge fields, characteristic classes. Two-component spinors.
Last update: Houfek Karel, doc. RNDr., Ph.D. (04.02.2021)
Course completion requirements -

Oral exam.

Last update: Krtouš Pavel, prof. RNDr., Ph.D. (21.04.2023)
Literature -
  • C. von Westenholz: Differential Forms in Mathematical Physics, North-Holland, 1978.
  • M. Fecko: Differential Geometry and Lie Groups for Physicists, Cambridge Univ. Press, 2011.
  • T. Frankel: The Geometry of Physics - An Introduction, Cambridge Univ. Press, 1999.
  • M. Nakahara: Geometry, Topology and Physics, Taylor&Francis, 2003.
  • Ch. Nash, S. Sen: Topology and Geometry for Physicists, Dover Publ., 2011.
  • J. A. de Azcárraga, J. M. Izquierdo: Lie Groups, Lie Algebras, Cohomology and some Applications in Physics, Cambridge Univ. Press, 1995.
  • Ch. J. Isham: Moddern Differential Geometry For Physicists, World Scientific, 1989.
  • E. W. Mielke: Geometrodynamics of Gauge Fields, Springer, 2017.
  • R. Penrose a W. Rindler: Spinors and space-time, Cambridge Univ. Press, 1999.
  • P. O'Donnell: 2-Spinors in General Relativity, World Scientific, 2003.
  • C. W. Misner, K. S. Thorne a J. A. Wheeler: Gravitation, Freedman, 1973.
  • S. Kobayashi a K. Nomizu: Foundations of Differential Geometry I, Interscience Publishers, 1963.
  • M. Spivak: A Comprehensive Introduction to Differential Geometry, Publish or Perish Press, 1970-1979.
Last update: Krtouš Pavel, prof. RNDr., Ph.D. (28.01.2021)
Requirements to the exam -

The oral exam. Students are examined from material in the syllabus and covered in lectures.

Last update: Krtouš Pavel, prof. RNDr., Ph.D. (21.04.2023)
Syllabus -
Hodge theory
Scalar product on forms, Hodge dual, coderivative, de Rham-Laplace and Beltrami-Laplace operators. Hodge decomposition, potential and copotential, harmonics, cohomology.
Topological methods
Cohomology a homology groups, homotopy, fundamental group, homotopy equivalence, homotopy operator, contraction, Poincare lemma.
Riemann geometry in terms of forms
Exterior calculus (overview). Maxwell theory. Othonormal frames, Cartan structure equations, Ricci coefficients. Bianchi identities. Calculation of the curvature, example - Vaidya metric.
Geometry of Lie groups and algebras
Lie groups, construction of Lie algebra, exponential mapping, Killing metric, structure constants. Bi-invariant metric, measure, covariant derivative. Adjoint representations. The action of Lie group on a manifold, flows and their generators. Representations on vector spaces.
Fibre bundles
Abstract fibre bundles. Vector bundles and their geometry, covariant derivative, vector potential and curvature. Objects on the gauge-algebra bundle.
Geometry of gauge fields
Inner degrees of freedom and their description in terms of vector bundles. Gauge symmetry. Gauge group and gauge algebra bundles. Gauge and Yang-Mills fields. The action and field equations. Electromagnetic and charged fields.
Characteristic classes
Invariant symmetric polynomials in curvature, Chern-Weil theorem, characteristic classes, Chern class and character, Pontrjagin class, Euler form, integral quantities.
Two-component spinors
Space of spinors, antisymmetric metric, soldering form. Relation between spinors and vectors. Geometric quantities and physical fields in terms of spinors. Electromagnetic field and curvature.
Last update: Krtouš Pavel, prof. RNDr., Ph.D. (28.01.2021)
 
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