Geometrical Methods of Theoretical Physics II

NTMF060

prof. RNDr. Jiří Bičák, DrSc.

prof. RNDr. Pavel Krtouš, Ph.D.

spring 2021: 3/0 Zk

Annotation:

Hodge theory (Hodge decomposition, de Rham-Laplace operator, harmonics). Topological methods (Cohomology a homology groups, homotopy, fundamental group, homotopy equivalence, Poincare lemma). Riemann geometry in terms of forms (Cartan structure equations, calculation of the curvature). Geometry of Lie groups and algebras (geometric structures on Lie groups, Lie algebra, the action of Lie group on a manifold, vector representations). Fibre bundles (vector bundles, covariant derivative). Geometry of gauge fields (inner degrees of freedom, gauge symmetry, the action and field equations). Characteristic classes (invariant symmetric polynomials, Chern-Weil theorem, characteristic classes, Euler form). Two-component spinors (relation between spinors and vectors, soldering form, physical fields in terms of spinors).

Knowledge of the differential geometry at the level of the course NTMF059 is assumed.

Information about lectures:

Lectures are scheduled each Wednesdays at 14:00–16:20.

Due to the epidemiological situation, lectures are given at a distance. Namely, the lectures are pre-recorded, and students can access them on the internal web page.

This year, the lectures are given in the Czech language.

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.

Literature: