Topological terms in condensed matter physics

Syllabus (under construction)

# Date Read Topic
1. Monday, May 21 Introduction. Particle on a ring. Quantum mechanics of the particle on a ring. Path integral. Topological sectors. Properties of theta terms. Effective field theory. Effective field theory approach in condensed matter physics. Low energy degrees of freedom from the spontaneous symmetry breaking and from the gauge symmetry. Nontrivial topology of the order parameter space. Exercises to lecture 1.
2. Wednesday, May 30 Basics of topology. Topology and topological spaces. Open sets. Metric topology. Relative topology. Examples of topological spaces. Continuous maps. Homeomorphism. Topological invariants. Paths and loops. Product of paths. Constant and inverse paths. Homotopy classes of loops. Fundamental or first homotopy group. Exercises to lecture 2.
3. Monday, June 4 Higher homotopy groups. Abelian nature of higher homotopy groups. Examples of topological spaces and their homotopy groups.Topological defects and textures in ordered media. Topological defects. Point and line defects. Domain walls. Topological textures. Examples: Euclidian spaces, projective planes, spheres. Hopf invariant. Quantum interference and topological terms in effective actions. Theta terms. Homotopy classes of spacetime configurations. Unitary, one-dimensional representations of corresponding homotopy groups. Complex weights of topological sectors in the path integral. Examples. Particle on a ring. Fractional statistics in 2+1. NLSM for spin chains with S>>1. Exercises to lecture 3.
4. Monday, June 11 Haldane's theta-term. Hopf term and spin-statistics of skyrmions. Topological terms induced by fermions. Fermionic model. Effective action for the fermion coupled to the unit vector. Topological term. Metric independence. Path integral representation of quantum spin. WZ terms. Multi-valued functionals. Correctness of the definition of WZ term. Reduction to theta term. General properties and some classification of topological terms. WZ terms, theta terms, and topological current terms. Calculation of topological terms. Topological terms induced by fermions. Calculation of WZ terms by variation of fermionic determinant and gradient expansion. Theta terms by reduction of other topological terms. Boundary-bulk correspondence of topological terms. Hopf term induced by fermions. Open problems. Exercises to lecture 4.