Supersymmetric quantum mechanics
Lecturer
Dmitri Bykov
Date
8th April ~ 30th May, 2025
Location
| Weekday | Time | Venue | Online | ID | Password |
|---|---|---|---|---|---|
| Tuesday,Friday | 13:30 - 15:05 | A3-2-301 | Zoom 16 | 468 248 1222 | BIMSA |
Prerequisite
Hamiltonian mechanics. Quantum mechanics. Complex analysis. Lie groups, Lie algebras. Basics of differential geometry. Homogeneous spaces. Recap of the basic concepts will be provided, depending on the audience.
Syllabus
1. Supersymmetry algebra and its representations. R-symmetry.
2. Supersymmetric harmonic oscillator. Superpotential. Spontaneous supersymmetry breaking.
3. Superspace and superfields ($N=1, 2, 4$). Chiral superfields ($N=2, 4$).
4. Witten’s model ($N=2$). 1D Wess-Zumino model ($N=4$). Reduction of $N=2$ superfields to $N=1$ superfields.
5. Supersymmetric particle in a magnetic field. Supersymmetric particle on a sphere in the field of a `Dirac monopole'. Quantization. Magnetic harmonics.
6. Magnetic harmonics on $\mathbb{CP}^n$ (complex projective space). Spectrum of the Dirac operator.
7. `Shape-invariant' potentials. Pöschl-Teller, Coulomb potential, etc. Exact solution for the spectrum.
8. Supersymmetric sigma models. $N=2$ supersymmetry: de Rham and Dolbeault complexes. Kähler target spaces and $N=4$ supersymmetry. Geometric intepretation of R-symmetry.
9. Witten index. Application to sigma models: the de Rham operator and its index (Euler characteristic). The Gauss-Bonnet formula in arbitrary dimension.
10. Sigma models with a superpotential. Morse theory and tunneling effects.
11. Supersymmetric Chern-Simons quantum mechanics. Nonlinear chiral multiplets. Truncations of 1D sigma models.
12. Gauge fields and symplectic reduction. Kähler quotients from supersymmetry.
13. Extended supersymmetry. Relation to hyper-Kähler geometry. Supersymmetry in higher dimensions. Dimensional reduction to 1D.
14. Superconformal quantum mechanics.
15. Parasupersymmetric quantum mechanics.
2. Supersymmetric harmonic oscillator. Superpotential. Spontaneous supersymmetry breaking.
3. Superspace and superfields ($N=1, 2, 4$). Chiral superfields ($N=2, 4$).
4. Witten’s model ($N=2$). 1D Wess-Zumino model ($N=4$). Reduction of $N=2$ superfields to $N=1$ superfields.
5. Supersymmetric particle in a magnetic field. Supersymmetric particle on a sphere in the field of a `Dirac monopole'. Quantization. Magnetic harmonics.
6. Magnetic harmonics on $\mathbb{CP}^n$ (complex projective space). Spectrum of the Dirac operator.
7. `Shape-invariant' potentials. Pöschl-Teller, Coulomb potential, etc. Exact solution for the spectrum.
8. Supersymmetric sigma models. $N=2$ supersymmetry: de Rham and Dolbeault complexes. Kähler target spaces and $N=4$ supersymmetry. Geometric intepretation of R-symmetry.
9. Witten index. Application to sigma models: the de Rham operator and its index (Euler characteristic). The Gauss-Bonnet formula in arbitrary dimension.
10. Sigma models with a superpotential. Morse theory and tunneling effects.
11. Supersymmetric Chern-Simons quantum mechanics. Nonlinear chiral multiplets. Truncations of 1D sigma models.
12. Gauge fields and symplectic reduction. Kähler quotients from supersymmetry.
13. Extended supersymmetry. Relation to hyper-Kähler geometry. Supersymmetry in higher dimensions. Dimensional reduction to 1D.
14. Superconformal quantum mechanics.
15. Parasupersymmetric quantum mechanics.
Video Public
Yes
Notes Public
Yes
Language
English
Lecturer Intro
Dmitri Bykov received his Ph.D. in theoretical physics from Trinity College Dublin in 2011. He held postdoc positions at Nordita (Stockholm, Sweden), Max-Planck-Institute for Gravitational Physics (Potsdam, Germany) and Max-Planck-Institute for Physics (Munich, Germany) and obtained his Habilitation (Dr. Phys.-Math. Sci.) in 2018. Since 2021 he is a leading scientific researcher at Steklov Mathematical Institute and at the Institute for Theoretical and Mathematical Physics (Moscow, Russia). His research interests include applications of differential-geometric methods in quantum field theory and his main results are related to 2D sigma models.