22nd February, 2027 ~ -

Students should be comfortable with classical mechanics (both Lagrangian and Hamiltonian formulations) and standard quantum mechanics. Familiarity with Lie groups and Lie algebras is expected. Basic knowledge of differential geometry, complex geometry, symplectic geometry, and vector bundles is helpful, especially in the second half of the course.

### Lecture 1. Introduction to supersymmetric quantum mechanics

Motivation for supersymmetric quantum mechanics. Definition of the \(N = 2\) supersymmetry algebra,

\[
H = \{Q,Q^{\dagger}\}, \qquad Q^{2} = 0.
\]

Complex and real supercharges; positivity of the Hamiltonian; conservation of the supercharges; consequences for the spectrum.

### Lecture 2. Representations of the \(N = 2\) SUSY algebra

Degeneracy of positive-energy states; supersymmetric doublets; the Clifford-algebra viewpoint; bosonic and fermionic sectors; \(\mathbb{Z}_2\) grading and fermion parity; structure of zero-energy states.

### Lecture 3. Extended supersymmetry and \(R\)-symmetry

Several supercharges and the \(N\)-extended supersymmetry algebra; real and complex formulations; automorphisms of the supercharges and \(R\)-symmetry; \(N = 1\) versus \(N = 2\) in one dimension.

### Lecture 4. Supersymmetry breaking and the Witten index

Broken and unbroken supersymmetry; zero-energy states; definition of the Witten index; cancellation of positive-energy states; invariance under smooth deformations and its limitations.

### Lecture 5. Supersymmetry algebra as a Lie superalgebra

\(\mathbb{Z}_2\)-graded vector spaces; even and odd elements; supercommutator and super-Jacobi identity; matrix realizations; Grassmann variables and the Grassmann-envelope construction. The algebra \(\mathfrak{gl}(1|1)\) as the basic example.

### Lecture 6. From superalgebras to supersymmetry algebras

The subalgebra \(\mathfrak{sl}(1|1)\) and its identification with the \(N = 2\) supersymmetry algebra; real forms such as \(\mathfrak{su}(1|1)\); brief comparison with \(\mathfrak{osp}(1|2)\); the idea of spacetime supersymmetry; the \((1+1)\)-dimensional super-Poincaré algebra as an example; dimensional reduction to \((0+1)\) dimensions.

### Lecture 7. Supersymmetric harmonic oscillator I

Construction using a two-component Hilbert space; bosonic and fermionic sectors; the Zeeman-type term; explicit spectrum; the zero-energy ground state and the Witten index.

### Lecture 8. Supersymmetric harmonic oscillator II

Bosonic and fermionic ladder operators; Grassmann realization; supersymmetric Fock space; relation between matrix and Grassmann descriptions; classical Hamiltonian formulation and moment maps.

### Lecture 9. Factorization of quantum-mechanical Hamiltonians

Schrödinger factorization; first-order operators \(A\) and \(A^{\dagger}\); the Riccati equation; partner Hamiltonians; spectral relations between the two partner systems.

### Lecture 10. Particle in a superpotential: the Witten model

Construction of the supersymmetric Hamiltonian from \(A\) and \(A^{\dagger}\); the superpotential \(\mathcal{W}\); supercharges; zero-energy wavefunctions \(e^{\pm \mathcal{W}}\); normalizability; elementary examples of broken and unbroken supersymmetry.

### Lecture 11. Classical formulation of supersymmetric mechanics

Classical Grassmann variables; the supersymmetric Lagrangian for the Witten model; canonical quantization; ordering ambiguities and Weyl ordering; relation between classical and quantum descriptions.

### Lecture 12. \(N = 1\) superspace and superfields

Superspace as the supergroup; Grassmann coordinates; supertranslations; supercharges and superderivatives; scalar superfields and component expansions.

### Lecture 13. Supersymmetric actions in \(N = 1\) superspace

Berezin integration; invariant superspace actions; passage to components; reconstruction of familiar supersymmetric particle models from compact superfield expressions.

### Lecture 14. \(N = 1\) supersymmetric sigma models

A particle moving on a curved target manifold; target-space metric; bosonic and fermionic fields; covariant form of the action; geometrical interpretation of the fermionic degrees of freedom.

### Lecture 15. Quantization of the \(N = 1\) sigma model and Dirac operators

Tetrads (vielbeins); fermion quantization and Clifford algebras; the quantum supercharge; the Dirac operator and the relation between the supersymmetric Hamiltonian and \(D^{2}\).

### Lecture 16. \(N = 2\) superspace and chiral superfields

Complex Grassmann coordinates; \(N = 2\) supercharges and covariant derivatives; real and chiral superfields; chirality constraints; invariant \(N = 2\) actions and Berezin integration.

### Lecture 17. \(2a\) and \(2b\) supersymmetric sigma models

Definition of the two basic classes; \(2a\) models from real superfields; \(2b\) models from chiral superfields; relation of the \(2a\) model to the Witten model; reduction from \(N = 2\) to \(N = 1\) superspace.

### Lecture 18. Complex geometry and extended supersymmetry

Complex structures arising from the second supersymmetry; Hermitian and Kähler geometry in \(2b\) models; geometric meaning of the additional SUSY transformations; brief discussion of conditions for further supersymmetry enhancement.

### Lecture 19. Landau problem on the plane as a supersymmetric model

Charged particle in a uniform magnetic field; ordinary Landau levels; supersymmetric extension; construction of supercharges; spectral pairing and zero modes.

### Lecture 20. Landau problem on the sphere

Magnetic monopole background on \(S^{2}\); magnetic charge and line bundles; geometric formulation; angular momentum algebra; supersymmetric spectrum and zero modes.

### Lecture 21. Quantization of general \(2b\) models

Classical supercharges; covariant quantization; ordering issues; quantum supercharges; Hermitian conjugation with respect to the natural scalar product; geometric interpretation of the resulting operators.

### Lecture 22. Dolbeault and Dirac operators in supersymmetric quantum mechanics

Identification of the supercharges with Dolbeault-type operators; \(Q \sim \bar{\partial}\) and \(Q^{\dagger} \sim \bar{\partial}^{\dagger}\); the supersymmetric Hamiltonian as a Laplacian; relation between Dolbeault and Dirac descriptions; spin versus spin\(^c\) structures.

### Lecture 23. Supersymmetric Landau problem on \(\mathbb{CP}^n\)

Geometry of complex projective space; homogeneous coordinates; magnetic line bundles; classical model and quantization; spectrum from symmetry and representation theory; bosonic and supersymmetric sectors.

### Lecture 24. Synthesis and outlook

Review of the main chain of ideas: SUSY algebra \(\to\) superfields \(\to\) geometry \(\to\) Dirac/Dolbeault operators \(\to\) Landau problems. Role of zero modes and index theory. Optional outlook on finite-dimensional truncations of sigma models and their relation to spin chains.

D. Bykov and V. Krivorol, Supersymmetric Quantum Mechanics and Sigma Models: An Introduction, lecture notes. Primary course text.
L. E. Gendenshtein and I. V. Krive, “Supersymmetry in Quantum Mechanics,” Soviet Physics Uspekhi 28 (1985), 645–666.
E. Witten, “Constraints on Supersymmetry Breaking,” Nuclear Physics B 202 (1982), 253.
E. Witten, “Supersymmetry and Morse Theory,” Journal of Differential Geometry 17 (1982), 661–692.
L. Frappat, P. Sorba, and A. Sciarrino, “Dictionary on Lie Superalgebras,” arXiv:hep-th/9607161 (1996).
W. Nahm, “Supersymmetries and Their Representations,” Nuclear Physics B 135 (1978), 149.
T. Frankel, The Geometry of Physics: An Introduction, 3rd ed., Cambridge University Press, 2011.
D. Tong, Supersymmetric Quantum Mechanics, online lecture notes.
A. Smilga, Differential Geometry through Supersymmetric Glasses, 2020.
G. Junker, Supersymmetric Methods in Quantum, Statistical and Solid State Physics, 2019.

Advanced Undergraduate , Graduate

Yes

Yes

English