**Prerequisite**

Hamiltonian mechanics. Quantum mechanics. $2D$ quantum field theory; renormalization. Basics of conformal field theory. Complex analysis. Lie groups, Lie algebras. Supersymmetry. Basics of Kähler geometry and differential geometry of homogeneous spaces. Recap of basic concepts will be provided, depending on the audience.

**Introduction**

The course introduces $2D$ sigma models with flag manifold target spaces. Models of this class arise in continuum limits of $SU(n)$ spin chains, but they are also of interest due to their resemblance to $4D$ gauge theories and because of their remarkable integrability properties. Examples such as the $CP^n$ model and Grassmannian models will be treated in some detail.

Upon reviewing the geometry of $SU(n)$ flag manifolds, I will explain how they serve as phase spaces for spin chain path integrals. Continuum limits of such spin chains are described by $2D$ sigma models with topological terms that affect the ground state properties. The second part of the course will be dedicated to the study of integrable sigma models and, in particular, to the duality with generalized Gross-Neveu models. I will introduce supersymmetry in this context and consider deformed models and their conformal limits.

**Syllabus**

1. Introduction to nonlinear sigma models. Minimal surfaces and sigma models in string theory. Geometric ingredients (metric, B-field, topological terms). $1D$ reductions: geodesics on manifolds.

2. Models with Kähler target spaces. The $CP^n$ model. Holomorphic maps as instanton solutions in $2D$.

3. Flag manifolds as (co)adjoint orbits. Coherent states. Elements of geometric quantization. Path integral for a single spin.

4. Invariant tensors on flag manifolds. Complex structures and Kähler metrics.

5. SU(n) representation theory via quantization of flag manifolds. Oscillator representations: Holstein-Primakoff, Schwinger-Wigner and Dyson-Maleev.

6. The large spin limit in quantum mechanics. The quantum particle on a flag manifold.

7. Spin chains. The Haldane limit. Topological terms and $Z_n$-anomalies. Large-n limit. The bilinear-biquadratic spin chain.

8. Zero curvature representations for models with symmetric target spaces. $Z_n$-graded homogeneous spaces.

9. Integrable models with complex homogeneous target spaces. The normal metric. Relation between two types of Lax connections.

10. Example of integrability: harmonic maps from $S^2$ to $CP^n$. Generalization to harmonic maps from $S^2$ to the complete flag manifold in $C^3$.

11. 'Derivation' of flag manifold sigma models from $4D$ Chern-Simons theory. The gauged linear sigma model.

12. Sigma models as Gross-Neveu models. Applications in quantum mechanics: construction of generalized spherical harmonics on $CP^n$.

13. Relation to nilpotent orbits. The Springer resolution. Quiver varieties. Orthogonal and symplectic Grassmannians.

14. Deformed (non-homogeneous) models. Calculation of beta function. Hermitian Ricci flow. Generalized Nahm equations.

15. Models with $N=(2,2)$ supersymmetry. Relation to Kähler geometry.

16. The 'sausage' model, its supersymmetric version and conformal limits.

17. The super-Thirring model. Duality with the super-cylinder sigma model. Calculation of correlation functions.

**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.