Nonlinear sigma model (NLSM) is a field theory, whose fields take values in Riemannian manifold M. NLSM field values can be understood as coordinates on M, for which metric is field-depenedent. NLSM is a general theoretical concept, whose geometric meaning is the reason for its successful applications in field theory, string theory and statistical mechanics. Usefullness of NLSM in QFT stems from the importance of symmetries in fundamental physics, which could be described via Standard model (SM), i.e. local QFT with gauge symmetry SU(3) × SU(2) × U(1). SM encompasses strong (QCD) and electroweak interaction and is consistent with numerous experimental data. Although perturbative approach successfully describes asymptotically free QCD, theoretical understanding of confinement of quarks and gluons in hadrons is still absent. The reason is the nonpertubative nature of confinement. In nonabelian gauge theories nonperturbative tools were used for the derivation of low energy effective action. For quite some time lattice QCD was the only way to study long distance phenomena. Recently Seiberg and Witten proposed supersymmetry (SUSY) and duality to solve the problem and understand the confinement in QCD. In many theoretical studies it is suggested that the building blocks of the ultimate QFT are 4D SUSY gauge theories, where the calculations are easier due to Feynman diagram cancellations. NLSM offers field theoretic laboratory for the study of 2D exactly solvable systems on a lattice, such as Ising model and Heisenberg antiferromagnet. O(N) 2D NLSM are frequent in condensed matter physics where they appear as antiferromagnetic spin chains or with regards to quantum Hall effect. Effective Lagrangian of superfuid He3 is NLSM as well. Strong argument for the study of NLSM is spontaneous symmetry breaking, which is crucial part of phenomenological QFT. Spontaneously broken symmetries (SBS) are not symmetries of physical state transformations, i.e. they do not leave vacuum invariant, however they are always connected with degenerate vacuum state. If continuous global symmetry is broken, we get Goldstone massless boson for every generator of broken symmetry. SBS determine nonlinear effective action of Goldstone fields. If fields are scalars, low energy effective action becomes NLSM. In the work we will present mathematical tools for the development of NLSM: supergeometry and BV formalism for the definition of AKSZ model (some generalization of NLSM). Specifically this includes Q-manifolds, P-structure, classical and quantum BRST formalism, odd-symplectic manifolds and BV formalism. Different NLSM models from particle and condensed matter physics are described. In the end topological field theories such as SUSY NLSM and Wess-Zumino-Witten model are shortly introduced.
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