Non-linear Sigma Model - O(3) Non-linear Sigma Model

O(3) Non-linear Sigma Model

One of the most famous examples, of particular interest due to its topological properties, is the O(3) nonlinear sigma model in 1 + 1 dimensions, with the Lagrangian density

where with the constraint and . This model allows for topological finite action solutions, as at infinite space-time the Lagrangian density must vanish, meaning at infinity. Therefore in the class of finite-action solutions we may identify the points at infinity as a single point, i.e. that space-time can be identified with a Riemann Sphere. Since the -field lives on a sphere as well, we have a mapping, the solutions of which are classified by the Second Homotopy group of a 2-sphere. These solutions are called the O(3) Instantons.