Toda field theory

In the study of field theory and partial differential equations, a Toda field theory (named after Morikazu Toda) is derived from the following Lagrangian: In the study of field theory and partial differential equations, a Toda field theory (named after Morikazu Toda) is derived from the following Lagrangian: Here x and t are spacetime coordinates, (,) is the Killing form of a real r-dimensional Cartan algebra h {displaystyle {mathfrak {h}}} of a Kac–Moody algebra over h {displaystyle {mathfrak {h}}} , αi is the ith simple root in some root basis, ni is the Coxeter number, m is the mass (or bare mass in the quantum field theory version) and β is the coupling constant. Then a Toda field theory is the study of a function φ mapping 2-dimensional Minkowski space satisfying the corresponding Euler–Lagrange equations. If the Kac–Moody algebra is finite, it's called a Toda field theory. If it is affine, it is called an affine Toda field theory (after the component of φ which decouples is removed) and if it is hyperbolic, it is called a hyperbolic Toda field theory. Toda field theories are integrable models and their solutions describe solitons. Liouville field theory is associated to the A1 Cartan matrix. The sinh-Gordon model is the affine Toda field theory with the generalized Cartan matrix