Toda field theory

In the study of field theory and partial differential equations, a Toda field theory is derived from the following Lagrangian:

mathcal{L}=frac{1}{2}left[left({partial phi over partial t},{partial phi over partial t}right)-left({partial phi over partial x}, {partial phi over partial x}right)right ]-{m^2 over beta^2}sum_{i=1}^r n_i e^{beta alpha_i(phi)}.

Here x and t are spacetime coordinates, (,) is the Killing form of a real r-dimensional Cartan algebra mathfrak{h} of a Kac-Moody algebra over 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.

The sinh-Gordon model is the affine Toda field theory with the generalized Cartan matrix

begin{pmatrix} 2&-2 -2&2 end{pmatrix}

and a positive value for β after we project out a component of φ which decouples.

The sine-Gordon model is the model with the same Cartan matrix but an imaginary β.


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