Definitions

# Toda field theory

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

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

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

$begin\left\{pmatrix\right\} 2&-2 -2&2 end\left\{pmatrix\right\}$

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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