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

- $mathcal\{L\}=frac\{1\}\{2\}left[left(\{partial\; phi\; over\; partial\; t\},\{partial\; phi\; over\; partial\; t\}right)-left(\{partial\; phi\; over\; partial\; x\},\; \{partial\; phi\; over\; partial\; x\}right)right\; ]-\{m^2\; over\; beta^2\}sum\_\{i=1\}^r\; n\_i\; e^\{beta\; alpha\_i(phi)\}.$

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 $mathfrak\{h\}$, α_{i} is the i^{th} simple root in some root basis, n_{i} 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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Last updated on Monday March 03, 2008 at 06:45:45 PST (GMT -0800)

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Last updated on Monday March 03, 2008 at 06:45:45 PST (GMT -0800)

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