Toda field theory

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In the study of field theory and partial differential equations, a Toda field theory (named after Morikazu Toda) 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 \cdot \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.


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

\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.


  • Mussardo, Giuseppe (2009), Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics, Oxford University Press, ISBN 0-199-54758-0