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:

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

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

Examples

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

${\displaystyle {\begin{pmatrix}2&-2\\-2&2\end{pmatrix}}}$

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.

References

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