Knot complement

In the study of field theory and partial differential equations, a Toda field theory (named after Morikazu Toda) is derived from the following Lagrangian:

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

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.

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

$\left(\begin{array}{cc}2& -2\\ -2& 2\end{array}\right)$

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.

References

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