Toda field theory

In mathematics and physics, specifically the study of field theory and partial differential equations, a Toda field theory, named after Morikazu Toda, is specified by a choice of Lie algebra and a specific Lagrangian.

Formulation
Fixing the Lie algebra to have rank $$r$$, that is, the Cartan subalgebra of the algebra has dimension $$r$$, the Lagrangian can be written

$$\mathcal{L}=\frac{1}{2}\left\langle \partial_\mu \phi, \partial^\mu \phi \right\rangle -\frac{m^2}{\beta^2}\sum_{i=1}^r n_i \exp(\beta \langle\alpha_i, \phi\rangle).$$

The background spacetime is 2-dimensional Minkowski space, with space-like coordinate $$x$$ and timelike coordinate $$t$$. Greek indices indicate spacetime coordinates.

For some choice of root basis, $$\alpha_i$$ is the $$i$$th simple root. This provides a basis for the Cartan subalgebra, allowing it to be identified with $$\mathbb{R}^r$$.

Then the field content is a collection of $$r$$ scalar fields $$\phi_i$$, which are scalar in the sense that they transform trivially under Lorentz transformations of the underlying spacetime.

The inner product $$\langle\cdot, \cdot\rangle$$ is the restriction of the Killing form to the Cartan subalgebra.

The $$n_i$$ are integer constants, known as Kac labels or Dynkin labels.

The physical constants are the mass $$m$$ and the coupling constant $$\beta$$.

Classification of Toda field theories
Toda field theories are classified according to their associated Lie algebra.

Toda field theories usually refer to theories with a finite Lie algebra. If the Lie algebra is an affine Lie algebra, it is called an affine Toda field theory (after the component of &phi; which decouples is removed). 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, which corresponds to the Lie algebra $$\mathfrak{su}(2)$$ in the classification of Lie algebras by Cartan matrices. The algebra $$\mathfrak{su}(2)$$ has only a single simple root.

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 &beta; after we project out a component of &phi; which decouples.

The sine-Gordon model is the model with the same Cartan matrix but an imaginary &beta;. This Cartan matrix corresponds to the Lie algebra $$\mathfrak{su}(2)$$. This has a single simple root, $$\alpha_1 = 1$$ and Coxeter label $$n_1 = 1$$, but the Lagrangian is modified for the affine theory: there is also an affine root $$\alpha_0 = -1$$ and Coxeter label $$n_0 = 1$$. One can expand $$\phi$$ as $$\phi_0 \alpha_0 + \phi_1 \alpha_1$$, but for the affine root $$\langle \alpha_0, \alpha_0 \rangle = 0$$, so the $$\phi_0$$ component decouples.

The sum is $$\sum_{i=0}^1 n_i\exp(\beta \alpha_i\phi) = \exp(\beta \phi) + \exp(-\beta\phi).$$ Then if $$\beta$$ is purely imaginary, $$\beta = ib$$ with $$b$$ real and, without loss of generality, positive, then this is $$2\cos(b\phi)$$. The Lagrangian is then $$\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi + \frac{2m^2}{b^2}\cos(b\phi),$$ which is the sine-Gordon Lagrangian.