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 <math>r</math>, that is, the Cartan subalgebra of the algebra has dimension <math>r</math>, the Lagrangian can be written
<math display=block>\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).</math>
The background spacetime is 2-dimensional Minkowski space, with space-like coordinate <math>x</math> and timelike coordinate <math>t</math>. Greek indices indicate spacetime coordinates.
For some choice of root basis, <math>\alpha_i</math> is the <math>i</math>th simple root. This provides a basis for the Cartan subalgebra, allowing it to be identified with <math>\mathbb{R}^r</math>.
Then the field content is a collection of <math>r</math> scalar fields <math>\phi_i</math>, which are scalar in the sense that they transform trivially under Lorentz transformations of the underlying spacetime.
The inner product <math>\langle\cdot, \cdot\rangle</math> is the restriction of the Killing form to the Cartan subalgebra.
The <math>n_i</math> are integer constants, known as Kac labels or Dynkin labels.
The physical constants are the mass <math>m</math> and the coupling constant <math>\beta</math>.
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 φ 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 A<sub>1</sub> Cartan matrix, which corresponds to the Lie algebra <math>\mathfrak{su}(2)</math> in the classification of Lie algebras by Cartan matrices. The algebra <math>\mathfrak{su}(2)</math> has only a single simple root.
The sinh-Gordon model is the affine Toda field theory with the generalized Cartan matrix
:<math>\begin{pmatrix} 2&-2 \\ -2&2 \end{pmatrix}</math>
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. This Cartan matrix corresponds to the Lie algebra <math>\mathfrak{su}(2)</math>. This has a single simple root, <math>\alpha_1 = 1</math> and Coxeter label <math>n_1 = 1</math>, but the Lagrangian is modified for the affine theory: there is also an affine root <math>\alpha_0 = -1</math> and Coxeter label <math>n_0 = 1</math>. One can expand <math>\phi</math> as <math>\phi_0 \alpha_0 + \phi_1 \alpha_1</math>, but for the affine root <math>\langle \alpha_0, \alpha_0 \rangle = 0</math>, so the <math>\phi_0</math> component decouples.
The sum is <math>\sum_{i=0}^1 n_i\exp(\beta \alpha_i\phi) = \exp(\beta \phi) + \exp(-\beta\phi).</math> Then if <math>\beta</math> is purely imaginary, <math>\beta = ib</math> with <math>b</math> real and, without loss of generality, positive, then this is <math>2\cos(b\phi)</math>. The Lagrangian is then
<math display=block>\mathcal{L} = \frac{1}{2}\partial_\mu \phi \partial^\mu \phi + \frac{2m^2}{b^2}\cos(b\phi),</math>
which is the sine-Gordon Lagrangian.