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.^[1]

Formulation

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

$${\displaystyle {\mathcal {L))={\frac {1}{2))\left\langle \partial _{\mu }\phi ,\partial ^{\mu }\phi \right\rangle -{\frac {m^{2)){\beta ^{2))}\sum _{i=1}^{r}n_{i}\exp(\beta \langle \alpha _{i},\phi \rangle ).}$$

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

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

Then the field content is a collection of ${\displaystyle r}$ scalar fields ${\displaystyle \phi _{i))$, which are scalar in the sense that they transform trivially under Lorentz transformations of the underlying spacetime.

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

The ${\displaystyle n_{i))$ are integer constants, known as Kac labels or Dynkin labels.

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

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 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 A₁ Cartan matrix, which corresponds to the Lie algebra ${\displaystyle {\mathfrak {su))(2)}$ in the classification of Lie algebras by Cartan matrices. The algebra ${\displaystyle {\mathfrak {su))(2)}$ has only a single simple root.

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. This Cartan matrix corresponds to the Lie algebra ${\displaystyle {\mathfrak {su))(2)}$. This has a single simple root, ${\displaystyle \alpha _{1}=1}$ and Coxeter label ${\displaystyle n_{1}=1}$, but the Lagrangian is modified for the affine theory: there is also an affine root ${\displaystyle \alpha _{0}=-1}$ and Coxeter label ${\displaystyle n_{0}=1}$. One can expand ${\displaystyle \phi }$ as ${\displaystyle \phi _{0}\alpha _{0}+\phi _{1}\alpha _{1))$, but for the affine root ${\displaystyle \langle \alpha _{0},\alpha _{0}\rangle =0}$, so the ${\displaystyle \phi _{0))$ component decouples.

The sum is ${\displaystyle \sum _{i=0}^{1}n_{i}\exp(\beta \alpha _{i}\phi )=\exp(\beta \phi )+\exp(-\beta \phi ).}$ Then if ${\displaystyle \beta }$ is purely imaginary, ${\displaystyle \beta =ib}$ with ${\displaystyle b}$ real and, without loss of generality, positive, then this is ${\displaystyle 2\cos(b\phi )}$. The Lagrangian is then

$${\displaystyle {\mathcal {L))={\frac {1}{2))\partial _{\mu }\phi \partial ^{\mu }\phi +{\frac {2m^{2)){b^{2))}\cos(b\phi ),}$$