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 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 ${\mathfrak {su}}(2)$ 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 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 ${\mathfrak {su}}(2)$. 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.

References

• Mussardo, Giuseppe (2009), Statistical Field Theory: An Introduction to Exactly Solved Models in Statistical Physics, Oxford University Press, ISBN 978-0-199-54758-6