Toda field theory

Template:Short description 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

$$ℒ=\frac{1}{2}⟨{\partial }_{\mu }\varphi ,{\partial }^{\mu }\varphi ⟩-\frac{m^2}{{\beta }^2}\sum _{i=1}^r{n}_i\exp (\beta ⟨{\alpha }_i,\varphi ⟩).$$

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 ${ℝ}^r$.

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

The inner product $⟨\cdot ,\cdot ⟩$ 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 $𝔰𝔲(2)$ in the classification of Lie algebras by Cartan matrices. The algebra $𝔰𝔲(2)$ has only a single simple root.

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. This Cartan matrix corresponds to the Lie algebra $𝔰𝔲(2)$. 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 $\varphi$ as ${\varphi }_0{\alpha }_0+{\varphi }_1{\alpha }_1$, but for the affine root $⟨{\alpha }_0,{\alpha }_0⟩=0$, so the ${\varphi }_0$ component decouples.

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

References

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