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 Kac–Moody algebra and a specific Lagrangian.

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

$\mathcal=\frac\left\langle \partial_\mu \phi, \partial^\mu \phi \right\rangle -\frac\sum_^r n_i \exp(\beta \langle\alpha_i, \phi\rangle).$

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 $\mathbb^r$.

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 Kac–Moody algebra.

Toda field theories usually refer to theories with a finite Kac–Moody algebra. If the Kac–Moody algebra is affine, 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 $\mathfrak(2)$ in the classification of Lie algebras by Cartan matrices. The algebra $\mathfrak(2)$ has only a single simple root.

The sinh-Gordon model is the affine Toda field theory with the generalized Cartan matrix

:$\begin 2&-2 \\ -2&2 \end$

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 $\mathfrak(2)$. 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 $n_0 = 1$. Due to the single root, there is a single field $\phi$, so $\langle \partial_\mu \phi, \partial^\mu \phi\rangle$ is simply $\partial_\mu \phi \partial^\mu \phi$.

The sum is $\sum_^1 n_i\exp(\beta \alpha_i\phi) = \exp(\beta \phi) + \exp(-\beta\phi).$ 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
$\mathcal = \frac\partial_\mu \phi \partial^\mu \phi + \frac\cos(b\phi),$
which is the Sine-gordon Lagrangian.

References

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Quantum field theory
Lattice models
Lie algebras
Exactly solvable models
Integrable systems

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