Toda field theory explained

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

r , that is, the Cartan subalgebra of the algebra has dimension r , the Lagrangian can be written

$$\backslash mathcal=\backslash frac\backslash left\backslash langle\; \backslash partial\_\backslash mu\; \backslash phi,\; \backslash partial^\backslash mu\; \backslash phi\; \backslash right\backslash rangle-\backslash frac\backslash sum\_^r\; n\_i\; \backslash exp(\backslash beta\; \backslash langle\backslash alpha\_i,\; \backslash phi\backslash rangle).$$

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 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 ⋅ , ⋅ \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

ak{su}(2) in the classification of Lie algebras by Cartan matrices. The algebra ak{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

ak{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 ^{1} _{i =0} 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$$\backslash mathcal\; =\; \backslash frac\backslash partial\_\backslash mu\; \backslash phi\; \backslash partial^\backslash mu\; \backslash phi\; +\; \backslash frac\backslash cos(b\backslash phi),$$which is the sine-Gordon Lagrangian.

Notes and References

1. Korff . Christian . Lie algebraic structures in integrable models, affine Toda field theory . 1 September 2000 . hep-th/0008200 .