Toda field theory - Alchetron, The Free Social Encyclopedia

In the study of field theory and partial differential equations, a Toda field theory (named after Morikazu Toda) is derived from the following Lagrangian:

L = 1 2 [ ( ∂ ϕ ∂ t , ∂ ϕ ∂ t ) − ( ∂ ϕ ∂ x , ∂ ϕ ∂ x ) ] − m 2 β 2 ∑ i = 1 r n i e β α i ⋅ ϕ .

Here x and t are spacetime coordinates, (,) is the Killing form of a real r-dimensional Cartan algebra h of a Kac–Moody algebra over h , α_i is the i^th simple root in some root basis, n_i is the Coxeter number, m is the mass (or bare mass in the quantum field theory version) and β is the coupling constant.

Then a Toda field theory is the study of a function φ mapping 2-dimensional Minkowski space satisfying the corresponding Euler–Lagrange equations.

If the Kac–Moody algebra is finite, it's called a Toda field theory. If it is affine, it is called an affine Toda field theory (after the component of φ which decouples is removed) and 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.

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

( 2 − 2 − 2 2 )

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.

References

Toda field theory Wikipedia

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