Toda field theory

Special quantum 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.[1]

## Formulation

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Fixing the Lie algebra to have rank  r {\displaystyle r} , that is, the Cartan subalgebra of the algebra has dimension  r {\displaystyle r} , the Lagrangian can be written

L = 1 2 ⟨ ∂ μ ϕ , ∂ μ ϕ ⟩ − m 2 β 2 ∑ i = 1 r n i exp ⁡ ( β ⟨ α i , ϕ ⟩ ) . {\displaystyle {\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 {\displaystyle x} and timelike coordinate  t {\displaystyle t} . Greek indices indicate spacetime coordinates.

For some choice of root basis,  α i {\displaystyle \alpha _{i}} is the  i {\displaystyle i} th simple root. This provides a basis for the Cartan subalgebra, allowing it to be identified with  R r {\displaystyle \mathbb {R} ^{r}} .

Then the field content is a collection of  r {\displaystyle r} scalar fields  ϕ i {\displaystyle \phi _{i}} , which are scalar in the sense that they transform trivially under Lorentz transformations of the underlying spacetime.

The inner product  ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } is the restriction of the Killing form to the Cartan subalgebra.

The  n i {\displaystyle n_{i}} are integer constants, known as Kac labels or Dynkin labels.

The physical constants are the mass  m {\displaystyle m} and the coupling constant β {\displaystyle \beta } .

## Classification of Toda field theories

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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

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Liouville field theory is associated to the A1 Cartan matrix, which corresponds to the Lie algebra  s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} in the classification of Lie algebras by Cartan matrices. The algebra  s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} has only a single simple root.

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

( 2 − 2 − 2 2 ) {\displaystyle {\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  s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} . This has a single simple root,  α 1 = 1 {\displaystyle \alpha _{1}=1} and Coxeter label  n 1 = 1 {\displaystyle n_{1}=1} , but the Lagrangian is modified for the affine theory: there is also an affine root α 0 = − 1 {\displaystyle \alpha _{0}=-1} and Coxeter label  n 0 = 1 {\displaystyle n_{0}=1} . One can expand  ϕ {\displaystyle \phi } as  ϕ 0 α 0 + ϕ 1 α 1 {\displaystyle \phi _{0}\alpha _{0}+\phi _{1}\alpha _{1}} , but for the affine root  ⟨ α 0 , α 0 ⟩ = 0 {\displaystyle \langle \alpha _{0},\alpha _{0}\rangle =0} , so the  ϕ 0 {\displaystyle \phi _{0}} component decouples.

The sum is  ∑ i = 0 1 n i exp ⁡ ( β α i ϕ ) = exp ⁡ ( β ϕ ) + exp ⁡ ( − β ϕ ) . {\displaystyle \sum _{i=0}^{1}n_{i}\exp(\beta \alpha _{i}\phi )=\exp(\beta \phi )+\exp(-\beta \phi ).} Then if  β {\displaystyle \beta } is purely imaginary,  β = i b {\displaystyle \beta =ib} with  b {\displaystyle b} real and, without loss of generality, positive, then this is  2 cos ⁡ ( b ϕ ) {\displaystyle 2\cos(b\phi )} . The Lagrangian is then  L = 1 2 ∂ μ ϕ ∂ μ ϕ + 2 m 2 b 2 cos ⁡ ( b ϕ ) , {\displaystyle {\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

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1. ^ Korff, Christian (1 September 2000). "Lie algebraic structures in integrable models, affine Toda field theory". arXiv:hep-th/0008200.

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

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