An outline of the original, 2-flavor model

The chiral model of Gürsey (1960; also see Gell-Mann and Lévy) is now appreciated to be an effective theory of QCD with two light quarks, u, and d. The QCD Lagrangian is approximately invariant under independent global flavor rotations of the left- and right-handed quark fields,

${\displaystyle {\begin{cases}q_{L}\mapsto q_{L}'=Lq_{L}=\exp {\left(-i{\boldsymbol {\theta }}_{L}\cdot {\tfrac {\boldsymbol {\tau }}{2}}\right)}q_{L}\\q_{R}\mapsto q_{R}'=Rq_{R}=\exp {\left(-i{\boldsymbol {\theta }}_{R}\cdot {\tfrac {\boldsymbol {\tau }}{2}}\right)}q_{R}\end{cases}}}$

where τ denote the Pauli matrices in the flavor space and θ_L, θ_R are the corresponding rotation angles.

The corresponding symmetry group ${\displaystyle {\text{SU}}(2)_{L}\times {\text{SU}}(2)_{R}}$ is the chiral group, controlled by the six conserved currents

${\displaystyle L_{\mu }^{i}={\bar {q}}_{L}\gamma _{\mu }{\tfrac {\tau ^{i}}{2}}q_{L},\qquad R_{\mu }^{i}={\bar {q}}_{R}\gamma _{\mu }{\tfrac {\tau ^{i}}{2}}q_{R},}$

which can equally well be expressed in terms of the vector and axial-vector currents

${\displaystyle V_{\mu }^{i}=L_{\mu }^{i}+R_{\mu }^{i},\qquad A_{\mu }^{i}=R_{\mu }^{i}-L_{\mu }^{i}.}$

The corresponding conserved charges generate the algebra of the chiral group,

${\displaystyle \left[Q_{I}^{i},Q_{I}^{j}\right]=i\epsilon ^{ijk}Q_{I}^{k}\qquad \qquad \left[Q_{L}^{i},Q_{R}^{j}\right]=0,}$

with I=L,R, or, equivalently,

${\displaystyle \left[Q_{V}^{i},Q_{V}^{j}\right]=i\epsilon ^{ijk}Q_{V}^{k},\qquad \left[Q_{A}^{i},Q_{A}^{j}\right]=i\epsilon ^{ijk}Q_{V}^{k},\qquad \left[Q_{V}^{i},Q_{A}^{j}\right]=i\epsilon ^{ijk}Q_{A}^{k}.}$

Application of these commutation relations to hadronic reactions dominated current algebra calculations in the early seventies of the last century.

At the level of hadrons, pseudoscalar mesons, the ambit of the chiral model, the chiral ${\displaystyle {\text{SU}}(2)_{L}\times {\text{SU}}(2)_{R}}$ group is spontaneously broken down to ${\displaystyle {\text{SU}}(2)_{V}}$, by the QCD vacuum. That is, it is realized nonlinearly, in the Nambu–Goldstone mode: The Q_V annihilate the vacuum, but the Q_A do not! This is visualized nicely through a geometrical argument based on the fact that the Lie algebra of ${\displaystyle {\text{SU}}(2)_{L}\times {\text{SU}}(2)_{R}}$ is isomorphic to that of SO(4). The unbroken subgroup, realized in the linear Wigner–Weyl mode, is ${\displaystyle {\text{SO}}(3)\subset {\text{SO}}(4)}$ which is locally isomorphic to SU(2) (V: isospin).

To construct a non-linear realization of SO(4), the representation describing four-dimensional rotations of a vector

${\displaystyle {\begin{pmatrix}{\boldsymbol {\pi }}\\\sigma \end{pmatrix}}\equiv {\begin{pmatrix}\pi _{1}\\\pi _{2}\\\pi _{3}\\\sigma \end{pmatrix}},}$

for an infinitesimal rotation parametrized by six angles

${\displaystyle \left\{\theta _{i}^{V,A}\right\},\qquad i=1,2,3,}$

is given by

${\displaystyle {\begin{pmatrix}{\boldsymbol {\pi }}\\\sigma \end{pmatrix}}{\stackrel {SO(4)}{\longrightarrow }}{\begin{pmatrix}{{\boldsymbol {\pi }}'}\\\sigma '\end{pmatrix}}=\left[\mathbf {1} _{4}+\sum _{i=1}^{3}\theta _{i}^{V}V_{i}+\sum _{i=1}^{3}\theta _{i}^{A}A_{i}\right]{\begin{pmatrix}{\boldsymbol {\pi }}\\\sigma \end{pmatrix}}}$

where

${\displaystyle \sum _{i=1}^{3}\theta _{i}^{V}V_{i}={\begin{pmatrix}0&-\theta _{3}^{V}&\theta _{2}^{V}&0\\\theta _{3}^{V}&0&-\theta _{1}^{V}&0\\-\theta _{2}^{V}&\theta _{1}^{V}&0&0\\0&0&0&0\end{pmatrix}}\qquad \qquad \sum _{i=1}^{3}\theta _{i}^{A}A_{i}={\begin{pmatrix}0&0&0&\theta _{1}^{A}\\0&0&0&\theta _{2}^{A}\\0&0&0&\theta _{3}^{A}\\-\theta _{1}^{A}&-\theta _{2}^{A}&-\theta _{3}^{A}&0\end{pmatrix}}.}$

The four real quantities (π, σ) define the smallest nontrivial chiral multiplet and represent the field content of the linear sigma model.

To switch from the above linear realization of SO(4) to the nonlinear one, we observe that, in fact, only three of the four components of (π, σ) are independent with respect to four-dimensional rotations. These three independent components correspond to coordinates on a hypersphere S³, where π and σ are subjected to the constraint

${\displaystyle {\boldsymbol {\pi }}^{2}+\sigma ^{2}=F^{2},}$

with F a (pion decay) constant of dimension mass.

Utilizing this to eliminate σ yields the following transformation properties of π under SO(4),

${\displaystyle {\begin{cases}\theta ^{V}:{\boldsymbol {\pi }}\mapsto {\boldsymbol {\pi }}'={\boldsymbol {\pi }}+{\boldsymbol {\theta }}^{V}\times {\boldsymbol {\pi }}\\\theta ^{A}:{\boldsymbol {\pi }}\mapsto {\boldsymbol {\pi }}'={\boldsymbol {\pi }}+{\boldsymbol {\theta }}^{A}{\sqrt {F^{2}-{\boldsymbol {\pi }}^{2}}}\end{cases}}\qquad {\boldsymbol {\theta }}^{V,A}\equiv \left\{\theta _{i}^{V,A}\right\},\qquad i=1,2,3.}$

The nonlinear terms (shifting π) on the right-hand side of the second equation underlie the nonlinear realization of SO(4). The chiral group ${\displaystyle {\text{SU}}(2)_{L}\times {\text{SU}}(2)_{R}\simeq {\text{SO}}(4)}$ is realized nonlinearly on the triplet of pions— which, however, still transform linearly under isospin ${\displaystyle {\text{SU}}(2)_{V}\simeq {\text{SO}}(3)}$ rotations parametrized through the angles ${\displaystyle \{{\boldsymbol {\theta }}_{V}\}.}$ By contrast, the ${\displaystyle \{{\boldsymbol {\theta }}_{A}\}}$ represent the nonlinear "shifts" (spontaneous breaking).

Through the spinor map, these four-dimensional rotations of (π, σ) can also be conveniently written using 2×2 matrix notation by introducing the unitary matrix

${\displaystyle U={\frac {1}{F}}\left(\sigma \mathbf {1} _{2}+i{\boldsymbol {\pi }}\cdot {\boldsymbol {\tau }}\right),}$

and requiring the transformation properties of U under chiral rotations to be

${\displaystyle U\longrightarrow U'=LUR^{\dagger },}$

where ${\displaystyle \theta _{L}=\theta _{V}-\theta _{A},\theta _{R}=\theta _{V}+\theta _{A}.}$

The transition to the nonlinear realization follows,

${\displaystyle U={\frac {1}{F}}\left({\sqrt {F^{2}-{\boldsymbol {\pi }}^{2}}}\mathbf {1} _{2}+i{\boldsymbol {\pi }}\cdot {\boldsymbol {\tau }}\right),\qquad {\mathcal {L}}_{\pi }^{(2)}={\frac {F^{2}}{4}}\langle \partial _{\mu }U\partial ^{\mu }U^{\dagger }\rangle ,}$

where ${\displaystyle \langle \ldots \rangle }$ denotes the trace in the flavor space. This is a non-linear sigma model.

Terms involving ${\displaystyle \textstyle \partial _{\mu }\partial ^{\mu }U}$ or ${\displaystyle \textstyle \partial _{\mu }\partial ^{\mu }U^{\dagger }}$ are not independent and can be brought to this form through partial integration. The constant F²/4 is chosen in such a way that the Lagrangian matches the usual free term for massless scalar fields when written in terms of the pions,

${\displaystyle {\mathcal {L}}_{\pi }^{(2)}={\frac {1}{2}}\partial _{\mu }{\boldsymbol {\pi }}\cdot \partial ^{\mu }{\boldsymbol {\pi }}+{\frac {1}{2F^{2}}}\left(\partial _{\mu }{\boldsymbol {\pi }}\cdot {\boldsymbol {\pi }}\right)^{2}+{\mathcal {O}}(\pi ^{6}).}$

Alternate Parametrization

An alternative, equivalent (Gürsey, 1960), parameterization

${\displaystyle {\boldsymbol {\pi }}\mapsto {\boldsymbol {\pi }}~{\frac {\sin(|\pi /F|)}{|\pi /F|}},}$

yields a simpler expression for U,

${\displaystyle U=\mathbf {1} \cos |\pi /F|+i{\widehat {\pi }}\cdot {\boldsymbol {\tau }}\sin |\pi /F|=e^{i~{\boldsymbol {\tau }}\cdot {\boldsymbol {\pi }}/F}.}$

Note the reparameterized π transform under

${\displaystyle LUR^{\dagger }=\exp(i{\boldsymbol {\theta }}_{A}\cdot {\boldsymbol {\tau }}/2-i{\boldsymbol {\theta }}_{V}\cdot {\boldsymbol {\tau }}/2)\exp(i{\boldsymbol {\pi }}\cdot {\boldsymbol {\tau }}/F)\exp(i{\boldsymbol {\theta }}_{A}\cdot {\boldsymbol {\tau }}/2+i{\boldsymbol {\theta }}_{V}\cdot {\boldsymbol {\tau }}/2)}$

so, then, manifestly identically to the above under isorotations, V; and similarly to the above, as

${\displaystyle {\boldsymbol {\pi }}\longrightarrow {\boldsymbol {\pi }}+{\boldsymbol {\theta }}_{A}F+\cdots ={\boldsymbol {\pi }}+{\boldsymbol {\theta }}_{A}F(|\pi /F|\cot |\pi /F|)}$

under the broken symmetries, A, the shifts. This simpler expression generalizes readily (Cronin, 1967) to N light quarks, so ${\displaystyle \textstyle {\text{SU}}(N)_{L}\times {\text{SU}}(N)_{R}/{\text{SU}}(N)_{V}.}$

Integrable chiral model

Introduced by Richard S. Ward,^[3] the integrable chiral model or Ward model is described in terms of a matrix-valued field ${\displaystyle J:\mathbb {R} ^{3}\rightarrow U(n)}$ and is given by the partial differential equation

$${\displaystyle \partial _{t}(J^{-1}J_{t})-\partial _{x}(J^{-1}J_{x})-\partial _{y}(J^{-1}J_{y})-[J^{-1}J_{t},J^{-1}J_{y}]=0.}$$

It has a Lagrangian formulation with the expected kinetic term together with a term which resembles a Wess–Zumino–Witten term. It also has a formulation which is formally identical to the Bogomolny equations but with Lorentz signature. The relation between these formulations can be found in Dunajski (2010).

Many exact solutions are known.^[4][5][6]

Two-dimensional principal chiral model

Here the underlying manifold ${\displaystyle M}$ is taken to be a Riemann surface, in particular the cylinder ${\displaystyle \mathbb {C} ^{*}}$ or plane ${\displaystyle \mathbb {C} }$, conventionally given real coordinates ${\displaystyle \tau ,\sigma }$, where on the cylinder ${\displaystyle \sigma \sim \sigma +2\pi }$ is a periodic coordinate. For application to string theory, this cylinder is the world sheet swept out by the closed string.^[7]

Global symmetries

The global symmetries act as internal symmetries on the group-valued field ${\displaystyle g(x)}$ as ${\displaystyle \rho _{L}(g')g(x)=g'g(x)}$ and ${\displaystyle \rho _{R}(g)g(x)=g(x)g'}$. The corresponding conserved currents from Noether's theorem are

$${\displaystyle L_{\alpha }=g^{-1}\partial _{\alpha }g,R_{\alpha }=\partial _{\alpha }gg^{-1}.}$$

The equations of motion turn out to be equivalent to conservation of the currents,

$${\displaystyle \partial _{\alpha }L^{\alpha }=\partial _{\alpha }R^{\alpha }=0{\text{ or in coordinate-free form }}d*L=d*R=0.}$$

The currents additionally satisfy the flatness condition,

$${\displaystyle dL+{\frac {1}{2}}[L,L]=0{\text{ or in coordinates }}\partial _{\alpha }L_{\beta }-\partial _{\beta }L_{\alpha }+[L_{\alpha },L_{\beta }]=0,}$$

and therefore the equations of motion can be formulated entirely in terms of the currents.

Lax formulation

Consider the worldsheet in light-cone coordinates ${\displaystyle x^{\pm }=t\pm x}$. The components of the appropriate Lax matrix are

$${\displaystyle L_{\pm }(x^{+},x^{-};\lambda )={\frac {j_{\pm }}{1\mp \lambda }}.}$$

The requirement that the zero-curvature condition on ${\displaystyle L_{\pm }}$ for all ${\displaystyle \lambda }$ is equivalent to the conservation of current and flatness of the current ${\displaystyle j=(j_{+},j_{-})}$, that is, the equations of motion from the principal chiral model (PCM).