Non-Abelian Twisted Integrable Models

• Non-Abelian Twisted Integrable Models are integrable field theories that employ a Zₙ branch cut and automorphism to decompose algebra-valued fields into distinct eigenspaces.
• They construct deformed η– and λ–sigma models by projecting group currents, thereby modifying global symmetries while preserving integrability.
• This framework utilizes modified Yang–Baxter equations and the choice between inner and outer automorphisms to yield models with unique quantum and classical properties.

is to choose a (twisted) trigonometric 1-form on the N-fold cover given in the paper by

$$\omega_{\text{trig}}^{(N)} = -c_0 \hat{h} \frac{(z^N - e^{N\alpha})(z^N - e^{-N\alpha})}{(z^N - 1)^2} \frac{dz}{z}.$$

Here $c_0$ (equal to 1 or i) depends on the choice of conjugation so that after imposing appropriate reality conditions, the two-dimensional models are real; $\hat{h}$ is a coupling parameter, and $\alpha$ parameterizes (via the positions of the zeros) the twist. In this setting, the automorphism σ (with σ^N = 1) is imposed on the algebra-valued fields, so a decomposition using projectors onto eigenspaces is defined as

$$P_a = \frac{1}{N} \sum_{b\in\mathbb{Z}_N} e^{-2\pi iab/N} \sigma^b \qquad (a = 0, 1, \ldots, N-1).$$

Then the group-valued field $g$ (or its Maurer–Cartan current) is decomposed as

$J_\pm^{(a)} = P_a (g^{-1}\partial_\pm g).$

This decomposition enters directly into the action. For example, one version of the twisted “η-model” has an action of the form

$$\mathcal{S}_\eta^{(N)}(g) = -\frac{4 c_0^2 \hat{h} \eta}{1-c_0^2 \eta^2} \int d^2\sigma \; \text{tr} \left[ J_+^{(0)} \frac{1}{1-\eta\mathcal{R}} J_-^{(0)} + \sum_{b\in\mathbb{Z}_N\setminus\{0\}} J_+^{(b)} J_-^{(N-b)} \right],$$

where $\mathcal{R}$ is an (antisymmetric) solution to the (modified) classical Yang–Baxter equation on $\mathfrak{g}$ (or its real form) and η is a deformation parameter. (A similar construction applies to the “λ-model” with an alternative set of boundary conditions.)

Significance of the Zₙ Branch Cut and Automorphism

• By introducing a Zₙ branch cut, the authors effectively “glue” together N copies of the original theory through the action of the automorphism σ. As a result, when one goes around the branch point (i.e., when $z \to e^{2\pi i/N} z$), the fields transform by σ. This procedure can be seen as a consistent truncation of the full set of degrees of freedom, and it guarantees that the resulting sigma model remains integrable.
• In the language of Lax connections (which guarantee the integrability of the sigma model), the twisted fields appear in an ansatz where the z-dependence, inherited from the meromorphic one-form ωₜʳᶦᵍ, is combined with the projection operators $P_a$. A typical form for the Lax connection is

$$L_\pm^{(N)}(z) = \mathcal{V}_\pm + e^{\frac{-(1\mp 1)N\alpha}{2}} \sum_{a\in\mathbb{Z}_N} V_\pm^{(a)} \frac{z^a}{e^{\pm N\alpha}z^N - 1},$$

which shows explicitly how the twist (α and the phase factors arising from the cyclic rotation) enters the construction.

• The automorphism σ, which may be inner (in the simplest cases such as SU(2)) or outer (as is possible for SU(n) with n > 2), determines the splitting of the algebra into eigenspaces. In the twisted model, the fields are decomposed into components $J^{(a)} = P_a(j)$ with different transformation properties. In many cases, the untwisted model (where σ is trivial) and the twisted model are related by parameter-dependent field redefinitions or dualities (for example Poisson–Lie duality) so that they are “equivalent” up to a closed B-field. However, inequivalent models arise if one uses an outer automorphism; for instance, for SU(n) (n > 2), the fixed-point subalgebra $\mathfrak{g}_0$ under an outer automorphism is strictly smaller than that coming from an inner automorphism. In such cases, the global symmetries of the σ-model change. (The paper shows, for example, that an untwisted SU(n) η–model has a right–acting symmetry group U(1)^n–1 while the twisted version may only have a U(1)^⌊n/2⌋ subgroup.)

Differences in Symmetries and Degrees of Freedom

• Although the counting of degrees of freedom remains identical between the twisted and untwisted models due to the gauge fixing, the global symmetry groups can be very different. In the twisted model, the fields at the fixed points are forced to lie in the fixed-point subgroup G₀ of G (i.e., the subgroup invariant under σ), while in the untwisted model there is no such restriction.
• For the (untwisted) Yang–Baxter deformation (the so-called η-model), the action takes a well–known form:

$$\mathcal{S}_\eta^{(1)}(g) = -4c_0^2 \hat{h} η \int d²σ \; \text{tr} \left[ j_+ \frac{1}{1-η\mathcal{R}} j_- \right],$$

but in the Z₂–twisted version, the action becomes

$$\mathcal{S}_\eta^{(2)}(g) = -4c_0^2 \hat{h} η \int d²σ \; \text{tr} \left[ j_+^{(0)} \frac{1}{1-η\mathcal{R}} j_-^{(0)} + j_+^{(1)} j_-^{(1)} \right].$$

It is the projection and inclusion of field components that alter the surviving global symmetry.

• In some cases – for instance, when the automorphism is inner (as with SU(2)) – the twisted and untwisted models are essentially equivalent. However, for groups with non-trivial outer automorphisms (like SU(n) for n > 2), the twisted models have different global symmetries and are genuinely new integrable models.

Novel Features and Implications

• Introducing the Zₙ twist via a branch cut in the trigonometric model represents a characteristic of “trigonometric” models obtained from a cylinder rather than directly from a sphere.
• Even though one might expect twisting to reduce degrees of freedom, the interplay among the defects, fixed-point boundary conditions, and gauge symmetry means the overall field content remains unchanged.
• The authors construct doubly–deformed models (YB– and CC–deformed versions), establishing both η– and λ–models as examples.
• When the twist involves an outer automorphism, twisted models with genuinely distinct symmetries from untwisted counterparts arise.

In summary, the Zₙ branch cut introduces new integrable sigma models with unique properties and symmetries. This new construction offers a broader exploration of integrable deformations and their quantum properties, yielding interesting applications in both classical and quantum integrable field theory.