On classical q-deformations of integrable sigma-models (1308.3581v1)

Abstract: A procedure is developed for constructing deformations of integrable sigma-models which are themselves classically integrable. When applied to the principal chiral model on any compact Lie group F, one recovers the Yang-Baxter sigma-model introduced a few years ago by C. Klimcik. In the case of the symmetric space sigma-model on F/G we obtain a new one-parameter family of integrable sigma-models. The actions of these models correspond to a deformation of the target space geometry and include a torsion term. An interesting feature of the construction is the q-deformation of the symmetry corresponding to left multiplication in the original models, which becomes replaced by a classical q-deformed Poisson-Hopf algebra. Another noteworthy aspect of the deformation in the coset sigma-model case is that it interpolates between a compact and a non-compact symmetric space. This is exemplified in the case of the SU(2)/U(1) coset sigma-model which interpolates all the way to the SU(1,1)/U(1) coset sigma-model.

• The paper presents a systematic construction for q-deformations that preserve the integrability of principal chiral and symmetric space sigma-models.
• It employs a generalized Poisson bracket and spectral parameter tuning, ensuring that the deformed models maintain exact solutions and symmetry properties.
• The approach offers new insights into classical symmetry deformations and lays groundwork for future research in higher-dimensional and supersymmetric integrable models.

An Examination of Classical $q$-Deformations in Integrable Sigma Models

The paper by Delduc, Magro, and Vicedo presents a comprehensive examination of deformations in the context of integrable $\sigma$-models, specifically focusing on classical $q$-deformations. Integrable $\sigma$-models hold intrinsic significance due to their rare and exact solutions, providing insights into a multitude of physical phenomena, particularly in two-dimensional field theories. However, a systematic procedure to determine the integrability of these models is absent. This paper proposes a method to construct integrable deformations of these models through a meticulous preservation of their integrability.

Constructing Classical $q$-Deformations

The procedure outlined extends to both the principal chiral models (PCMs) over compact Lie groups and symmetric space $\sigma$-models. The paper delineates a method to deform the integrable structures of these models while sustaining their intrinsic properties through the introduction of a compatible Poisson bracket. The deformation incorporates a generalized Faddeev-Reshetikhin bracket, ensuring the preservation of integrability.

A noteworthy output of this methodology is the Yang-Baxter $\sigma$-model upon applying the procedure to the PCM over a compact Lie group $F$. The deformed model is characterized by the appearance of a classical $q$-deformed Poisson-Hopf algebra, which modifies the initial left symmetry of these models.

Practical and Theoretical Implications

From a symmetry perspective, these deformations pivot around a classical $q$-deformed symmetry structure. The methodology's cornerstone resides in the concept of spectral parameter deformation and manipulation of twist functions, ensuring compatibility with Poisson algebra. This embodies the Yang-Baxter $\sigma$-model which not only maintains but accentuates integrability under deformation.

In the landscape of symmetric spaces, deformations interpolate between compact and non-compact symmetric spaces, leveraging the inherent characteristics of the models. The approach guarantees that the $q$-deformation retains the symmetry structures deemed pivotal in integrability scenarios.

Models and Symmetry Analysis

The robust examination of deformation in the context of the $SU(2)/U(1)$ coset $\sigma$-model serves as a profound example. The deformed model effortlessly transitions between compact and non-compact symmetric spaces, illustrating the versatility and depth of the proposed deformations. Moreover, models such as these witness a linearization in their equations of motion, ensuring sustained Lorentz invariance and conformability to prescribed geometric settings.

Potential and Future Directions

The work paves the path for future research directions in different integrable models pertinent to higher-dimensional manifolds and super-symmetric constructs. The overlay of classical $q$-deformations on the $AdS_5 \times S^5$ superstring is promising, potentially amalgamating the principles of $q$-deformations with string theory.

Furthermore, as highlighted in the text, the interplay between modified Yang-Baxter equations provides a template for crafting new integrability paradigms. These aspects collectively suggest a broader applicability, encompassing quantum integrability and providing insights into non-linear sigma models within the quantum field theory domain. The future embellishment of these methods can result in unveiling the integrable nature of various unsolved problems in theoretical physics.

In conclusion, this research sets forth a framework to systematically construct integrable deformations in $\sigma$-models, harnessing the harmony of mathematical structures like Poisson algebras and classical $q$-deformations. It signifies a step forward in understanding symmetry and integrability within the field of mathematical physics.