Deforming Integrable Sigma-models – DefIS

A central tool used by the DefIS collaboration is that of a rational function, the twist function. In short, Twist function = Integrability! DefIS has constructed and studied integrable sigma models associated with increasingly «complex« twist functions. The DefIS collaboration had already shown that deforming the twist function of a known integrable sigma models makes it possible to obtain new sigma models while preserving integrability. To start, the collaboration determined the twist function of a sigma model previously known as an integrable deformation of a classical integrable sigma model. Then certain symmetries of integrable sigma models have been precisely related to this twist function. Depending on the type of symmetry, this in-depth analysis has been carried out either on a specific example or in a generic way. Most of integrable sigma models known for a long time or recently discovered, including the ones constructed by the DefIS collaboration, essentially correspond to twist functions with one site. A difficulty has been overcome when DefIS has obtained completely new integrable sigma models which correspond to a twist function with an arbitrary number of sites.

The «landscape« of known integrable sigma models has been greatly enriched thanks to the DefIS collaboration.

What characterizes an integrable field theory is the existence of an infinite number of symmetries. A first class of important results obtained by DefIS corresponds to a generic description of some of these symmetries, the ones which are associated with some particular local charges.

Until recently, one had to travel all the way to the heart of integrability in order to uncover the twist function of a given integrable sigma model. This is the case when the sigma model is defined by its action, such as the action of the string theory mentioned above. An unexpected result which was not forecast is that the twist function is also present at the level of the action. It is also the twist function which enables to make the link with a very recent approach called four dimensional Chern-Simons theory. In particular, the models built by the DefIS collaboration have been recovered within this Chern-Simons approach. Therefore, the approach of DefIS opens new promising perspectives.

A future prospect is to continue studying the links with the Chern-Simons formulation in order to understand all its aspects and to continue exploring the family of integrable sigma models. Using jointly two methods, that is to say the Hamiltonian and the Lagrangian ones corresponding respectively to the Affine Gaudin approach and the Chern-Simons one looks promising. One important question is to understand to what extent it would be possible to assemble coset-type models.

Another subject is the study of integrable sigma models at the quantum level by using their reinterpretation as realizations of affine Gaudin models. The results obtained by DefIS are first steps in this direction. It requires significant investment over the long term. Also at the quantum level, renormalization properties of the models built by DefIS are being investigated.

On the Hamiltonian integrability of the bi-Yang-Baxter sigma-model, F. Delduc, S. Lacroix, M. Magro, B. Vicedo, JHEP 1603 (2016) 104.

On q-deformed symmetries as Poisson-Lie symmetries and application to Yang-Baxter type models, F. Delduc, S. Lacroix, M. Magro, B. Vicedo, J.Phys. A49 (2016) no.41.

Cyclotomic Gaudin models, Miura opers and flag varieties, S. Lacroix, B. Vicedo, [Annales Henri Poincare 19 (2018) no.1.

Generalized IIB supergravity from exceptional field theory, A. Baguet, M. Magro. H. Samtleben, JHEP 1703 (2017).

Affine q-deformed symmetry and the classical Yang-Baxter sigma-model, F. Delduc, T. Kameyama, M. Magro, B. Vicedo, JHEP 1703 (2017).

Local charges in involution and hierarchies in integrable sigma-models, S. Lacroix, M. Magro, B. Vicedo, JHEP 1709 (2017).

Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT trabsformations in one integrable sigma-model, F. Delduc, B. Hoare, T. Kameyama, M. Magro, JHEP 1710 (2017).

Affine Gaudin models and hypergeometric functions on affine opers, S. Lacroix, B. Vicedo, C.A.S. Young, Adv. Math. 350 (2019).

Cubic hypergeometric integrals of motion in affine Gaudin models, S. Lacroix, B. Vicedo, C.A.S. Young, Adv. Theor. Math. Phys. Volume 24, Number 1.

Three-parameter integrable deformation of Z4 permutation supercosets, F. Delduc, B. Hoare, T. Kameyama, S. Lacroix, M. Magro, JHEP 1901 (2019).

Integrable coupled s-models, F. Delduc, S. Lacroix, M. Magro, B. Vicedo, Phys. Rev. Lett. 122, 041601 (2019).

Assembling integrable s-models as affine Gaudin models, F. Delduc, S. Lacroix, M. Magro, B. Vicedo, JHEP 1901 (2019).

Utralocal Lax connection for para-complex ZT cosets, F. Delduc, T. Kameyama, S. Lacroix, M. Magro, B. Vicedo, Nucl. Phys. B949, (2019), 114821.

A unifying 2d action for integrable sigma-models from 4d Chern-Simons theory, F. Delduc, S. Lacroix, M. Magro, B. Vicedo, Lett. Math. Phys. (2020).

DefIS has been conceived on purpose as a low-cost proposal (163 904 EUR) in theoretical high energy physics with very high expectations. It takes place in the arena of Integrability in Gauge and String Theory. This is a domain of research which is very active. For instance, there are at the present time three Advanced ERC Grantees (V. Kazakov, A. Tseytlin, K. Zarembo) working on this subject. The purpose of DefIS is to obtain new results related to one topic of this field. What characterises this topic is that its emergence is very recent. Furthermore, it relies on previous achievements of the members of the DefIS collaboration. Finally, interest of the community working on Integrability in Gauge and String Theory for this subject of research is expanding fast.

The primary goal of DefIS is to push forward the current frontier of knowledge on integrable field theories. Few field theories have the property of being integrable. Their study has nevertheless proved to be of great importance for theoretical physics. The reason is that integrability allows for the use of specific techniques in order to compute exact results. This property has been extensively used recently in the context of Integrability in Gauge and String Theory where the dynamics is obtained from an integrable sigma-model.

DefIS aims to construct the full landscape of integrable sigma-models showing up in this context and to determine its characteristics. This will be achieved by deforming integrable sigma-models while preserving their integrability. DefIS will also unravel dualities within this landscape. Consequences for the AdS/CFT correspondence and its extension will also be studied.

There is no other French team devoted to the study of deformations of integrable sigma-models and its consequences for the AdS/CFT correspondence and beyond. But there is at the same time an ever increasing international competition on this line of research. Furthermore, the interest in the underlying theme of DefIS is emerging from the recent results (including one publication in Physical Review Letters) obtained by members of DefIS. Therefore, to maintain France as a prominent actor at the forefront of this competitive and emerging field, the only possibility is to strengthen this existing collaboration. This is one important motivation for proposing DefIS to ANR.