Integrable deformations of sigma models

In this pedagogical review we introduce systematic approaches to deforming integrable 2-dimensional sigma models. We use the integrable principal chiral model and the conformal Wess-Zumino-Witten model as our starting points and explore their Yang-Baxter and current-current deformations. There is an intricate web of relations between these models based on underlying algebraic structures and worldsheet dualities, which is highlighted throughout. We finish with a discussion of the generalisation to other symmetric integrable models, including some original results related to ZT cosets and their deformations, and the application to string theory. This review is based on notes written for lectures delivered at the school "Integrability, Dualities and Deformations," which ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.

4-dimensional Chern-Simons theory and integrable field theories

These lecture notes concern the semi-holomorphic 4d Chern-Simons theory and its applications to classical integrable field theories in 2d and in particular integrable sigma-models. After introducing the main properties of the Chern-Simons theory in 3d, we will define its 4d analogue and explain how it is naturally related to the Lax formalism of integrable 2d theories. Moreover, we will explain how varying the boundary conditions imposed on this 4d theory allows to recover various occurences of integrable sigma-models through this construction, in particular illustrating this on two simple examples: the Principal Chiral Model and its Yang-Baxter deformation. These notes were written for the lectures delivered at the school “Integrability, Dualities and Deformations”, that ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.

Radiation Modulated Spin coupling in double-stranded DNA model

The spin activity in macromolecules such as DNA and oligopeptides, in the context of the Chiral Induced Spin Selectivity (CISS) has been proposed to be due to the atomic Spin-Orbit Coupling (SOC) and the associated chiral symmetry of the structures. This coupling, associated with carbon, nitrogen, and oxygen atoms in biological molecules, albeit small (meV), can be enhanced by the geometry, and strong local polarization effects such as hydrogen bonding (HB). A novel way to manipulate the spin degree of freedom is by modifying the spectrum using a coupling to the appropriate electromagnetic radiation field. Here we use the Floquet formalism in order to show how the half-filled band Hamiltonian for DNA, can be modulated by the radiation to produce up to a tenfold increase of the effective SOC once the intrinsic coupling is present. On the other hand, the chiral model, once incorporating the orbital angular momentum of electron motion on the helix, opens a gap for different helicity states (helicity splitting) that chooses spin polarization according to transport direction and chirality, without breaking time-reversal symmetry. The observed effects are feasible in physically reasonable parameter ranges for the radiation field amplitude and frequency.

A chiral model for sterile neutrino

A model, which extends the standard model with a new chiral U(1)′ gauge symmetry sector, for the eV-mass sterile neutrino is constructed. It is basically fixed by anomaly free conditions. The lightness of the sterile neutrino has a natural explanation. As a by product, this model provides a WIMP-like dark matter candidate.

$\eta$ mesons in hot magnetized nuclear matter

The $\eta N$ interactions are investigated in the hot magnetized asymmetric nuclear matter using chiral SU(3) model and chiral perturbation theory (ChPT). In the chiral model, the in-medium properties of $\eta$-meson are calculated by the medium modified scalar densities under the influence of an external magnetic field. Further, in the combined approach of chiral model and ChPT, off-shell contributions of $\eta N$ interactions are evaluated from the ChPT effective $\eta N$ Lagrangian, and the in-medium effect of scalar densities are incorporated from the chiral SU(3) model. We observe a significant effect of magnetic field on the in-medium mass and optical potential of $\eta$ meson. We observe a deeper mass-shift in the combined approach of ChPT and chiral model compared to the effect of solo chiral SU(3) model. In both approaches, no additional mass-shift is observed due to the uncharged nature of $\eta$ mesons in the presence of magnetic field.

Superconductivity of twisted double bi-layer graphene in chiral model

Taking into account the sp -hybridization effect for valence electrons in carbon atoms, a very simple chiral model of graphene was suggested some years ago [1]. This model used as a special order parameter the unitary SU(2) -matrix U with the kink-like (or domain wall) structure for the description of electrons in a mono-atomic graphene sheet. However, later the new graphene physics began since studying twisted multilayer configurations revealing unconventional superconducting properties (twist-tronics) [2]. Within the scope of the chiral model these twisted configurations can be described by the so-called “product ansatz” U = U1U2 ▪ ▪ ▪ Un . As an example, the special case of twisted double bi-layer graphene (TDBG) configuration is studied and the corresponding twist angles, for which the superconductivity takes place, are found.

New Anisotropic Exact Solution in Multifield Cosmology

In the case of two-scalar field cosmology, and specifically for the Chiral model, we determine an exact solution for the field equations with an anisotropic background space. The exact solution can describe anisotropic inflation with a Kantowski–Sachs geometry and can be seen as the anisotropic analogue of the hyperbolic inflation. Finally, we investigate the stability conditions for the exact solution.

On the Zakharov–Mikhailov action: $$4\hbox {d}$$ Chern–Simons origin and covariant Poisson algebra of the Lax connection

We derive the $$2\hbox {d}$$ 2 d Zakharov–Mikhailov action from $$4\hbox {d}$$ 4 d Chern–Simons theory. This $$2\hbox {d}$$ 2 d action is known to produce as equations of motion the flatness condition of a large class of Lax connections of Zakharov–Shabat type, which includes an ultralocal variant of the principal chiral model as a special case. At the $$2\hbox {d}$$ 2 d level, we determine for the first time the covariant Poisson bracket r-matrix structure of the Zakharov–Shabat Lax connection, which is of rational type. The flatness condition is then derived as a covariant Hamilton equation. We obtain a remarkable formula for the covariant Hamiltonian in terms of the Lax connection which is the covariant analogue of the well-known formula “$$H={{\,\mathrm{Tr}\,}}L^2$$ H = Tr L 2 ”.