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Register a new userAlgebraic integrability of ${cal PT}$-deformed Calogero models
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We review some recents developments of the algebraic structures and spectral properties of non-Hermitian deformations of Calogero models. The behavior of such extensions is illustrated by the $A_2$ trigonometric and the $D_3$ angular Calogero models. Features like intertwining operators and conserved charges are discussed in terms of Dunkl operators. Hidden symmetries coming from the so-called algebraic integrability for integral values of the coupling are addressed together with a physical regularization of their action on the states by virtue of a $mathcal{PT}$-symmetry deformation.
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For the rational quantum Calogero systems of type $A_1{oplus}A_2$, $AD_3$ and $BC_3$, we explicitly present complete sets of independent conserved charges and their nonlinear algebras. Using intertwining (or shift) operators, we include the extra `odd charges appearing for integral couplings. Formulae for the energy eigenstates are used to tabulate the low-level wave functions.
We consider new Abelian twists of Poincare algebra describing non-symmetric generalization of the ones given in [1], which lead to the class of Lie-deformed quantum Minkowski spaces. We apply corresponding twist quantization in two ways: as generating quantum Poincare-Hopf algebra providing quantum Poincare symmetries, and by considering the quantization which provides Hopf algebroid describing the class of quantum relativistic phase spaces with built-in quantum Poincare covariance. If we assume that Lorentz generators are orbital i.e.do not describe spin degrees of freedom, one can embed the considered generalized phase spaces into the ones describing the quantum-deformed Heisenberg algebras.
We consider several classes of $sigma$-models (on groups and symmetric spaces, $eta$-models, $lambda$-models) with local couplings that may depend on the 2d coordinates, e.g. on time $tau$. We observe that (i) starting with a classically integrable 2d $sigma$-model, (ii) formally promoting its couplings $h_alpha$ to functions $h_alpha(tau)$ of 2d time, and (iii) demanding that the resulting time-dependent model also admits a Lax connection implies that $h_alpha(tau)$ must solve the 1-loop RG equations of the original theory with $tau$ interpreted as RG time. This provides a novel example of an integrability - RG flow connection. The existence of a Lax connection suggests that these time-dependent $sigma$-models may themselves be understood as integrable. We investigate this question by studying the possibility of constructing non-local and local conserved charges. Such $sigma$-models with $D$-dimensional target space and time-dependent couplings subject to the RG flow naturally appear in string theory upon fixing the light-cone gauge in a $(D+2)$-dimensional conformal $sigma$-model with a metric admitting a covariantly constant null Killing vector and a dilaton linear in the null coordinate.
It has been recently shown that every SCFT living on D3 branes at a toric Calabi-Yau singularity surprisingly also describes a complete integrable system. In this paper we use the Master Space as a bridge between the integrable system and the underlying field theory and we reinterpret the Poisson manifold of the integrable system in term of the geometry of the field theory moduli space.
Compactifying type $A_{N-1}$ 6d ${cal N}{=}(2,0)$ supersymmetric CFT on a product manifold $M^4timesSigma^2=M^3timestilde{S}^1times S^1times{cal I}$ either over $S^1$ or over $tilde{S}^1$ leads to maximally supersymmetric 5d gauge theories on $M^4times{cal I}$ or on $M^3timesSigma^2$, respectively. Choosing the radii of $S^1$ and $tilde{S}^1$ inversely proportional to each other, these 5d gauge theories are dual to one another since their coupling constants $e^2$ and $tilde{e}^2$ are proportional to those radii respectively. We consider their non-Abelian but non-supersymmetric extensions, i.e. SU($N$) Yang-Mills theories on $M^4times{cal I}$ and on $M^3timesSigma^2$, where $M^4supset M^3=mathbb R_ttimes T_p^2$ with time $t$ and a punctured 2-torus, and ${cal I}subsetSigma^2$ is an interval. In the first case, shrinking ${cal I}$ to a point reduces to Yang-Mills theory or to the Skyrme model on $M^4$, depending on the method chosen for the low-energy reduction. In the second case, scaling down the metric on $M^3$ and employing the adiabatic method, we derive in the infrared limit a non-linear SU($N$) sigma model with a baby-Skyrme-type term on $Sigma^2$, which can be reduced further to $A_{N-1}$ Toda theory.
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