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Register a new userIntegrability, Duality and Sigma Models
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We introduce and study conformal field theories specified by $W-$algebras commuting with certain set of screening charges. These CFTs possess perturbations which define integrable QFTs. We establish that these QFTs have local and non-local Integrals of Motion and admit the perturbation theory in the weak coupling region. We construct factorized scattering theory which is consistent with non-local Integrals of Motion and perturbation theory. In the strong coupling limit the $S-$matrix of this QFT tends to the scattering matrix of the $O(N)$ sigma model. The perturbation theory, Bethe anzatz technique, renormalization group approach and methods of conformal field theory are applied to show, that the constructed QFTs are dual to integrable deformation of $O(N)$ sigma-models.
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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.
We study dualities in off-shell 4D N = 2 supersymmetric sigma-models, using the projective superspace approach. These include (i) duality between the real O(2n) and polar multiplets; and (ii) polar-polar duality. We demonstrate that the dual of any superconformal sigma-model is superconformal. Since N = 2 superconformal sigma-models (for which target spaces are hyperkahler cones) formulated in terms of polar multiplets are naturally associated with Kahler cones (which are target spaces for N = 1 superconformal sigma-models), polar-polar duality generates a transformation between different Kahler cones. In the non-superconformal case, we study implications of polar-polar duality for the sigma-model formulation in terms of N = 1 chiral superfields. In particular, we find the relation between the original hyperkahler potential and its dual. As an application of polar-polar duality, we study self-dual models.
The $O(d,d)$ invariant worldsheet theory for bosonic string theory with $d$ abelian isometries is employed to compute the beta functions and Weyl anomaly at one-loop. We show that vanishing of the Weyl anomaly coefficients implies the equations of motion of the Maharana-Schwarz action. We give a self-contained introduction into the required techniques, including beta functions, the Weyl anomaly for two-dimensional sigma models and the background field method. This sets the stage for a sequel to this paper on generalizations to higher loops and $alpha$ corrections.
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.
Supersymmetric non-linear sigma-models are described by a field dependent Kaehler metric determining the kinetic terms. In general it is not guaranteed that this metric is always invertible. Our aim is to investigate the symmetry structure of supersymmetric models in four dimensional space-time in which metric singularities occur. For this purpose we study a simple anomaly-free extension of the supersymmetric CP^1 model from a classical point of view. We show that the metric singularities can be regularized by the addition of a soft supersymmetry-breaking mass parameter.
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