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Register a new userThe Dressing Method as Non Linear Superposition in Sigma Models
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We apply the dressing method on the Non Linear Sigma Model (NLSM), which describes the propagation of strings on $mathbb{R}times mathrm{S}^2$, for an arbitrary seed. We obtain a formal solution of the corresponding auxiliary system, which is expressed in terms of the solutions of the NLSM that have the same Pohlmeyer counterpart as the seed. Accordingly, we show that the dressing method can be applied without solving any differential equations. In this context a superposition principle emerges: The dressed solution is expressed as a non-linear superposition of the seed with solutions of the NLSM with the same Pohlmeyer counterpart as the seed.
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We have introduced Faddeev-Niemi type variables for static SU(3) Yang-Mills theory. The variables suggest that a non-linear sigma model whose sigma fields take values in SU(3)/(U(1)xU(1)) and SU(3)/(SU(2)xU(1)) may be relevant to infrared limit of the theory. Shabanov showed that the energy functional of the non-linear sigma model is bounded from below by certain functional. However, the Shabanovs functional is not homotopy invariant, and its value can be an arbitrary real number -- therefore it is not a topological charge. Since the third homotopy group of SU(3)/(U(1)xU(1)) is isomorphic to the group of integer numbers, there is a non-trivial topological charge (given by the isomorphism). We apply Novikovs procedure to obtain integral expression for this charge. The resulting formula is analogous to the Whiteheads realization of the Hopf invariant.
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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.
In four-dimensional N=1 Minkowski superspace, general nonlinear sigma models with four-dimensional target spaces may be realised in term of CCL (chiral and complex linear) dynamical variables which consist of a chiral scalar, a complex linear scalar and their conjugate superfields. Here we introduce CCL sigma models that are invariant under U(1) duality rotations exchanging the dynamical variables and their equations of motion. The Lagrangians of such sigma models prove to obey a partial differential equation that is analogous to the self-duality equation obeyed by U(1) duality invariant models for nonlinear electrodynamics. These sigma models are self-dual under a Legendre transformation that simultaneously dualises (i) the chiral multiplet into a complex linear one; and (ii) the complex linear multiplet into a chiral one. Any CCL sigma model possesses a dual formulation given in terms of two chiral multiplets. The U(1) duality invariance of the CCL sigma model proves to be equivalent, in the dual chiral formulation, to a manifest U(1) invariance rotating the two chiral scalars. Since the target space has a holomorphic Killing vector, the sigma model possesses a third formulation realised in terms of a chiral multiplet and a tensor multiplet. The family of U(1) duality invariant CCL sigma models includes a subset of N=2 supersymmetric theories. Their target spaces are hyper Kahler manifolds with a non-zero Killing vector field. In the case that the Killing vector field is triholomorphic, the sigma model admits a dual formulation in terms of a self-interacting off-shell N=2 tensor multiplet. We also identify a subset of CCL sigma models which are in a one-to-one correspondence with the U(1) duality invariant models for nonlinear electrodynamics. The target space isometry group for these sigma models contains a subgroup U(1) x U(1).
We revise the use of 8-dimensional conformal, complex (Cartan) domains as a base for the construction of conformally invariant quantum (field) theory, either as phase or configuration spaces. We follow a gauge-invariant Lagrangian approach (of nonlinear sigma-model type) and use a generalized Dirac method for the quantization of constrained systems, which resembles in some aspects the standard approach to quantizing coadjoint orbits of a group G. Physical wave functions, Haar measures, orthonormal basis and reproducing (Bergman) kernels are explicitly calculated in and holomorphic picture in these Cartan domains for both scalar and spinning quantum particles. Similarities and differences with other results in the literature are also discussed and an extension of Schwingers Master Theorem is commented in connection with closure relations. An adaptation of the Borns Reciprocity Principle (BRP) to the conformal relativity, the replacement of space-time by the 8-dimensional conformal domain at short distances and the existence of a maximal acceleration are also put forward.
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