Integrability and Vesture for Harmonic Maps into Symmetric Spaces

Beheshti, Shabnam; Tahvildar-Zadeh, A. Shadi

doi:10.48550/arxiv.1209.1383

Cited by 2 publications

(10 citation statements)

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“…Motivated by previous work, we prove a pair of theorems in [7] which unify the integrability and solution-generating mechanism (vesture or dressing technique) of special cases considered in the literature for the Einstein vacuum, Einstein-Maxwell and Chiral Field models; the results further establish integrability and vesture for a broader class of geometric field theories. We shall outline the main points surrounding their proof, after which the second half of this communication will focus on control of the solution-generating mechanism.…”

Section: Integrability and Vesture Of Axially Symmetric Harmonic Mapsmentioning

confidence: 68%

“…Compare with Figure 1. In our case, the dressing procedure results in axisymmetric harmonic maps q possessing ring singularities [7].…”

Section: 2mentioning

confidence: 85%

“…The symmetry conditions in (35) ensure that the resulting axially symmetric harmonic map q(x) = χ(0, x)q 0 (x), found by setting λ = 0 in the generating matrix Ψ, takes its values in the appropriate target space, G/K. We show that there always exists an element of the equivalence class of Ψ which satisfies such imposed symmetries [7].…”

Section: 2mentioning

confidence: 91%

“…This definition generalizes the notion of a geodesic to higher-dimensional domains: letting (M, g) = (R 1 , id), the Euler-Lagrange equations for the harmonic map action reduce precisely to the equations of an energy-minimizing geodesic on N . Harmonic maps also generalize harmonic functions to nonlinear targets: letting (N , h) = (R 1 , id), the Euler-Lagrange equation for (24) corresponds to that of a harmonic function on the base manifold M. With this notation in place, we can cast some of our prior examples from the previous sections in the context of harmonic maps 7 .…”

Section: Ernst Formulation Of the Vacuum Equationsmentioning

confidence: 99%

“…The mapping T x : C → C, T x (λ) = −ρ 2 /λ is a deck transformation on the universal cover of R x and T 2 x = id for all x ∈ R 2 + . We prove in [7] that the compatibility condition of the D µ is equivalent to a zero curvature condition dΩ + Ω ∧ Ω = 0, for the 1-form Ω on the domain (cf. [24], p. 54).…”

Section: Integrability and Vesture Of Axially Symmetric Harmonic Mapsmentioning

confidence: 99%

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Dressing with Control: using integrability to generate desired solutions to Einstein's equations

Beheshti¹,

Tahvildar-Zadeh²

2013

Preprint

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Motivated by integrability of the sine-Gordon equation, we investigate a technique for constructing desired solutions to Einstein's equations by combining a dressing technique with a control-theory approach. After reviewing classical integrability, we recall two well-known Killing field reductions of Einstein's equations, unify them using a harmonic map formulation, and state two results on the integrability of the equations and solvability of the dressing system. The resulting algorithm is then combined with an asymptotic analysis to produce constraints on the degrees of freedom arising in the solution-generation mechanism. The approach is carried out explicitly for the Einstein vacuum equations. Applications of the technique to other geometric field theories are also discussed.

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Section: Integrability and Vesture Of Axially Symmetric Harmonic Mapsmentioning

confidence: 68%

“…Compare with Figure 1. In our case, the dressing procedure results in axisymmetric harmonic maps q possessing ring singularities [7].…”

Section: 2mentioning

confidence: 85%

Section: 2mentioning

confidence: 91%

Section: Ernst Formulation Of the Vacuum Equationsmentioning

confidence: 99%

Section: Integrability and Vesture Of Axially Symmetric Harmonic Mapsmentioning

confidence: 99%

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Dressing with Control: using integrability to generate desired solutions to Einstein's equations

Beheshti¹,

Tahvildar-Zadeh²

2013

Preprint

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Inverse scattering and the Geroch group

Katsimpouri

Kleinschmidt

Virmani

2013

J. High Energ. Phys.

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We study the integrability of gravity-matter systems in D = 2 spatial dimensions with matter related to a symmetric space G/K using the well-known linear systems of Belinski-Zakharov (BZ) and Breitenlohner-Maison (BM). The linear system of BM makes the group structure of the Geroch group manifest and we analyse the relation of this group structure to the inverse scattering method of the BZ approach in general.Concrete solution generating methods are exhibited in the BM approach in the socalled soliton transformation sector where the analysis becomes purely algebraic. As a novel example we construct the Kerr-NUT solution by solving the appropriate purely algebraic Riemann-Hilbert problem in the BM approach.The best developed technique for generating solutions is that of Belinski and Zakharov [2,4], henceforth BZ for short and often called the inverse scattering method. It has been very successful in constructing novel solutions in both four-and five-dimensional vacuum gravity. The method involves some rather special adjustments to certain quantities before new physical solutions can be obtained. However, there are problems when applying this method to different gravity-matter systems like those of interest in supergravity where the implementation of the same adjustments fails and it is not guaranteed that an inverse scattering transformation preserves all features required of a solution to the gravity-matter equations [15]. In the group theoretical framework of Breitenlohner and Maison (BM) [3] this problem does not arise since the solution generating transformations form the so-called Geroch group (an affine group) and by the group property any transformation will generate a new solution. The drawback of the BM method is that it is not as easy to implement and does not always operate directly on the physical quantities. Despite these shortcomings, the promise of the BM approach is that it can be applied to various general settings of interest.In order to illustrate this point, we consider for concreteness D = 4 gravity with a spacelike and a time-like Killing vector such that the system is effectively two-dimensional and we are in the realm of so-called stationary axisymmetric solutions. The infinite-dimensional affine symmetry group can be viewed as the closure of two finite-dimensional symmetry groups that act on the space of solutions [6,3]. The first one is typically called the Matzner-Misner group SL(2, R) MM and consists of area preserving diffeomorphisms of the orbit space of the two Killing vectors (the 'torus' that one reduced on, the Killing vector orbits are not necessarily compact). The other group is called the Ehlers group SL(2, R) E and is a hidden symmetry already of the three-dimensional model. The fields that it acts on are partly formed from dualising a Kaluza-Klein vector field (in D = 3) to a scalar field; therefore it does not act directly on the metric components. Combining SL(2, R) MM and SL(2, R) E yields the infinite-dimensional affine Geroch group [6,3].The linear systems of BZ and BM d...

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Integrability and Vesture for Harmonic Maps into Symmetric Spaces

Cited by 2 publications

References 0 publications

Dressing with Control: using integrability to generate desired solutions to Einstein's equations

Dressing with Control: using integrability to generate desired solutions to Einstein's equations

Inverse scattering and the Geroch group

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