Morse theory and the Skyrmion-Skyrmion potential

Isler, K.; Letourneux, J.; Paranjape, M. B.

doi:10.1103/physrevd.43.1366

Cited by 5 publications

(14 citation statements)

References 31 publications

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“…The problem was analyzed in greater detail by Isler et al [35], for the B = 0 and B = 2 situation together. Let U S (x) be the eld of a Skyrmion.…”

Section: Search For a Sphaleron In The Skyrme Model: Morse Theorymentioning

confidence: 99%

“…For instance, a physical quantity such as the amount of energy stored in the body by the deformation should not depend on these choices. (35) and T is the transpose of . Therefore A depends strongly on the choice of the bases.…”

Section: Non-linear Deformation Of a Bodymentioning

confidence: 99%

“…To do this we rst write U 1 and U 2 as functions of the matrices A and B and the position vectorsR 1 andR 2 using (351) and the following identity: further we will use a standard method for this type of calculation (for a very complete description of this method, its subtleties and its application to the computation of the Skyrmion-Skyrmion potential in the product ansatz see [35]). …”

Section: Lagrangian Of the Skyrmion-skyrmion Systemmentioning

confidence: 99%

“…T o obtain (363), we used the fact that expanding the integration bounds from C to the whole space only adds higher order contributions to the integral, but does not alter the value of the leading order terms (see [35] for details). The last line (364) is obtained by shifting the integration variable by R 1 and absorbing 3 factors of d in the measure.…”

Section: Lagrangian Of the Skyrmion-skyrmion Systemmentioning

confidence: 99%

“…We give in subsection 3.4 a short introduction to Morse theory and how it could be used to nd new solutions to the equations of motion [35]. Morse theory relates the existence of critical points of a function de ned on a manifold to non-trivial topological aspects of the manifold.…”

Section: Introductionmentioning

confidence: 99%

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Recent mathematical developments in the Skyrme model

Gisiger

1998

Physics Reports

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In this review we present a pedagogical introduction to recent, more mathematical developments in the Skyrme model. Our aim is to render these advances accessible to mainstream nuclear and particle physicists. We start with the static sector and elaborate on geometrical aspects of the de nition of the model. Then we review the instanton method which yields an analytical approximation to the minimum energy con guration in any sector of xed baryon number, as well as an approximation to the surfaces which join together all the low energy critical points. We present some explicit results for B = 2 . W e then describe the work done on the multibaryon minima using rational maps, on the topology of the con guration space and the possible implications of Morse theory. Next we turn to recent w ork on the dynamics of Skyrmions. We focus exclusively on the low energy interaction, speci cally the gradient o w method put forward by Manton. We illustrate the method with some expository toy models. We end this review with a presentation of our own work on the semi-classical quantization of nucleon states and low energy nucleon-nucleon scattering.

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“…The problem was analyzed in greater detail by Isler et al [35], for the B = 0 and B = 2 situation together. Let U S (x) be the eld of a Skyrmion.…”

Section: Search For a Sphaleron In The Skyrme Model: Morse Theorymentioning

confidence: 99%

Section: Non-linear Deformation Of a Bodymentioning

confidence: 99%

Section: Lagrangian Of the Skyrmion-skyrmion Systemmentioning

confidence: 99%

Section: Lagrangian Of the Skyrmion-skyrmion Systemmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Recent mathematical developments in the Skyrme model

Gisiger

1998

Physics Reports

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Topological and Non-Topological Solitons in Scalar Field Theories

Shnir

2018

134

126

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Solitons emerge in various non-linear systems as stable localized configurations, behaving in many ways like particles, from non-linear optics and condensed matter to nuclear physics, cosmology and supersymmetric theories. This book provides an introduction to integrable and non-integrable scalar field models with topological and non-topological soliton solutions. Focusing on both topological and non-topological solitons, it brings together debates around solitary waves and construction of soliton solutions in various models and provides a discussion of solitons using simple model examples. These include the Kortenweg-de-Vries system, sine-Gordon model, kinks and oscillons, and skyrmions and hopfions. The classical field theory of scalar field in various spatial dimensions is used throughout the book in presentation of related concepts, both at the technical and conceptual level. Providing a comprehensive introduction to the description and construction of solitons, this book is ideal for researchers and graduate students in mathematics and theoretical physics.

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Sphalerons in the Skyrme model

Krusch¹,

Sutcliffe²

2004

J. Phys. A: Math. Gen.

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Numerical methods are used to compute sphaleron solutions of the Skyrme model. These solutions have topological charge zero and are axially symmetric, consisting of an axial charge n Skyrmion and an axial charge −n antiSkyrmion (with n > 1), balanced in unstable equilibrium. The energy is slightly less than twice the energy of the axially symmetric charge n Skyrmion. A similar configuration with n = 1 does not produce a sphaleron solution, and this difference is explained by considering the interaction of asymptotic pion dipole fields. For sphaleron solutions with n > 4 the positions of the Skyrmion and antiSkyrmion merge to form a circle, rather than isolated points, and there are some features in common with Hopf solitons of the Skyrme-Faddeev model.

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Morse theory and the Skyrmion-Skyrmion potential

Cited by 5 publications

References 31 publications

Recent mathematical developments in the Skyrme model

Recent mathematical developments in the Skyrme model

Topological and Non-Topological Solitons in Scalar Field Theories

Sphalerons in the Skyrme model

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