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M. Nitta (Keio University), Domain-wall Skyrmions in QCD and Chiral Magnets
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In this talk, I will review domain wall-Skyrmions, that is composite states of domain walls and Skyrmions, appearing in two subjects. The first is those in QCD at finite density and magnetic field (or rotation). In this case, these states give a new phase in QCD and a surprising formula expressing the effective nucleon mass analytically in terms of the pion's decay constant and pion mass (to be about 1 Gev) [1]. The second is those in chiral magnets [2,3] which may be useful for nanotechnology such as magnetic memories. I will also introduce a string theory realization [3].

[1] M. Eto, K. Nishimura, M. Nitta, e-Print:
2304.02940 [hep-ph]

[2] C. Ross and M. Nitta, Phys.Rev.B 107 (2023) 2, 024422, e-Print:
2205.11417 [cond-mat.mes-hall]

[3] Y. Amari and M. Nitta, e-Print:
2307.11113 [hep-th]

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K. Lee (KIAS), Breaking instantons to magnetic monopoles
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In this talk, I would like to review the topic and also present our recent related work in 6d little string theories. The T-duality relates the instanton strings in (1,1) LSTs to self-dual strings in (2,0) LSTs. The twisting along the T-duality exhibits a rich structure of magnetic monopole strings as instanton string cartons.

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S. Cherkis (University of Arizona), Tesserons from Monopoles
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A tesseron is a complete hyperhahler manifold with a square integrable Riemann curvature tensor. We discuss which of the tesserons, in the recently emerged classification, are moduli spaces of monopoles. The resulting moduli space perspective leads to a global view of the universal parameter space of all tesserons.

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M. Amano (Yamagata University), Modes of the Sakai-Sugimoto soliton
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The instanton in the Sakai-Sugimoto model corresponds to the Skyrmion on the holographic boundary - which is asymptotically flat - and is fundamentally different from the flat Minkowski space Yang-Mills instanton. The Atiyah-Patodi-Singer index theorem and a series of transformations are used to show that there are 6k zeromodes - or moduli - in the limit of infinite 't Hooft coupling of the Sakai-Sugimoto model. The implications for the low-energy baryons - the Skyrmions - on the holographic boundary, is a scale separation between 2k "heavy" massive modes and 6k-9 "light" massive modes for k>1; the 9 global transformations that correspond to translations, rotations and isorotations remain as zeromodes. For k=1 there are 2 "heavy" modes and 6 zeromodes due to degeneracy between rotations and isorotations.

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S. Cotterill (University of Manchester), Spontaneous Hopf Fibration in the 2HDM
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I will present how energetic considerations enforce a "spontaneous Hopf fibration" of the standard model topology within the two Higgs-Doublet model, whose potential has either an SO(3) or U(1) Higgs-family symmetry that can lead to monopole and vortex solutions. We find these solutions, characterise their basic properties and demonstrate the nature of the fibration along with the connection to Nambu's monopole solution. We point out that the breaking of the electromagnetic U(1) in the core of the defect can be a feature, which leads to a localised non-zero photon mass.

In liquid crystals they experimentally obtain topological charges with long-range e.g. Coulomb-like interactions. It brings question how far can we take its resemblance with particle physics. I will talk about such looking promising approach based on Landau-de Gennes model, with EM-like Lagrangian interpreting field curvature as dual EM field (Faber's approach). This way Gauss law counts topological charge for its quantization, also thanks to Higgs-like potential regularizing charge to finite energy in agreement with the running coupling effect. Then there appear further particle-like topological defects, like topological vortices with knots resembling baryons (e.g. having proton lighter than neutron), and nuclei knotted against Coulomb repulsion. Extending to 4D field adding boosts, their dynamics turns out governed by second set of Maxwell equations for gravity. Article: https://arxiv.org/pdf/2108.07896

Classical kink-(anti)kink scattering has always been a popular research topic, whereas the simpler kink-meson scattering has been relatively less studied. Kink-meson scattering can be roughly divided into two situations: classical and quantum. In classical kink-meson scattering, two processes are allowed: meson fusion and meson reflection. In the quantum case, kink-meson scattering exhibits richer phenomena than in the classical case, including elastic and inelastic scattering. The subject of this talk is the quantum theory of inelastic scattering of a meson with a kink. In this talk, we first review the linearized soliton perturbation theory developed in recent years, which is particularly simple in the one-kink sector. Using it, the amplitude and probability of kink-meson inelastic scattering can be simplified into a perturbative problem in the kink frame. Although the Sine-Gordon soliton and $\phi^4$ kink, which people are usually interested in, are both reflectionless kinks, in order to consider more general cases, we study quantum reflective kinks and find that the amplitude of meson wave packets at different positions during propagation corresponds to the reflection and transmission coefficients of particles scattered by symmetric potential barriers or potential wells in quantum mechanics. We calculate the reduced inner product of the kink states to solve the infrared divergence problem of non-normalizable states. Then we consider the inelastic scattering of a meson off of a kink in a (1+1)-dimensional scalar quantum field theory. At leading order there are three inelastic scattering processes: (1) meson multiplication (the final state is two mesons and a kink); (2) Stokes scattering (the final state is a meson and an excited kink); (3) anti-Stokes scattering (the initial kink is excited and the final kink is de-excited). For the first time, we calculate the leading-order probabilities of these three processes and the differential probabilities for final-state mesons with different momenta. We first obtain general results for arbitrary scalar kinks and then applied them to the kinks of the $\phi^4$ double well model to obtain analytical and numerical results. Finally, we believe that our method can be generalized to higher dimensions, such as the case of monopoles.

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Y. Amari (Keio University), Skyrmion crystals and their relatives in SU(3) chiral magnets
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In this talk, we will numerically show that Skyrmion crystals and other exotic ordered states appear as the ground state in SU(3) chiral magnets, which are SU(3) magnets endowed with a generalization of the Dzyaloshinskii-Moriya interaction. The system could be implemented using spin-1 spinor Bose-Einstein condensates with an artificial spin-orbit coupling in an optical lattice. We discuss the ground state structure of the SU(3) magnets. In addition, we clarify the properties of the ground states using several physical quantities like energy, topological charge, and structure factors.

With the idea to find geometric formulations of particle physics we investigate the predictions of a three dimensional generalisation of the Sine-Gordon model, very close to the Wu-Yang description of Dirac monopoles and to the Skyrme model. With three rotational degrees of freedom of spatial Dreibeins we formulate a Lagrangian and confront the predictions to electromagnetic phenomena. Stable solitonic excitations we compare with the lightest fundamental electric charges, electrons and positrons. Two Goldstone bosons we relate to the properties of photons. These particles are characterised by three topological quantum numbers, which we compare to charge, spin and photon number.

In this talk we explore the dynamics of a wobbling kink in a two-component coupled $\phi^4$ scalar field theory. Firstly, we will study the linear stability of this solution, which will allow us to calculate both the vibration spectrum and the corresponding vibration modes associated with the first (longitudinal) and second field components (orthogonal). Through this analysis, it will be determined that the shape mode and spectrum structure of the orthogonal channel depend on the value of the coupling constant. Additionally, it was discovered that when one orthogonal shape mode is initially triggered, the sole shape mode existing in the longitudinal field is immediately activated due to the coupling between both field components. Furthermore, it will be found that the vibrations of the shape modes act as a source of radiation, which will be emitted at three different frequencies. All the analytical results obtained will be compared with the outcomes obtained via numerical simulations.

The index theorem states that the number of fermionic zero energy modes corresponds to the topological charge Q of the background soliton. In this context, zero modes and localized modes are often identified. In other words, the number of localized modes that appear is typically equal to the topological charge. However, in situations where multiple solitons are mixed, this relationship is not trivial. We study Dirac fermions coupled with a background field that is a topologically trivial soliton, which has a net topological charge Q = 0 but has an internal structure. As a tractable example, we employ an analytical soliton solution constructed by Din and Zakrzewski in the CP2 nonlinear sigma model. This soliton is a mixture of instantons and anti-instantons. To clarify the relationship between the number of zero modes and the topological charge, we perform a spectral flow analysis using this configuration. This allows us to observe how zero modes manifest when Q = 0. In addition, by analyzing the wave functions of each mode, we can verify the occurrence of localized modes. Our results suggest the number of localized modes does not necessarily coincide with the number of zero modes and the topological charge.

Using the BPS Lagrangian method, we show that gravity theory coupled to matter in various dimensions may possess Bogomolny-like equations, which are first-order differential equations, satisfying the Einstein equations and the Euler-Lagrange equations of matter fields (scalar and U(1) gauge fields). We consider static and spherically symmetric ansatzes for gravitational field and the matter fields. Using proper ansatzes we write an effective Lagrangian density that can reproduce the Einstein equations and the Euler-Lagrange equations of matter fields. We define the BPS Lagrangian density to be linear function of first-order derivative of the fields. From these two Lagrangian desities we are able to obtain the Bogomolny-like equations whose some of solutions are well-knon such as Reissner-Nordstrom and Tangherlini black holes. We are also able to find the recent black holes with scalar hair in three dimensions [Phys. Rev. D 107, 124047] and show that there are other possible hairy black holes.

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R. Close (Clark College (retired)), Elastic Wave Model of Magnetic Flux and Electric Charge
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Elastic solid models of vacuum have been influential in the historical development of physics. MacCullagh first derived an equation for light waves from a "rotationally elastic" solid. Maxwell derived the equations of electromagnetism from a crude model of rotating elastic cells. More recently, it has been shown that elastic waves satisfy a nonlinear Dirac equation with the usual operators for both spin and orbital angular momentum. This presentation describes how elastic solitons can satisfy the Pauli exclusion principle and interact via electromagnetic potentials. Solitons are assumed to have a time-varying azimuthal phase characteristic of spherical harmonics. Magnetic flux is proportional to the azimuthal quantum number, while the electrostatic potential is proportional to the time derivative of the azimuthal phase.

Skyrmion is known as a suitable model of baryonic particles in low energy regimes and, in recent decades, its interaction with gravitational field has been actively studied. The coupling between Skyrmions and gravity has been studied in various setups especially the spherically symmetric case where gravitating Skyrmions is used to model star-like objects and hairy black holes. In this talk, I am going to discuss some important results related to the spherically symmetric gravitating Skyrmions and then followed by discussing the well-posedness of the problem of dynamical Einstein-Skyrme system in Bondi Coordinates.

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K. Shiozawa (Kitasato University), Born Sigma Models and Worldsheet Instantons in T-fold
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String worldsheet instantons are governed by the complex structures of the target space. In a space called T-fold (where local charts are patched by T-duality), the metric, complex structures, and so on, have non-trivial T-duality monodromies. Worldsheet instantons acting on the T-fold are thus multivalued and ill-defined. To resolve this issue, we introduce the doubled formalism developed recently by Hull and Zwiebach. The target space becomes T-duality covariant in this formalism. The two-dimensional sigma model whose target space is doubled is called Born sigma model. We explicitly show that the worldsheet instantons on the T-fold are well-defined in the Born sigma model and comment on their properties.

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N. Sogabe (University of Illinois at Chicago), Novel transition dynamics of topological solitons
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Continuous phase transitions can be classified into ones characterized by local order parameters and others that need additional topological constraints. The critical dynamics near the former transitions have been extensively studied, but the latter is less understood. We fill this gap in knowledge by studying the transition dynamics to a parity-breaking topological ground state called the chiral soliton lattice in quantum chromodynamics at finite temperature, baryon chemical potential, and an external magnetic field. We find a slowing down of the soliton¡Çs translational motion as the critical magnetic field approaches while the local dissipation rate remains finite. Therefore, the characteristic time it takes to converge to the stationary state associated with a finite topological number strongly depends on the initial configuration: whether it forms a solitonic structure or not. This talk is based on arXiv:2304.01264 [hep-th].

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L. Disney-Hogg (University of Edinburgh), Symmetries of Monopole Spectral Curves
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The study of magnetic monopoles in gauge theory and the connection with algebraic geometry has been ongoing since the 1970s, and in 1983 Hitchin provided the criteria for certain algebraic curves to correspond to monopole solutions. These constraints are hard to solve, being transcendental in nature, and in the ensuing 40 years only a limited number of spectral curves have been discovered, typically requiring large symmetry groups. In this talk, I will present one approach to finding spectral curves using Nahm's equations and the standard Lax approach to integrable systems, highlighting how computer algebra software can dramatically simplify the non-linear part of the process. Finally, I will discuss how this combination of ideas has let us classify certain charge-3 monopole spectral curves. This is based on joint work with Harry Braden ``Towards a classification of charge-3 monopoles with symmetry''. Lett Math Phys 113, 87 (2023). https://doi.org/10.1007/s11005-023-01708-5

Tesserons are Riemannian solutions of the Einstein Field Equations from General Relativity. ALH* Tesserons are ones such that the asymptotic volume growth of balls of radius R approaches R^{4/3}. Thanks to recent work in 2021, all ALH* Tesserons were classified by Hein, Sun, Viaclovsky, Zhang. They fall into types by their intersection form, $E_0, E_1 \ldots, E_6, E_7, E_8$. Cherkis and Cross showed that the $E_0, E_1, \ldots E_6$ case arise from Doubly Periodic Monopoles. My work is in completing this to the $E_7$ and $E_8$ case.

Understanding the intricate dynamics of topological solitons often requires the application of effective approaches to reduce the complexity. In this talk, we will present an effective theory employing the Collective Co-ordinate Method (CCM) for the \phi^4 model including genuine radiation modes as collective coordinates and examine their role in certain dynamical processes. Particularly, the energy transfer between radiation, translation and shape modes is carefully investigated in the single-kink sector. We will also explore various descriptions of radiation in the study of topologically non-trivial long-lived structures that deserve special attention, the oscillons. This will lead to relevant phenomena such as the oscillon decay and the kink-antikink creation.

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L. Stepien (The Pedagogical University of Cracow),
On some Bogomolny equations and Cauchy problems
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Some exact solutions of Cauchy problems associated with the Bogomolny equations (derived by using the strong necessary conditions and associated with some ordinary equation and some partial differential equations) will be presented. Certain properties associated with one of these solutions will be shown.

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A. Vrba (Independent Researcher), Classical Electromagnetic Soliton Theory
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Last Update: September 23, 2023.