Geometric Models of Matter Conference

Conference Program

Here you can find the four day programme, each speaker has a 40 minute slot, roughly 30 mins to speak and 10 mins for questions.
Recordings of talks are being posted at Solitons@Work

Monday, August 19th

Nicholas Manton: Skyrmions - Shapes and Dynamics

This talk is on two Skyrmion topics. 1. Based on an analysis of the small r and large r asymptotics of the B=1 hedgehog Skyrmion profile f(r), I propose a universal profile (up to a scale) with continuous curvature. This could be useful as an approximation for Skyrmions in general pion EFTs, including those with higher-order space- and time-derivatives. 2. I draw attention to two important and well-known nuclear reactions, involving Nitrogen-14 and Boron-10, and suggest that they are worth studying through numerical Skyrmion collisions.

Katarzyna Sławińska: Amplitude modulations and resonant decay of excited oscillons

We show that the decay of strongly excited oscillons in a single vacuum model reveals a chaotic, fractal-like pattern very much like one found in kink-antikink collision in the \phi^4 model. This structure can be attributed to the resonant energy transfer mechanism triggered by the modulations of amplitudes of constituent oscillons which form the excited oscillon. We also find evidence that such modulations arise as a motion of two quasi-breathers inside the constituent oscillon.

Jarah Evslin: Quantum kink - meson Scattering

I will describe the scattering of a quantum kink with a meson, which is a quantum of radiation. At leading order, only two nontrivial processes are allowed. The first is meson multiplication, in which the initial meson splits into two mesons. The second is (anti-)Stokes scattering, in which the meson is energy is (increased) reduced while the kink's shape mode is (de-)excited. Elastic kink-meson scattering arises at the next order, and the corresponding scattering amplitude has a peak corresponding to the unstable double-excitation of the kink's shape mode. None of these interactions are allowed in the Sine-Gordon model, but all occur in the phi^4 model.

Kinga Gawrych: Circumventing Derrick's Theorem - Constructing topological solitons in Electroweak Theory

It is well known that instantons give rise to baryon-number-violating interactions in the Standard Model (SM). However, in electroweak (EW) theory, the presence of the Higgs field breaks the scale invariance, and, by Derrick's Theorem, instantons cease to exist in the EW sector. It was speculated that some approximate constrained instanton solutions still exist and give a non-negligible contribution to the path integral. Such solutions would give rise to a new source of baryon number violation in the SM. In this talk, I will discuss how to bypass Derrick's Theorem and construct the constrained instanton configurations in the EW sector. I will also demonstrate the existence of constrained instantons in a simple toy model by presenting specific examples of such solutions.

Stefano Bolognesi: The monopole-fermion problem in a chiral gauge theory

The scattering of electrically charged fermions on magnetic monopole leads to the Callan-Rubakov problem. We discuss some aspects of this problem for Abelian gauge theories with chiral fermions in a Dirac monopole background. In some cases, it is possible to embed the theory in a non-Abelian gauge theory where the monopole is regularized as a 't Hooft-Polyakov monopole. One theory of this kind is the SU(N) chiral gauge theory with fermions in the symmetric, anti-antisymmetric and anti-fundamental representations. We examine this model in detail and provide a possible solution for the condensates around the monopole, the symmetry preserving boundary conditions, and discuss the particle scattering problem. We discuss in general the problem of the fractional OUT states, sometimes called semiton problem in the GUT model, how it emerges in our model, and what are the proposals for the solution.

Yakov Shnir: Self-gravitating monopoles and Skyrmions with localized fermions

Fermionic modes localized on solitons are discussed. We construct self-consistent solutions of the corresponding system of coupled integral-differential equations and study dependencies of solitons and normalizable fermion bound states on the values of the parameters of the model. In particular, we consider solutions of (1+1)-dimensional scalar models with self-interaction potentials and the fermionic modes localized by the Skyrmions and monopoles. Considering the effects of gravity we study fermions localized on the static spherically symmetric non-Abelian monopole in the Einstein-Dirac-Yang-Mills-Higgs theory and on the self-gravitating Skyrmion and discuss how the corresponding spectral flows depend on the effective gravitational coupling constant.

Maximilian Bachmaier: The slingshot effect: Scattering monopoles from Coulomb into the confining vacuum

The slingshot effect describes magnetic monopoles traversing the boundary between Coulomb and confining phases. This passage causes the gauge field to be confined in a cosmic string connected to the domain wall separating the two phases. I will present the results of our numerical simulations. Furthermore, since this phenomenon could be important for first-order phase transitions in the early universe, it may give significant imprints to cosmology. Here I will discuss the characteristics of the gravitational radiation emitted during a single slingshot effect.

Tuesday, August 20th

Alberto García Martín-Caro: Moduli space dynamics of near extremal black holes

In this talk I will present a review of earlier results on the moduli space approximation (MSA) to the dynamics of extremally charged black holes, and how it can be extended by dropping off the extremality condition. Although there are no static solutions of the dynamical equations in the non-extremal case, we are inspired by the success of the MSA for the dynamics of non-BPS solitons to construct an approximate moduli space in terms of solutions of the constraint equations of the Einstein-Maxwell system for a collection of charged, non-extremal black holes of equal charge-to-mass ratio (which where first obtained by Alcubierre et al.) and study the dynamics of collective coordinates in such space.

Jaime Mendizabal: Hyper-Kähler moduli spaces of monopoles with arbitrary symmetry breaking

Moduli spaces for SU(2) monopoles on R³ have been extensively studied, and are known, among other things, to have smooth complete hyper-Kähler structures. In this talk we will present a differential-geometric construction for a smooth hyper-Kähler structure on the moduli spaces of monopoles with arbitrary (compact) gauge group, mass and charge, and hence symmetry breaking. We will furthermore discuss some decay properties of the monopoles themselves which can be deduced from this construction, and we will point to further properties which can be investigated for these moduli spaces.

Sergey Cherkis: Gravitational instantons as monopole moduli spaces

Among all gravitational instantons, we focus on those that are complete hyperkaehler four-manifolds, also called tesserons. We aim to realize each tesseron as a monopole moduli space. Using this realization we get a convenient description of the parameter space of all tesserons, with each type (ALE, ALF, ALG/ALG*, and ALH/ALH*) appearing as a stratum in that parameter space.

Andy Royston: Solitons and the extended Bogomolny equations with jumping data

The extended Bogomolny equations are a system of PDE's for a connection and a triplet of Higgs fields on a three-dimensional space. They are a hybrid of the Bogomolny equations and the Nahm equations. After reviewing how these latter systems arise in the study of magnetic monopoles, I will present an energy functional for which solutions of the extended Bogomolny equations are minimizers in a fixed topological class. In this construction, the connection and Higgs triplet are defined on all of R^3 and couple to additional dynamical fields localized on a two-plane that are analogous to jumping data in the Nahm equations. Solutions can therefore be interpreted as finite-energy BPS solitons in a three-dimensional theory with a planar defect. This talk is based on work done in collaboration with Sophia Domokos.

Harry Braden: Monopoles with symmetry

I will describe some old and new results for su(2) Euclidean BPS monopoles with symmetry reporting on joint work with Linden Disney-Hogg.

Christopher Lang: Instantons with continuous symmetries

In this talk, we use a theorem regarding fixed points on moduli spaces to study instantons with continuous symmetries. In doing so, we find linear constraints for these symmetric instantons, which both greatly simplify the non-linear constraints that instantons satisfy and reduce the problem to representation theory. Moreover, we analyze the connection between these instantons and other gauge theoretic objects, identify novel instantons with higher rank structure groups, and even find all instantons with certain conformal symmetries.

Wednesday, August 21st

Paul Sutcliffe: Skyrmions and ADHM

The ADHM construction provides a method that requires only linear algebra to obtain Yang-Mills instantons from quaternionic matrix data. A similar construction to obtain approximate Skyrmions will be discussed.

Josh Cork: Finkelstein-Rubinstein constraints from ADHM data and rational maps

A key step in quantisation of the Skyrme model is to constrain the wavefunction; this in order to calculate possible spin/isospin quantum numbers of nuclei. Mathematically these constraints arise due to the nontriviality of the fundamental group $\mathbb{Z}_2$ of configuration space Maps$_N(S^3,S^3)$, with the action of a loop on the wavefunction leading to a sign, and thus a constraint. These are called Finkelstein-Rubinstein (FR) constraints. Various methods are known for calculating these in limited scenarios (such as product Skyrme fields, or loops arising from symmetries of rational maps), but in order to extract more meaningful quantum effects, more sophisticated families of Skyrme fields are required. In this talk we shall present new (and remarkably simple) methods for calculating FR constraints for any Skyrme field generated from two popular and powerful schemes: instanton ADHM data, and rational maps.

Benoit Charbonneau: Lowest charge symmetric instantons

Maciej Dunajski: Twistor theory of Chen-Teo instantons

Toric Ricci--flat metrics in dimension four correspond to certain holomorphic vector bundles over a twistor space. We construct these bundles explicitly, by exhibiting and characterising their patching matrices, for the five--parameter family of Riemannian ALF metrics constructed by Chen and Teo. The Chen--Teo family contains a two--parameter family of asymptotically flat gravitational instantons. The patching matrices for these instantons take a simple rational form. This is based on a joint work with Paul Tod https://arxiv.org/abs/2405.08170

Tathagata Ghosh: Instantons and monopoles in dimensions 7 and 8

In this talk I will gently introduce the notion of Yang--Mills instantons in higher dimensions, in particular, in dimensions 7 and 8. I will also briefly discuss the current research in this area, including my own, and how it fits into the bigger picture. After reviewing the 4-dimensional instantons, I will discuss the main physical motivations behind higher dimensional instantons, by following the historical development of the subject. Then, I will introduce G\"unaydin--Nicolai instantons and Fairlie-Nuyts-Fubini-Nicolai (FNFN) instantons on $\mathbb{R}^7$ and $\mathbb{R}^8$ respectively. These are the earliest examples of instantons in dimensions 7 and 8 respectively, analogous to the BPST instantons on $\mathbb{R}^4$. Finally, I will briefly explain how my own research on the deformation theory of instantons on asymptotically conical manifolds can provide many important properties of these instantons. Following the spirit of the theme of the conference, in my talk, I will also very briefly describe the (very new) notion of monopoles in dimension 7.

Ragini Singhal: Instantons on nearly parallel G2 manifolds

We will formulate the deformation theory of G_2-instantons on nearly G_2 manifolds. There is a one-to-one correspondence between nearly parallel G_2 structures and real Killing spinors, thus the deformation theory can be formulated in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to describe the deformation space of the canonical connection on the four normal homogeneous nearly G_2 manifolds.

Theodora Ioannidou: The SU(2) Lie-Poisson algebra & Its descendants

Novel discrete algebra is presented which follows by combining the SU(2) Lie-Poisson bracket with the discrete Frenet equation. Physically, the construction describes a discrete piecewise linear string in R^3. The starting point of our derivation is the discrete Frenet frame assigned at each vertex of the string. Then the link vector that connects the neighboring vertices is assigned the SU(2) Lie-Poisson bracket. Moreover, the same bracket defines the transfer matrices of the discrete Frenet equation which relates two neighboring frames along the string. The procedure extends in a self-similar manner to an infinite hierarchy of Poisson structures. As an example, the first descendant of the SU(2) Lie-Poisson structure is presented in detail. For this, the spinor representation of the discrete Frenet equation is employed, as it converts the brackets into a computationally more manageable form. The final result is a nonlinear, nontrivial, and novel Poisson structure that engages four neighboring vertices.

Thursday, August 22nd

Luis Alvarez-Consul: Gravitating vortices and symplectic reduction by stages

I will explain an approach to study gravitating vortices based on finite-energy pluripotential theory, as recently applied to constant scalar curvature Kähler metrics, combined with symplectic reduction by stages. The gravitating vortices I will consider are solutions to self-dual Einstein-Maxwell-Higgs equations coupling a Kähler metric on a compact Riemann surface with a hermitian metric on a holomorphic line bundle equipped with a fixed global section, in the critical Bogomol'nyi phase. As an application, we obtain GIT polystability of the effective divisor defined by the zeroes of the Higgs field as an algebro-geometric obstruction to the existence of gravitating vortices on the 2-sphere, and find new existence and uniqueness results in positive genus. This is joint work with Mario Garcia-Fernandez, Oscar Garcia-Prada, Vamsi Pritham Pingali and Chengjian Yao (arXiv:2406.03639).

Andrzej Wereszczynski: Collective coordinate model for BPS 2-vortex with shape mode

Nora Gavrea: Multi-vortices from the Sinh-Gordon and Tzitzeica equations

We construct multi-vortices arising as radial solutions to the elliptic sinh-Gordon and Tzitzeica equations. These can be interpreted as abelian vortices on certain surfaces of revolution. These surfaces have a conical excess angle at infinity, and thus they cannot be embedded in the Euclidean 3-space. We show that they can be globally embedded in the hyperbolic space. The existence of these hyperbolic embeddings follows from the asymptotic analysis of a Painleve III ODE.

Nuno Romão: Vortex moduli in toric fibre bundles

I will describe moduli spaces of abelian vortices, targeted in any compact Kähler toric manifold $X$, on a compact Kähler manifold of arbitrary complex dimension. A key step for this result is the proof of a 20-year-old conjecture by J. Baptista (established by him and other authors for $X=\mathbb{P}^n$) concerning an identification of moduli between linear and nonlinear vortices. This is joint work with Marcel Bökstedt.

Bernd Schroers: Geometry and dynamics of chiral magnetic Skyrmions

A key step in defining and analysing the Bogomol’nyi models of chiral magnetic skyrmions is the interpretation of the DMI term in terms of a non-abelian connection. In this talk I’ll revisit this analysis, discuss the physical interpretation of the connection and draw conclusions from a fully geometrical point of view for the coupling of chiral skyrmions to an external current.

Claire Cisowski: Paraxial optical skyrmions

The observation of topological skyrmionics structures in evanescent electromagnetic fields in 2018 has prompted the emergence of a new line of research in optics. Powerful analogies can be drawn between non-trivial topologies of matter and light, from defect formation and annihilation in structured light fields to the emergence of geometric phases. Research on optical skyrmions is still in its infancy, and while initial works have focused on emulating the "baby-skyrmion" textures found in magnetism using the polarization degree of freedom of light, recent developments indicate that the full potential of optical skyrmions has yet to be realized. In this presentation, I will give a state-of-the-art overview of optical skyrmion research and will discuss how these structures are generated and characterized experimentally.

Christoph Adam: Skyrme Crystals, Nuclear Matter and Compact Stars

In this talk, a review of recent results on the different crystalline solutions of the Skyrme model and some of its generalizations is presented. We also discuss the application of these Skyrme crystals to the study of cold nuclear matter at finite density, and the resulting neutron stars. We highlight both the improved results for dense nuclear matter implied by this recent progress and some of the open questions and problems which still must be resolved for a completely satisfactory description of nuclear matter within the Skyrme model framework.

Participants

This is the current list of participants, which will be updated periodically as we get closer to the date of the conference.

Name	Institution
Alberto García Martín-Caro	University of the Basque Country (UPV/EHU)
Alberto José Balseyro Sebastián	University of Salamanca
Alex Colling	University of Cambridge
Andrzej Wereszczynski	Jagiellonian University
Andy Royston	Penn State Fayette
Azadeh Mohammadi	Federal University of Pernambuco
Benoit Charbonneau	University of Waterloo
Bernd Schroers	University of Edinburgh
Bruno Barton-Singer	Foundation for Research and Technology - Hellas
Calum Ross	Edge Hill University
Carlos Garzón Sánchez
Carlos Naya	University of Alcalá
Christoph Adam	Universidad de Santiago de Compostela
Christopher Lang	University of Waterloo
Claire Cisowski	Unviersity of Glasgow
Danial Saadatmand
Daniel Canillas Martínez	University of Salamanca
David Miguélez Caballero	University of Valladolid
Derek Harland	University of Leeds
Filip Blaschke	Silesian University in Opava
Gautam Chaudhuri	University of Leeds
Graeme Wilkin	University of York
Haoyu Sun
Harry Braden	University of Edinburgh
Jack McKenna	City University of London
Jaime Mendizabal	University College London
James Bradshaw	University of Kent
Jarah Evslin	Institute of Modern Physics, CAS
Josh Cork	University of Leicester
Katarzyna Oles	Jagiellonian University
Kinga Gawrych	Imperial College London
Linden Disney-Hogg	University of Leeds
Luis Alvarez-Consul	ICMAT
Maciej Dunajski	University of Cambridge
Manu Paranjape	Université de Montréal
Martin Speight	University of Leeds
Matthew Dales	University of Leicester
Maximilian Bachmaiers	University of Munich & Max Planck Institute for Physics
Michael Singer	University College London
Morgan Rees	University of Kent
Nick Manton	University of Cambridge
Nora Gavrea	University of Cambridge
Nuno Romao	Université Paris-Saclay
Paul Leask	KTH Royal Institute of Technology
Paul Sutcliffe	Durham University
Ragini Singhal	Université Libre de Bruxelles
Sergey Cherkis	Unviersity of Arizona
Sergio Navarro Obregón	University of Valladolid
Seungho Lee	Korea Advanced Institute of Science and Technology
Spencer Whitehead	Duke University
Stefano Bolognesi	University of Pisa
Steffen Krusch	University of Kent
Tathagata Ghosh	University of Leeds
Theodora Ioannidou	Aristotle University of Thessaloniki
Thomas Winyard	University of Edinburgh
Tom Galvin	University of Leeds
Vatsalya Vaibhav	University of Edinburgh
Yakov Shnir
Yuki Amari	Keio University