Analytic (3+1)-dimensional gauged Skyrmions, Heun and Whittaker-Hill equations and resurgence

Abstract

We show that one can reduce the coupled system of seven field equations of the (3+1)-dimensional gauged Skyrme model to the Heun equation (which, for suitable choices of the parameters, can be further reduced to the Whittaker-Hill equation) in two non-trivial topological sectors. Hence, one can get a complete analytic description of gauged solitons in (3+1) dimensions living within a finite volume in terms of classic results in the theory of differential equations and Kummer's con uent functions. We consider here two types of gauged solitons: gauged Skyrmions and gauged time-crystals (namely, gauged solitons periodic in time, whose time-period is protected by a winding number). The dependence of the energy of the gauged Skyrmions on the Baryon charge can be determined explicitly. The theory of Kummer's confluent functions leads to a quantization condition for the period of the time-crystals. Likewise, the theory of Sturm-Liouville operators gives rise to a quantization condition for the volume occupied by the gauged Skyrmions. The present analysis also discloses that resurgent techniques are very well suited to deal with the gauged Skyrme model as well. In particular, we discuss a very nice relation between the electromagnetic perturbations of the gauged Skyrmions and the Mathieu equation which allows to use many of the modern resurgence techniques to determine the behavior of the spectrum of these perturbations.

28 pages, 6 figures, accepted in Phys. Rev. D, New references added

References in corpus (37)

• Observation of a Discrete Time Crystal
• Observation of discrete time-crystalline order in a disordered dipolar many-body system
• Floquet Time Crystals
• On the phase structure of driven quantum systems
• Quantum Time Crystals
• Discrete time crystals: rigidity, criticality, and realizations
• Absence of Quantum Time Crystals
• Classical Time Crystals
• Modeling spontaneous breaking of time-translation symmetry
• Impossibility of Spontaneously Rotating Time-Crystals: A No-Go Theorem
• Comment on "Quantum Time Crystals": a new paradigm or just another proposal of perpetuum mobile?
• Analytic self-gravitating Skyrmions, cosmological bounces and AdS wormholes
• Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lam'e Systems
• Superfluidity and Space-Time Translation Symmetry Breaking
• Hedgehog ansatz and its generalization for self-gravitating Skyrmions
• Multi-Instantons and Multi-Cuts
• Non-linear superposition law and Skyrme crystals
• Comment on "Space-Time Crystals of Trapped Ions": And Yet it Moves Not !
• Static solutions in the U(1) gauged Skyrme model
• Exact multi-soliton solutions in the four dimensional Skyrme model
• Integrability and chemical potential in the (3+1)-dimensional Skyrme model
• Spinning U(1) gauged Skyrmions
• Resurgence in $\eta$-deformed Principal Chiral Models
• Exact Kink Solitons in Skyrme Crystals
• Analytic topologically non-trivial solutions of the (3+1)-dimensional $U(1)$ gauged Skyrme model and extended duality
• Expansions of the solutions to the confluent Heun equation in terms of the Kummer confluent hypergeometric functions
• Whittaker-Hill equation and semifinite-gap Schroedinger operators
• Gravitational catalysis of merons in Einstein-Yang-Mills theory
• Cosmological dynamics of gravitating hadron matter
• Multi-Skyrmions with orientational moduli
• Skyrme model and Isospin Chemical Potential
• On the possibility of spinorial quantization in the Skyrme model
• Skyrmions in the presence of isospin chemical potential
• Solitons in a cavity for the Einstein-SU(2) Non-linear Sigma Model and Skyrme model
• Constraining Monopoles by Topology: an Autonomous System
• $SU(N)$ BPS Monopoles in ${\mathcal{M}}^2\times S^2$
• WKB and Resurgence in the Mathieu Equation

Cited by in corpus (21)

• Ordered arrays of Baryonic tubes in the Skyrme model in (3+1) dimensions at finite density
• Crystals of superconducting Baryonic tubes in the low energy limit of QCD at finite density
• Analytic self-gravitating $4$-Baryons, traversable NUT-AdS wormholes, flat space-time multi-Skyrmions at finite volume and a novel transition in the $SU(3)$-Skyrme model
• Analytic crystals of solitons in the four dimensional gauged non-linear sigma model
• Pion crystals hosting topologically stable baryons
• Analytic SU(N) Skyrmions at finite Baryon density
• Analytic multi-Baryonic solutions in the SU(N)-Skyrme model at finite density
• Gravitating superconducting solitons in the (3+1)-dimensional Einstein gauged non-linear sigma-model
• Cooking pasta with Lie groups
• Resurgence in Lorentzian quantum cosmology: No-boundary saddles and resummation of quantum gravity corrections around tunneling saddle points
• Quantum phase transition and Resurgence: Lessons from 3d $\mathcal{N}=4$ SQED
• On the topological charge of $SO(2)$ gauged Skyrmions in $2+1$ and $3+1$ dimensions
• Exact \textit{pp}-waves, (A)dS waves and Kundt spaces in the Abelian-Higgs model
• Crystals of gauged solitons, force free plasma and resurgence
• Analytic meronic black holes, gravitating solitons and higher spins in the Einstein $SU(N)$-Yang-Mills theory
• Thermodynamics and Phase Transition in Shapere-Wilczek {\it fgh} model: Cosmological Time Crystal in Quadratic Gravity
• Analytic non-homogeneous condensates in the $(2+1)$-dimensional Yang-Mills-Higgs-Chern-Simons theory at finite density
• Analytic Baby Skyrmions at Finite Density
• On the robustness of solitons crystals in the Skyrme model
• Backreacted Coulomb energy in the Skyrme model
• Existence of a supersymmetric extension of the gauged non-linear $\Sigma$-model and the transition from integrability to chaos