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Register a new userFermions, Skyrmions and the 3-Sphere
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This paper investigates a background charge one Skyrme field chirally coupled to light fermions on the 3-sphere. The Dirac equation for the system commutes with a generalised angular momentum or grand spin. It can be solved explicitly for a Skyrme configuration given by the hedgehog form. The energy spectrum and degeneracies are derived for all values of the grand spin. Solutions for non-zero grand spin are each characterised by a set of four polynomials. The paper also discusses the energy of the Dirac sea using zeta function regularization.
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We find an analytic solution of the backreacted coupled fermion-baby-Skyrmion system valid at all values of the coupling parameter. The solution, built on a finite cylinder, is generally given in terms of the Heun functions and satisfies the physical requirements of finite energy. For a special value of the coupling parameter, the solution becomes a periodic crystal of baby-Skyrmions and fermions defined on the plane $mathbb{R}^2$. These solutions are trivially extended to multi-solitonic branches of higher Baryon number.
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 Kummers 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 Kummers 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.
In the Skyrme model atomic nuclei are modelled as quantized soliton solutions in a nonlinear field theory of pions. The mass number is given by the conserved topological charge $B$ of the solitons. Conventionally, Skyrmions are semiclassically quantized within the rigid body approach. In this approach Skyrmions are effectively treated as rigid rotors in space and isospace that is it is assumed that Skyrmions do not deform at all when they spin and isospin. This approximation resulted in qualitative and encouraging quantitative agreement with experimental nuclear physics data. In this talk, we point out that the theoretical agreement could be further improved by allowing classical Skyrmion solutions to deform as they spin and isospin. As a first step towards a better understanding of how nuclei can be approximated by classically spinning and isospinning soliton solutions, we study how classical Skyrmion solutions of topological charges $B=1-4,8$ deform when classical isospin is added.
In this note we consider whether there could be Swampland constraints associated to the presence of fermions in the theory. We propose that any fermion must couple to an infinite tower of states, and that the mass scale of this tower, in Planck units, is set by the strength of the Yukawa coupling to the tower. This is a type of fermionic version of the (magnetic) Weak Gravity Conjecture. We also find that supersymmetry plays a natural part in this fermionic realisation, which motivates a further proposal that supersymmetry can only be broken below the scale set by this Yukawa coupling. We perform some preliminary checks in string theory of these ideas.
We compute the nuclear spin-orbit coupling from the Skyrme model. Previous attempts to do this were based on the product ansatz, and as such were limited to a system of two well-separated nuclei. Our calculation utilises a new method, and is applicable to the phenomenologically important situation of a single nucleon orbiting a large nucleus. We find that, to second order in perturbation theory, the coefficient of the spin-orbit coupling induced by pion field interactions has the wrong sign, but as the strength of the pion-nucleon interactions increases the correct sign is recovered non-perturbatively.
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