arXiv subjects
Publications and source records attributed to J. M. Speight.
It is proved that the normalized $L^2$ metric on the moduli space of $n$-vortices on a two-sphere, endowed with any Riemannian metric, converges uniformly in the Bradlow limit to the Fubini-Study metric. This establishes, in a rigorous setting, a longstanding informal conjecture of Baptista and Manton.
It is shown that the addition of a single chiral symmetry breaking term to the standard omega meson variant of the nuclear Skyrme model can reproduce the proton-neutron mass difference.
Existence of Maxwell-Chern-Simons-Higgs (MCSH) vortices in a Hermitian line bundle $Ł$ over a general compact Riemann surface $Σ$ is proved by a continuation method. The solutions are proved to be smooth both spatially and as functions of the Chern-Simons deformation parameter $κ$, and exist for all $|κ|<κ_*$, where $κ_*$ depends, in principle, on the geometry of $Σ$, the degree $n$ of $Ł$, which may be interpreted as the vortex number, and the vortex positions. A simple upper bound on $κ_*$, depending only on $n$ and the volume of $Σ$, is found. Further, it is proved that a positive {\em lower} bound on $κ_*$, depending on $Σ$ and $n$, but independent of vortex positions, exists. A detailed numerical study of rotationally equivariant vortices on round two-spheres is performed. We find that $κ_*$ in general does depend on vortex positions, and, for fixed $n$ and radius, tends to be larger the more evenly vortices are distributed between the North and South poles. A generalization of the MCSH model to compact Kähler domains $Σ$ of complex dimension $k\geq 1$ is formulated. The Chern-Simons term is replaced by the integral over spacetime of $A\wedge F\wedge ω^{k-1}$, where $ω$ is the Kähler form on $Σ$. A topological lower bound on energy is found, attained by solutions of a deformed version of the usual vortex equations on $Σ$. Existence, uniqueness and smoothness of vortex solutions of these generalized equations is proved, for $|κ|<κ_*$, and an upper bound on $κ_*$ depending only on the Kähler class of $Σ$ and the first Chern class of $Ł$ is obtained.
We demonstrate that the geometric volume of a soliton coincides with the thermodynamical volume also for field theories with higher-dimensional vacuum manifolds (e.g., for gauged scalar field theories supporting vortices or monopoles). We apply this observation to understand Bradlow type bounds for general abelian gauge theories supporting vortices. In the case of SDiff BPS models (being examples of perfect fluid models) we show that the geometric "volume" (area) of the vortex, which is base-space independent, is exactly equal to the Bradlow volume (a minimal volume for which a BPS soliton solution exists). This can be finite for compactons or infinite for infinitely extended solitons (in flat Minkowski space-time).
The two-sphere valued wave map flow on a Lorentzian domain R x Sigma, where Sigma is any flat two-torus, is studied. The Cauchy problem with initial data tangent to the moduli space of holomorphic maps Sigma -> S^2 is considered, in the limit of small initial velocity. It is proved that wave maps, in this limit, converge in a precise sense to geodesics in the moduli space of holomorphic maps, with respect to the L^2 metric. This establishes, in a rigorous setting, a long-standing informal conjecture of Ward.
Motivated by a class of near BPS Skyrme models introduced by Adam, Sánchez-Guillén and Wereszczyński, the following variant of the harmonic map problem is introduced: a map $ϕ:(M,g)\rightarrow (N,h)$ between Riemannian manifolds is restricted harmonic (RH) if it locally extremizes $E_2$ on its $SDiff(M)$ orbit, where $SDiff(M)$ denotes the group of volume preserving diffeomorphisms of $(M,g)$, and $E_2$ denotes the Dirichlet energy. It is conjectured that near BPS skyrmions tend to RH maps in the BPS limit. It is shown that $ϕ$ is RH if and only if $ϕ^*h$ has exact divergence, and a linear stability theory of RH maps is developed, whence it follows that all weakly conformal maps, for example, are stable RH. Examples of RH maps in every degree class $R^3\to SU(2)$ and $R^2\to S^2$ are constructed. It is shown that the axially symmetric BPS skyrmions on which all previous analytic studies of near BPS Skyrme models have been based, are not RH, so each such field can be deformed along $SDiff(R^3)$ to yield BPS skyrmions with lower $E_2$, casting doubt on the predictions of such studies. The problem of minimizing $E_2$ for $ϕ:R^k\to N$ over all linear volume preserving diffeomorphisms is solved explicitly, and a deformed axially symmetric family of Skyrme fields constructed which are candidates for approximate near BPS skyrmions at low baryon number. The notion of restricted harmonicity is generalized to restricted $F$-criticality where $F$ is any functional on maps $(M,g)\to (N,h)$ which is, in a precise sense, geometrically natural. The case where $F$ is a linear combination of $E_2$ and $E_4$, the usual Skyrme term, is studied in detail, and it is shown that inverse stereographic projection $R^3\to S^3\equiv SU(2)$ is stable restricted $F$-critical for every such $F$.
We present a concrete model of a low energy effective field theory of QCD, the well-known Skyrme Model. Specifically, we will work with the BPS submodel in order to describe the binding energies of nuclei. This BPS Skyrme model is characterized by having a saturated bound for the energy proportional to the baryon number of the nuclei. After presenting this classical result, we will proceed with a semi-classical quantization of the coordinates of spin and isospin. Then, with the further inclusion of the Coulomb interaction as well as a small explicit breaking of the isospin symmetry, we finally calculate the binding energies of nuclei, where an excellent agreement has been found for the nuclei with high baryon number. Besides this, we also apply this model to the study of some thermodynamic properties and to neutron stars.
Ricci magnetic geodesic (RMG) motion in a kähler manifold is the analogue of geodesic motion in the presence of a magnetic field proportional to the ricci form. It has been conjectured to model low-energy dynamics of vortex solitons in the presence of a Chern-Simons term, the kähler manifold in question being the $n$-vortex moduli space. This paper presents a detailed study of RMG motion in soliton moduli spaces, focusing on the cases of hyperbolic vortices and spherical $\mathbb{C}P^1$ lumps. It is shown that RMG flow localizes on fixed point sets of groups of holomorphic isometries, but that the flow on such submanifolds does not, in general, coincide with their intrinsic RMG flow. For planar vortices, it is shown that RMG flow differs from an earlier reduced dynamics proposed by Kim and Lee, and that the latter flow is ill-defined on the vortex coincidence set. An explicit formula for the metric on the whole moduli space of hyperbolic two-vortices is computed (extending an old result of Strachan's), and RMG motion of centred two-vortices is studied in detail. Turning to lumps, the moduli space of static $n$-lumps is $Rat_n$, the space of degree $n$ rational maps, which is known to be kähler and geodesically incomplete. It is proved that $Rat_1$ is, somewhat surprisingly, RMG complete (meaning that that the initial value problem for RMG motion has a global solution for all initial data). It is also proved that the submanifold of rotationally equivariant $n$-lumps, $Rat_n^{eq}$, a topologically cylindrical surface of revolution, is intrinsically RMG incomplete for $n=2$ and all $n\geq 5$, but that the extrinsic RMG flow on $Rat_2^{eq}$ (defined by the inclusion $Rat_2^{eq}\hookrightarrow Rat_2$) is complete.
One problem in the application of the Skyrme model to nuclear physics is that it predicts too large a value for the compression modulus of nuclear matter. Here we investigate the thermodynamics of the BPS Skyrme model at zero temperature and calculate its equation of state. Among other results, we find that classically (i.e. without taking into account quantum corrections) the compressibility of BPS skyrmions is, in fact, infinite, corresponding to a zero compression modulus. This suggests that the inclusion of the BPS submodel into the Skyrme model lagrangian may significantly reduce this too large value, providing further evidence for the claim that the BPS Skyrme model may play an important role in the description of nuclei and nuclear matter.
Necessary conditions for a soliton on a torus $M=\R^m/Λ$ to be a soliton crystal, that is, a spatially periodic array of topological solitons in stable equilibrium, are derived. The stress tensor of the soliton must be $L^2$ orthogonal to $\ee$, the space of parallel symmetric bilinear forms on $TM$, and, further, a certain symmetric bilinear form on $\ee$, called the hessian, must be positive. It is shown that, for baby Skyrme models, the first condition actually implies the second. It is also shown that, for any choice of period lattice $Λ$, there is a baby Skyrme model which supports a soliton crystal of periodicity $Λ$. For the three-dimensional Skyrme model, it is shown that any soliton solution on a cubic lattice which satisfies a virial constraint and is equivariant with respect to (a subgroup of) the lattice symmetries automatically satisfies both tests. This verifies in particular that the celebrated Skyrme crystal of Castillejo {\it et al.}, and Kugler and Shtrikman, passes both tests.
It is well known that the low-energy classical dynamics of solitons of Bogomol'nyi type is well approximated by geodesic motion in M_n, the moduli space of static n-solitons. There is an obvious quantization of this dynamics wherein the wavefunction evolves according to the Hamiltonian H_0 equal to (half) the Laplacian on M_n. Born-Oppenheimer reduction of analogous mechanical systems suggests, however, that this simple Hamiltonian should receive corrections including k, the scalar curvature of M_n, and C, the n-soliton Casimir energy, which are usually difficult to compute, and whose effect on the energy spectrum is unknown. This paper analyzes the spectra of H_0 and two corrections to it suggested by work of Moss and Shiiki, namely H_1=H_0+k/4 and H_2=H_1+C, in the simple but nontrivial case of a single CP^1 lump moving on the two-sphere. Here M_1=TSO(3), a noncompact kaehler 6-manifold invariant under an SO(3)xSO(3) action, whose geometry is well understood. The symmetry gives rise to two conserved angular momenta, spin and isospin. A hidden isometry of M_1 is found which implies that all three energy spectra are symmetric under spin-isospin interchange. The Casimir energy is found exactly on the zero section of TSO(3), and approximated numerically on the rest of M_1. The lowest 19 eigenvalues of H_i are found for i=0,1,2, and their spin-isospin and parity compared. The curvature corrections in H_1 lead to a qualitatively unchanged low-level spectrum while the Casimir energy in H_2 leads to significant changes. The scaling behaviour of the spectra under changes in the radii of the domain and target spheres is analyzed, and it is found that the disparity between the spectra of H_1 and H_2 is reduced when the target sphere is made smaller.
A conventional superconductor is described by a single complex order parameter field which has two fundamental length scales, the magnetic field penetration depth λand the coherence length ξ. Their ratio κdetermines the response of a superconductor to an external field, sorting them into two categories as follows; type-I when κ<1/\sqrt{2} and type-II when κ>1/\sqrt{2} . We overview here multicomponent systems which can possess three or more fundamental length scales and allow a separate "type-1.5" superconducting state when, e.g. in two-component case ξ_1<\sqrt{2}λ<ξ_2. In that state, as a consequence of the extra fundamental length scale, vortices attract one another at long range but repel at shorter ranges. As a consequence the system should form an additional Semi-Meissner state which properties we discuss below. In that state vortices form clusters in low magnetic fields. Inside the cluster one of the component is depleted and the superconductor-to-normal interface has negative energy. In contrast the current in second component is mostly concentrated on the cluster's boundary, making the energy of this interface positive. Here we briefly overview recent developments in Ginzburg-Landau and microscopic descriptions of this state.
In an influential paper of 2002, Babaev, Faddeev and Niemi conjectured that two-component Ginzburg-Landau (TCGL) theory in three dimensions should support knot solitons, where the projective equivalence class of the pair of complex condensate fields [psi_1,psi_2]:R^3 -> CP^1 has non-zero Hopf degree. The conjecture was motivated by a certain truncation of the TCGL model which reduced it to the Faddeev-Skyrme model, long known to support knot solitons. Physically, the truncation amounts to ignoring the coupling between [psi_1,psi_2] and the supercurrent of the condensates. The current paper presents a direct test of the validity of this truncation by numerically tracking the knot solitons as the supercurrent coupling is turned back on. It is found that the knot solitons shrink and disappear as the true TCGL model is reached. This undermines the reasoning underlying the conjecture and, when combined with other negative numerical studies, suggests the conjecture, in its original form, is very unlikely to be true.
It is shown that the quantum ground state energy of particle of mass m and electric charge e moving on a compact Riemann surface under the influence of a constant magnetic field of strength B is E_0=eB/2m. Remarkably, this formula is completely independent of both the geometry and topology of the Riemann surface. The formula is obtained by reinterpreting the quantum Hamiltonian as the second variation operator of an associated classical variational problem.
The static baby Skyrme model is investigated in the extreme limit where the energy functional contains only the potential and Skyrme terms, but not the Dirichlet energy term. It is shown that the model with potential $V=\frac12(1+ϕ_3)^2$ possesses solutions with extremely unusual localization properties, which we call semi-compactons. These minimize energy in the degree 1 homotopy class, have support contained in a semi-infinite rectangular strip, and decay along the length of the strip as $x^{-\log x}$. By gluing together several semi-compactons, it is shown that every homotopy class has linearly stable solutions of arbitrarily high, but quantized, energy. For various other choices of potential, compactons are constructed with support in a closed disk, or in a closed annulus. In the latter case, one can construct higher winding compactons, and complicated superpositions in which several closed string-like compactons are nested within one another. The constructions make heavy use of the invariance of the model under area-preserving diffeomorphisms, and of a topological lower energy bound, both of which are established in a general geometric setting. All the solutions presented are classical, that is, they are (at least) twice continuously differentiable and satisfy the Euler-Lagrange equation of the model everywhere.
Let $Σ$ be a compact Riemann surface and $\h_{d,k}(Σ)$ denote the space of degree $d\geq 1$ holomorphic maps $Σ\ra \CP^k$. In theoretical physics this arises as the moduli space of charge $d$ lumps (or instantons) in the $\CP^k$ model on $Σ$. There is a natural Riemannian metric on this moduli space, called the $L^2$ metric, whose geometry is conjectured to control the low energy dynamics of $\CP^k$ lumps. In this paper an explicit formula for the $L^2$ metric on of $\h_{d,k}(Σ)$ in the special case $d=1$ and $Σ=S^2$ is computed. Essential use is made of the kähler property of the $L^2$ metric, and its invariance under a natural action of $G=U(k+1)\times U(2)$. It is shown that {\em all} $G$-invariant kähler metrics on $\h_{1,k}(S^2)$ have finite volume for $k\geq 2$. The volume of $\h_{1,k}(S^2)$ with respect to the $L^2$ metric is computed explicitly and is shown to agree with a general formula for $\h_{d,k}(Σ)$ recently conjectured by Baptista. The area of a family of twice punctured spheres in $\h_{d,k}(Σ)$ is computed exactly, and a formal argument is presented in support of Baptista's formula for $\h_{d,k}(S^2)$ for all $d$, $k$, and $\h_{2,1}(T^2)$.
The variational problem for the functional $F=\frac12\|ϕ^*ω\|_{L^2}^2$ is considered, where $ϕ:(M,g)\to (N,ω)$ maps a Riemannian manifold to a symplectic manifold. This functional arises in theoretical physics as the strong coupling limit of the Faddeev-Hopf energy, and may be regarded as a symplectic analogue of the Dirichlet energy familiar from harmonic map theory. The Hopf fibration $π:S^3\to S^2$ is known to be a locally stable critical point of $F$. It is proved here that $π$ in fact minimizes $F$ in its homotopy class and this result is extended to the case where $S^3$ is given the metric of the Berger's sphere. It is proved that if $ϕ^*ω$ is coclosed then $ϕ$ is a critical point of $F$ and minimizes $F$ in its homotopy class. If $M$ is a compact Riemann surface, it is proved that every critical point of $F$ has $ϕ^*ω$ coclosed. A family of holomorphic homogeneous projections into Hermitian symmetric spaces is constructed and it is proved that these too minimize $F$ in their homotopy class.
Motivated by the sigma model limit of multicomponent Ginzburg-Landau theory, a version of the Faddeev-Skyrme model is considered in which the scalar field is coupled dynamically to a one-form field called the supercurrent. This coupled model is investigated in the general setting where physical space is an oriented Riemannian manifold and the target space is a Kaehler manifold. It is shown that supercurrent coupling destroys the topological stability enjoyed by the usual Faddeev-Skyrme model, so that there can be no globally stable knot solitons in this model. Nonetheless, local energy minimizers may still exist. The first variation formula is derived and used to construct three families of static solutions of the model, all on compact domains. In particular, a coupled version of the unit-charge hopfion on a three-sphere of arbitrary radius is found. The second variation formula is derived, and used to analyze the stability of some of these solutions. A family of stable solutions is identified, though these may exist only in spaces of even dimension. Finally, it is shown that, in contrast to the uncoupled model, the coupled unit hopfion on the three-sphere of radius R is unstable for all R. This gives an explicit, exact example of supercurrent coupling destabilizing a stable solution of the uncoupled Faddeev-Skyrme model, and casts doubt on the conjecture of Babaev, Faddeev and Niemi that knot solitons should exist in the low-energy regime of two-component superconductors.