Quantum Lump Dynamics on the Two-Sphere

Abstract

It is well known that the low-energy classical dynamics of solitons of Bogomol’nyi type is well approximated by geodesic motion in ${{\sf M}_n}$, the moduli space of static n-solitons. There is an obvious quantization of this dynamics wherein the wavefunction ${\psi : {\sf M}_n \rightarrow \mathbb{C}}$ evolves according to the Hamiltonian ${H_0 = \frac{1}{2} \triangle}$, where ${\triangle}$ is the Laplacian on ${{\sf M}_n}$ . Born-Oppenheimer reduction of analogous mechanical systems suggests, however, that this simple Hamiltonian should receive corrections including ${\kappa}$, the scalar curvature of ${{\sf M}_n}$, and ${\fancyscript{C}}$, the n-soliton Casimir energy, which are usually difficult or impossible 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 ${H_1 = H_0 + \frac{1}{4} \kappa}$ and ${H_2 = H_1 + \fancyscript{C}}$, in the simple but nontrivial case of a single ${\mathbb{C}P^1}$ lump moving on the two-sphere. Here ${{\sf M}_1 = {\sf Rat}_1}$, a noncompact kähler 6-manifold invariant under an ${SO(3)\times SO(3)}$ action, whose geometry is well understood. The symmetry gives rise to two conserved angular momenta, spin and isospin. By exploiting the diffeomorphism ${{\sf Rat}_1\cong TSO(3)}$, a hidden isometry of ${{\sf Rat}_1}$ is found which implies that all three energy spectra are symmetric under spin-isospin interchange. The Casimir energy is found exactly on an SO(3) submanifold of ${{\sf Rat}_1}$, using standard results from harmonic map theory and zeta function regularization, and approximated numerically on the rest of ${{\sf Rat}_1}$. The lowest 19 eigenvalues of H _i are found, and their spin-isospin and parity compared for i =  0, 1, 2. It is found that 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.