Some Global Minimizers of a Symplectic Dirichlet Energy

Abstract

Abstract. The variational problem for the functional F = 1 2 M ϕ * ω 2 is considered, where ϕ : (M, g) → (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 → 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.

Supercurrent coupling in the Faddeev–Skyrme model

“…The power 3 4 is believed to be sharp, and is certainly consistent with numerics. The optimal constant c is not known.…”

“…An interesting fact, which does not seem to have been appreciated previously, is that the Faddeev-Skyrme energy E FS grows at least linearly with |Q | in this case, in contrast to the |Q | 3 4 growth found on R 3 . The essential proof has appeared previously for the case M = S 3 [3], but adapts readily to the case of general M. Proof. Since ϕ is algebraically inessential, ϕ * ω is exact, and there exists A ∈ Ω 1 (M) such that dA = ϕ * ω.…”

“…with C = 0) in the coupled model. A less trivial family can be adapted from [3]. Let M be the space of full flags in C k and N be the Grassmannian of l-planes in C k .…”

“…We will see that supercurrent coupling destroys the topological stability enjoyed by knot solitons, in that configurations of arbitrarily small energy can be found in every homotopy class. This should be contrasted with the usual Faddeev-Skyrme model where one has the Vakulenko-Kapitanski bound on R 3 [2], or a linear energy bound on compact domains [3]. We will develop the variational calculus for the coupled model in a rather general geometric setting, and use our results to show explicitly and exactly that supercurrent coupling destabilizes the unit charge ''hopfion'' on the three-sphere of small radius.…”

On the geometrized Skyrme and Faddeev models

“…It is moreover a minimizer in its homotopy class, cf. [40]. Notice that this example will be essentially unique, among semiconformal harmonic submersions from S 3 onto a Riemann surface, cf.…”