We study geometric variational problems for a class of nonlinear sigma-models in quantum field theory. Mathematically, one needs to minimize an energy functional on homotopy classes of maps from closed 3-manifolds into compact homogeneous spaces G/H. The minimizers are known as Hopfions and exhibit localized knot-like structure. Our main results include proving existence of Hopfions as finite energy Sobolev maps in each (generalized) homotopy class when the target space is a symmetric space. For more general spaces we obtain a weaker result on existence of minimizers in each 2-homotopy class.
Our approach is based on representing maps into G/H by equivalence classes of flat connections. The equivalence is given by gauge symmetry on pullbacks of G-->G/H bundles. We work out a gauge calculus for connections under this symmetry, and use it to eliminate non-compactness from the minimization problem by fixing the gauge.
Global minimality of the Hopf map in the Faddeev-Skyrme model with large coupling constant
Department of Pure Mathematics and Mathematical Statistics, University of Cambridge（剑桥大学纯数学与数学统计系）；Institut de Mathématiques de Toulouse, Université de Toulouse（图卢兹数学研究所，图卢兹大学）；Institute for Applied Mathematics, University of Bonn（波恩大学应用数学研究所）
We point out a duality between steady incompressible Euler flows and solutions of the strongly coupled Faddeev-Skyrme sigma model with potential (mass) term. We supplement this result with various applications and several explicit examples.
Firstly, we consider Yang-Mills theory on ${\mathbb R}^{3,1}$ with an adjoint Higgs field spontaneously breaking a compact gauge group $G$ to a subgroup $H$, so that the Higgs vacuum manifold forms the coset $G/H$. It is shown that in the low-energy limit, when the Higgs vacuum value is large, the 4d Yang-Mills-Higgs theory reduces to the Faddeev sigma model on ${\mathbb R}^{3,1}$ with $G/H$ as target. Its action contains the standard two-derivative sigma-model term as well as the four-derivative Skyrme-type term, which stabilizes solutions against scaling. Secondly, we put the Higgs field in …
We discuss the existence of knot solitons (Hopfions) in a Skryme-Faddeev-Niemi-type model on the target space $SU(3)/U(1)^2$, which can be viewed as an effective theory of both the $SU(3)$ Yang-Mills theory and the $SU(3)$ anti-ferromagnetic Heisenberg model. We derive the knot solitons with two different types of ansatz: the first is a trivial embedding configuration of $SU(2)$ into $SU(3)$, and the second is a non-embedding configuration that can be generated through the Bäcklund transformation. The resulting Euler-Lagrange equations for both ansatz reduce exactly to those of the $CP^1$ Skyr…
Compact Q-balls and Q-shells within a $CP^N$ Skyrme-Faddeev type model
Universidade Federal de Santa Catarina（圣卡塔琳娜联邦大学）；Universidade de São Paulo（圣保罗大学）；Jagiellonian University（雅盖隆大学）；Tokyo University of Science（东京科学大学）；National Institute of Technology, Toyota College（国立丰田工学院）
We consider 5d Yang-Mills-Higgs theory with a compact ADE-type gauge group $G$ and one adjoint scalar field on $\mathbb{R}^{3,1}\times\mathbb{R}_+$, where $\mathbb{R}_+=[0,\infty)$ is the half-line. The maximally supersymmetric extension of this model, with five adjoint scalars, appears after a reduction of 6d ${\cal N}{=}\,(2,0)$ superconformal field theory on $\mathbb{R}^{3,1}\times\mathbb{R}_+\times S^1$ along the circle $S^1$. We show that in the low-energy limit, when momenta along $\mathbb{R}^{3,1}$ are much smaller than along $\mathbb{R}_+$, the 5d Yang-Mills-Higgs theory reduces to a n…
Hopf solitons in the Skyrme-Faddeev system on $R^3$ typically have a complicated structure, in particular when the Hopf number Q is large. By contrast, if we work on a compact 3-manifold M, and the energy functional consists only of the Skyrme term (the strong-coupling limit), then the picture simplifies. There is a topological lower bound $E\geq Q$ on the energy, and the local minima of E can look simple even for large Q. The aim here is to describe and investigate some of these solutions, when M is $S^3$, $T^3$ or $S^2 \times S^1$. In addition, we review the more elementary baby-Skyrme syste…
We carry out an investigation imposing a chiral constraint in the phase space of vector and axial-vector Schwinger model. We find that resulting model is identical to gauge non-invariant model which was obtained by the imposition of chiral constraint in the phase-space of in Chiral Schwinger model with the parameter-free Faddeevian anomaly. Three different models having different types of interaction between the matter and gauge field become indistinguishable under a chiral constraint at the quantum mechanical level. The resulting gauge non-invariant model has an equivalent gauge-invariant ver…
