On a low energy bound in a class of chiral field theories with solitons

A low energy bound in a class of chiral solitonic field theories related the infrared physics of the SU(N) Yang-Mills theory is established. 1. The model. Consider N−1 smooth fields na = na(x) in spacetime taking their values in the Lie algebra of SU(N). The fields are chosen to be commutative [na, nb] = 0 and orthonormal (na, nb) = δab with respect the Cartan-Killing form in the Lie algebra. For any two Lie algebra elements ξ and η, the Cartan-Killing form is defined as (ξ, η) = tr (ξ̂η̂) where the operator ξ̂ acts on the Lie algebra as a Lie derivative ξ̂η = [ξ, η]. There can only be N − 1 mutually commutative and linearly independent elements in the Lie algebra of SU(N) because the rank of SU(N) is r = N − 1 (the dimension of the Cartan subalgebra). If ha form an orthonormal basis in the Cartan subalgebra in a matrix representation of SU(N), then na(x) = U †(x)haU(x) , (1) where U(x) ∈ SU(N). In Eq. (1) U(x) is defined modulo the left multiplication by elements from the Cartan subgroup generated by ha (the maximal Abelian subgroup T = U(1) N−1). So, in fact, U(x) ∈ SU(N)/T since any group element can be represented as a product of an element of T and an element of the quotient SU(N)/T . Under the condition that na approach fixed constant values at the spatial infinity, na(x) → ha, i.e., U(x) approaches the group unity, the fields na define a map of a spatial three-sphere S 3 into the manifold SU(N)/T for every moment of time. The third homotopy group of this map is nontrivial π3(G/T ) ∼ Z, G = SU(N). When N = 2, the only field n1 can be regarded as a unit 3-vector. It is a Hopf map: S → S ∼ SU(2)/U(1). The corresponding topological number is the Hopf invariant which can also be interpreted as a linking number of two curves in S being preimages of two distinct points of S. The two-forms F a = F a jkdx j ∧ dx, j, k = 1, 2, 3, where F a jk = iN ∑ b(na, [∂jnb, ∂knb]) , (2) are closed, that is, F a jk = ∂jC a k − ∂kC j . This is proved at the end of next section. The forms F a may not be exact. This follows from the fact that the cohomology ring H∗(G/T ) is rationally generated by H(G/T ) [1]. The topological number of the map S → G/T should be constructed out of the 2-forms F a = iN(na, ∑ b[dnb, dnb]) on G/T . Introducing the field on leave from Laboratory of Theoretical Physics, JINR, Dubna, Russia