2 2 Ju n 20 00 Clebsch ( String ) Parameterization of 3 - Vectors and Their Actions

We discuss some properties of the intrinsically nonlinear Clebsch decomposition of a vector field into three scalars in d = 3. In particular, we note and account for the incom-pleteness of this parameterization when attempting to use it in variational principles involving Maxwell and Chern-Simons actions. Similarities with string decomposition of metrics and their actions are also pointed out. The decomposition of vectors, as well as higher rank tensors, into irreducible parts is an ancient and extremely useful tool in fluid mechanics, electrodynamics, and gravity. The longitudinal/transverse split separates an arbitrary d = 3 Euclidean vector into a scalar plus a transverse vector, A ≡ A L + A T = ∇λ + ∇ × W T , (1) the vector W T being defined up to a gradient. This linear orthogonal (upon spatial integration) parameterization naturally decomposes the 3-vector A into two, with 1 and 2 components, respectively. The completeness of (∇λ, W T) in representing A is evidenced by their uniqueness and invertibility (up to zero modes); in particular, any variational principle yields the same Euler-Lagrange system whether we vary A or first decompose it and then vary (λ, W T) separately. It is easy to check that since the field strength is B≡∇ × A = −∇ 2 W T , (2) the Chern-Simons (first used physically in [1]) and Maxwell actions