Lie-Dragging Sections Vectorially. Thought of the Day 149.0

Generalized vector fields over a bundle are not vector fields on the bundle in the standard sense; nevertheless, one can drag sections along them and thence define their Lie derivative. The formal Lie derivative on a bundle may be seen as a generalized vector field. Furthermore, generalized vector fields are objects suitable to describe generalized symmetries.

Let B = (B, M, π; F) be a bundle, with local fibered coordinates (x^μ; yⁱ). Let us consider the pull-back of the tangent bundle  τ_B: T_B → B along the map π^k₀: J^kB → B:

A generalized vector field of order k over B is a section Ξ of the fibre bundle π^* : π^k₀ : ^*T_B → J^kB, i.e.

for each section σ: M → B, one can define Ξ_σ = i ○ Ξ ○ j^kσ: M → TB, which is a vector field over the section σ. Generalized vector fields of order k = 0 are ordinary vector fields over B. Locally, Ξ(x^μ, yⁱ, …, yⁱ_μ1,…μk) is given the form:

Ξ = ξ^μ(x^μ, yⁱ, …, yⁱ_μ1,…μk)∂_μ + ξⁱ(x^μ, yⁱ, …, yⁱ_μ1,…μk)∂_i

which, for k ≠ 0, is not an ordinary vector field on B due to the dependence of the components (ξ^μ, ξⁱ) on the derivative of fields. Once one computes it on a section σ, then the pulled-back components depend just on the basic coordinates (x^μ) so that Ξ_σ is a vector field over the section σ, in the standard sense. Thus, generalized vector fields over B do not preserve the fiber structure of B.

A generalized projectable vector field of order k over the bundle B is a generalized vector field Ξ over B which projects on to an ordinary vector field ξ = ξ^μ(x)∂_μ on the base. Locally, a generalized projectable vector field over B is in the form:

Ξ = ξ^μ(x^μ)∂_μ + ξⁱ(x^μ, yⁱ, …, yⁱ_μ1,…μk)∂_i

As a particular case, one can define generalized vertical vector fields (of order k) over B, which are locally of the form:

Ξ = ξⁱ(x^μ, yⁱ, …, yⁱ_μ1,…μk)∂_i

In particular, for any section σ of B and any generalized vertical vector field Ξ over B, one can define a vertical vector field over σ given by:

Ξ_σ = ξⁱ(x^μ, σⁱ(x),…, ∂_{μ1,…, μk}σⁱ(x))∂_i

If Ξ = ξ^μ∂_μ + ξⁱ∂_i is a generalized projectable vector field, then Ξ_(v) = (ξⁱ – yⁱ_μξ^μ)∂_i = ξⁱ_(v)∂_i is a generalized vertical vector field, where Ξ_(v) is called the vertical part of Ξ.

If σ’: ℜ x M → B is a smooth map such that for any fixed s ∈ ℜ σ_s(x) = σ'(s, x): M → B is a global section of B. The map σ’ as well as the family {σ_s}, is then called a 1-parameter family of sections. In other words, a suitable restriction of the family σ_s, is a homotopic deformation with s ∈ ℜ of the central section σ = σ₀. Often one restricts it to a finite (open) interval, conventionally (- 1, 1) (or (-ε, ε) if “small” deformations are considered). Analogous definitions are given for the homotopic families of sections over a fixed open subset U ⊆ M or on some domain D ⊂ M (possibly with values fixed at the boundary ∂D, together with any number of their derivatives).

A 1-parameter family of sections σ_s is Lie-dragged along a generalized projectable vector field Ξ iff

(Ξ_(v))_σs = d/ds σ_s

thus dragging the section.

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