Loops, Knots, Gauge Theories and Quantum Gravity 2026, 17 - Generators of the Loop Group

The way, a continuous, complex-valued function Ψ(γ) of L_o changes under variation δγ makes the difference of loop derivative. A two-parameter family of infinitesimal loops containing a particular coordinate chart, traversing u^a, v^a is assumed. For π, γ loop differentiable functions depend only on infinitesimal vectors ϵ₁u^a and ϵ₂v^a, assume Ψ expands using the path dependent antisymmetric operator Δ and the element area σ of δγ.

Δ_ab is the loop derivative (1.3.1). The loop derivative has tensor character, by its invariance to transformations containing the end point of the path π, and contracts with σ^ab. They're non-commutative. A loop dependent operator U(α) with U(α)Ψ(γ) = Ψ(α○γ) with natural inverse and composition law. It relates to the loop derivative as Δ_ab(α○π_o^x) = U(α)Δ_ab(π_o^x)U(α)^-1 which describes the transformation property of the loop derivative under finite deformations of γ. The loop derivative has the typical Bianchi and Ricci identities of the Yang-Mills theories. Finally, it also serves as a generator for the group of loops through superposition of loop derivatives (1.3.2). A derivative with similar properties to those of the connection appears as an intermediate step in the construction of gauge theories from the group of loops. A covering of the manifold with overlapping coordinate, where a path π is attached to each coordinate patch P going from the loop origin to some point y in P, a continuous function with support on the points of the patch's chart associates each point x on the patch to π. Each vector u at x has a connection derivative of a continuous function of a loop Ψ(γ) is obtained through deformation of the loop given by the variated path. This notion can be extended to functions of open paths, similarly to the extension of the loop derivative. The connection derivative provides connections or vector potentials similar to those in gauge theories, expressed through loops. The connection in gauge theory is gauge-dependent, which is analogous to the fact that the connection derivative is dependent on the choice of path used for computation (1.3.3). A functional derivative acting on functionals of parametrized curves is the contact derivative C_a(x).

which can be considered a projection of the loop derivative on the tangent to the loop γ. It's the generator of diffeomorphisms on functions of loops (1.3.4).