Loops, Knots, Gauge Theories and Quantum Gravity 2026, 18 - Extended Group of Loops
All gauge invariant information in a gauge field can be retrieved from holonomy, so all relevant information from loops are also those that emerge from the definition of the holonomy.
(Eq. 2.1 - 2.2) where the multitangents X are loop-dependent objects of rank n. Write μ i = (a i x i ) for easier notation. X transform as multivector densities under coordinate transformations with fixed base points. They're not coordinates and not independent. They're constrainted by algebraic and differential relations. The former stems from the relations satisfied by the generalized Heaviside function in the definition of X and the latter ensures that the holonomy transforms correctly under gauge transformations (p. 49). The group of loops is not a Lie group, since it has no one-component subgroup. The group of loops is however a subgroup of a Lie group called the special extended group of loops (SeL group) (p. 51). The arbitrary multitensor densities give rise to an arbitrary multivector density field E with product law
which is associative and distributive w.r.t. vector addition. The null element is the trivial null vector and identity is I = (1, 0, ...). An inverse element exists for all vectors with non-vanishing zeroth rank component through E-1 = E-1I + ∑∞i=1(-1)iE-i-1(E - E I)i. The set of all vectors with non-vanishing zeroth rank component forms a group with this product as composition law. For multitangents, this product acts as regular loop composition (p. 52). A set 𝕏 of multitensors with zeroth rank component equal to one is closed under this product law, and satisfies both constraints, which finally identifies a group structure with the multitangents as a subgroup.
The generators of the one-parameter subgroup {Xλ} changes under infinitesimal changes to λ as Xλ + dλ = Xλ × Xdλ = Xλ + dXλ/dλ dλ, which for λ = 0 simplifies to Xdλ = I + F dλ (Eqs. 2.29 - 2.30). The vector F is the generator of the one-parameter subgroup {Xλ} (p. 55). The set of all F's satisfying the differential and algebraic constraints spans a vector space 𝔽, with a bilinear operation [f, g] = f × g - g × f ∀ f, g ∈ 𝔽 (Eq. 2.38).