Loops, Knots, Gauge Theories and Quantum Gravity 2026, 19 - Loop Coordinates

The multitangents X are not freely specifiable, as elements of a group. The associated algebra's free parameters enable separation of the part of the multitangents that can be freely specified. The F's had the advantage of being constrained not by the algebraic constraint, but the homogeneous algebraic constraint. It's easily solvable by requiring some symmetries on F, so the solution for multitangent's constraints is simply X = exp(F) (p. 56). Transversality is well defined for vector densities. For multivector density, this requires construction of a two-form W_ab = ε_abcW^c, which is curl-free due to its transversality. Then, a one-form A_a^W can be defined as ∂_[b]A_a]^W = W_ab. This is defined up to the addition of a gradient (p. 57). The inner product gives rise to a covariant metric on the space of transverse vectors g(V, W) = g_{0 axby}V^axW^by (Eq. 2.42). Transverse and longitudinal projectors are written without a backgroun metric g^axby = ε^abc∂_cδ(x - y) (Eq. 2.46). Define the transverse and longitudinal Dirac deltas for ease of dealing with projectors: δ_T^ax_by = g^{ax cz}g_{cz by}; δ_L^ax_by = δ^ax_by - δ_T^ax_by; where δ^ax_by = δ^a_bδ(x - y). A multivector density E gives rise to a multivector density E_T = E_T = δ_T E. The soldering quantity σ defined through E_D = σ E_T depend only on ϕ that characterizes the choice of decomposition in transverse and longitudinal parts. It has to satisfy the differential constraint in its upper indices (Eqs. 2.54, 2.56 - 2.58). They have definite transversality properties. Given an arbitrary E, σE is a solution of the differential constraint. For an arbitrary transverse multitensor E_T, σE_T satisfies the differential constraint, but not the homogeneous algebraic constraint. Instead, introduce a matrix Ω that does in the upper indices:

(Eq. 2.66) Its product with arbitrary vectors is an algebraic-free object. The set of vectors S = (σΩ) combine the action of σ converting an arbitrary multitensor into a solution of the differential constraint. The loop coordinates provide an explicit representation in terms to describe the action of the differential operators. The loop derivative on a multitangent field is thus defined

(Eqs. 2.75 - 2.76). The multitangent can be evaluated order by order, through expansion of the Dirac-deltas. At first order, all linear terms cancel (2.5).

Any vector F ∈ SeL behaves as a multivector density under diffeomorphisms with fixed basepoint. The transformation law for transverse algebraic-free vectors Y is Y' = δ_TF' = L_DY (Eq. 2.94). The definition of L_D includes the non-diagonal matrix σ, which through isomorphism exhibits σ = Λ_D σ L_D^-1 (Eq. 2.96), and defines it as the soldering quantity between the fundamental representation Λ_D and the adjoint representation L_D. The subspaces 𝕐 and 𝔽 are invariant under diffeomorphisms (p.48).