Loops, Knots, Gauge Theories and Quantum Gravity 2026, 16 - Holonomies and Loop Group

I haven't looked properly into quantum loop gravity yet, and I haven't actually found very strong defenders of it, but I know it exists. I'm expressing the same interest in it as I did in String Theory. This book probably isn't meant to be a primer to it, but it starts from a corner of mathematics that I'm decently familiar with. It begins with Connections and parallel transport as should be familiar from Yang-Mills theory, and defines holonomies through their connection to closed curves, and from there works up to groups of topological loops through their invariance under gauge transformations acting trivially on the base point. I'm slightly surprised to find the mention of an SU(2) Yang-Mills theory in here as well, which also used holonomies to model the common geometrical framework for all fundamental forces. Generally, holonomies can be defined without connections through homomorphisms from closed curves, mapped onto a Lie group. Equivalent closed curves are loops, and those are associated by group structures. Non-local formulations of gauge theories are made on the basis of holonomies. When quantizing the theory, wave functions expressed through loops are dependent on the elements of these groups, which leads to quantum loop representation. Groups of loops aren't Lie groups, but they can be equipped with infinitesimal generators. When they are represented in the space of functions of loops, they give rise to differential operators in loop space. Loop space is often formulated with parametrized curves. The arising differential operators are written through functional derivatives (1.1).

Any set of continuous, piecewise smooth parametrized curves on a manifold M has a natural composition by setting the end-point of one as the origin-point of the other. Write the "opposite curve" p^-1(s) = p (1 - s) as the curve p traversed in the opposite orientation (Eq. 1.3). The composition of unparametrized curves is well defined and independent of the members of their equivalence classes. Closed curves have the same end-point and origin-point, and their set are noted by L_o. L_o is a semi-group under the composition law with identity element of the constant curve (a point). Since the group inverse is not given through the opposite curve, it doesn't have full group structure. This can be corrected through parallel transport, which enables association of closed curves to the same holonomy of smooth connections. This equivalence relation is the loop, denoted in this book by Greek letters. Given H_A: L_o → G be the holonomy map of a connection A defined on a bundle P(M, G), so that l, m ∈ L_o, l ~ m ⇔ H_A(l) = H_A(m) for every bundle P and smooth connection A (Def. 1). Closed loops are defined as tree or thin, if a homotopy between l and the null curve exists, with image included in image(l). Such curves don't enclose any area of M. l ~ m iff l ○ m^-1 is thin. Thin curves then are equivalent to the null curve (Def. 2). Given closed curves l, m and open curves p, q, r: l = p ○ q; m = p ○ r ○ r^-1 ○ q, then l ~ m (Def. 3). The set of loops basepointed at o L_o is non-Abelian under the composition law. Any homomorphism from L₀ to any Lie group G defines a holonomy associated with a generalized connection, which doesn't necessarily need t be a smooth function. Extra smoothness conditions may be imposed (1.2).

A representation of the group of loops through operators acting on continuous functions under this topology emerges from differential operators acting on the connections, which themselves relate to the infinitesimal group generators. These are the loop derivative (1.3.1), the connection derivative (1.3.3) and the contact and functional derivatives (1.3.4)