Holonomy and Wilson Loops

Holonomy is parallel transport around a closed loop: the linear map a fiber undergoes on returning to its starting point. It is the global, gauge-invariant content of a connection — the observable in gauge theory, the phase in the Aharonov–Bohm effect, and the order parameter of confinement on the lattice. This page builds on connections.md and curvature.md.

Holonomy

Given a connection and a loop $\gamma$ based at $p$, parallel transport around $\gamma$ returns an isomorphism of the fiber $E_p$ to itself — the holonomy $$\mathrm{Hol}(\gamma )=\mathcal{P}\exp (-{\oint }_{\gamma }A)\in G.$$ Its deviation from the identity records how curved the connection is. The set of all holonomies (over all loops at $p$) forms the holonomy group ${\mathrm{Hol}}_p\subseteq G$, a subgroup measuring the connection's "twist". For a flat connection the holonomy of a contractible loop is trivial, so holonomy factors through the fundamental group, giving a homomorphism ${\pi }_1(M)\to G$ — a purely topological object.

Ambrose–Singer: holonomy is integrated curvature

The infinitesimal and global pictures are tied together by:

Ambrose–Singer theorem. The Lie algebra of the holonomy group is spanned by the curvature $F$ evaluated at all points (parallel-transported back to the base point). Holonomy is trivial for all loops iff the connection is flat.

For an infinitesimal loop bounding an area element $d{S}^{\mu \nu }$, the holonomy is $$\mathrm{Hol}\approx 1-F_{\mu \nu }d{S}^{\mu \nu }+\cdots ,$$ so curvature is holonomy per unit area — the non-abelian Stokes theorem. This is the precise sense in which "curvature = field strength" is what you measure by transporting a charge around a small loop.

Wilson loops

The gauge-invariant trace of the holonomy is the Wilson loop $$W(\gamma )=\mathrm{tr}\mathcal{P}\exp (-{\oint }_{\gamma }A).$$ Taking the trace removes the residual gauge freedom (holonomy transforms by conjugation $g^{-1}\mathrm{Hol}g$ under a change of base-point gauge), leaving a genuine observable. Wilson loops are:

• the natural gauge-invariant observables of a non-abelian gauge theory (they, and their products, generate the gauge-invariant functions of $A$);
• the order parameter for confinement: an area-law $⟨W⟩\sim e^{-\sigma \mathrm{Area}}$ signals a linearly confining potential, as computed in lattice gauge theory;
• the variables of loop quantizations of gauge theory and gravity.

The Aharonov–Bohm effect

The cleanest physical holonomy is flat but nontrivial. Outside an infinite solenoid the electromagnetic field vanishes ($F=0$), yet the vector potential does not, and a charged particle encircling the solenoid picks up the phase $$\mathrm{Hol}(\gamma )=\exp (-iq{\oint }_{\gamma }A)=\exp (-iq\Phi ),$$ $\Phi$ the enclosed flux. The effect is nonzero because the loop is non-contractible in the field-free region: $F=0$ makes it locally pure gauge, but $H^1\ne 0$ (de-rham.md) makes the holonomy a real, gauge-invariant, topological observable. It is the experimental proof that the connection — not just the field strength — is physical, and that the right variables are holonomies.

Flux quantization

Single-valuedness of the wavefunction forces the Aharonov–Bohm phase around any loop to be consistent, which quantizes magnetic flux through a non-simply-connected region: $q\Phi \in 2\pi \mathbb{Z}$ for a $U(1)$-bundle. This is the same integer as the first Chern number (characteristic-classes.md) and the Dirac monopole quantization — three faces of ${\pi }_1(U(1))=\mathbb{Z}$.

References

• Kobayashi & Nomizu, Foundations of Differential Geometry, Vol. 1, Ch. 2 §7–8.
• Nakahara, Geometry, Topology and Physics, Ch. 10.
• Peskin & Schroeder, An Introduction to QFT, §15 (Wilson loops).