Module 08 — Connections and Curvature
Why this module: The technical heart. A connection is the geometric object whose local coordinate expression is Landau's $A_i$; its curvature is $F_{ik}$. After this module, the translation table in Module 04 - Landau Reread - Electromagnetism as Geometry stops being a promise and becomes a theorem.
Core idea
Fibers over nearby points are not canonically identified — a connection supplies the missing identification, infinitesimally: it splits each tangent space of $P$ into vertical (along the fiber) ⊕ horizontal (the connection's choice of "constant" directions). Equivalently, it is a $\mathfrak{g}$-valued Connection 1-Form $\omega$ on $P$. Its failure to be integrable — parallel transport around a small loop returning rotated — is the Curvature 2-Form $\Omega = d\omega + \tfrac{1}{2}[\omega, \omega]$.
Physics anchor: For $U(1)$, $[\omega,\omega] = 0$, so $F = dA$ — precisely Landau §23. The Bianchi identity $d\Omega + [\omega, \Omega] = 0$ reduces to $dF = 0$ — Landau §26, now revealed as an identity true for any connection.
Concepts
- Connection on a Principal Bundle — horizontal subspaces; three equivalent definitions
- Connection 1-Form — $\omega$ upstairs, local potentials $A_\alpha$ downstairs, the gluing law $A_\beta = t_{\alpha\beta}^{-1}A_\alpha t_{\alpha\beta} + t_{\alpha\beta}^{-1}dt_{\alpha\beta}$
- Covariant Derivative — the induced derivative on any Associated Bundle
- Parallel Transport and Holonomy — integrating the connection along curves
- Curvature 2-Form — structure equation, Bianchi identity
Reading
- Sternberg - Lectures on Differential Geometry Ch. VII (connections on G-structures)
- Woit - Quantum Theory, Groups and Representations — the gauge fields / covariant derivative chapters, as a physics-side check
Self-check before moving on
State the three definitions of a connection and sketch why they're equivalent. Why does curvature measure "path-dependence of parallel transport"? Why is $dF = 0$ automatic while $d\!\star\!F = J$ is physics?
Concepts in this module
Study tasks
- Read the five concept notes slowly; this is the longest module
- Exercise: for $G = U(1)$ verify the gluing law reduces to $A_\beta = A_\alpha + d\theta_{\alpha\beta}$ (log of transition function) — i.e., Landau's §18 gauge shift *is* the transition between patches
- Exercise: derive $F = dA + \tfrac12 [A,A]$ in a local trivialization from $\Omega = d\omega + \tfrac12[\omega,\omega]$
- Exercise: compute the holonomy of a $U(1)$ connection around a loop: $\exp\left(i\oint_\gamma A\right)$ — memorize this formula; it's the Aharonov–Bohm phase
- Exercise (geometric warm-up): connection on the frame bundle of $S^2$ = Levi-Civita; holonomy around a spherical triangle = its area. Verify for an octant ($\pi/2$)