A Notion To Ponder

Definition. Given a Principal Bundle $P \to M$ with group $G$ and a Group Representation $\rho: G \to GL(V)$, define $$E = P \times_\rho V = (P \times V)\,/\,\big[(p, v) \sim (p \cdot g,\, \rho(g^{-1}) v)\big].$$ A Vector Bundle over $M$ with fiber $V$ and Transition Functions $\rho(t_{\alpha\beta})$.

Intuition. "$V$-valued quantities whose meaning depends on a convention." A point of $E$ is a pair (convention, components) with the equivalence: change convention ⇒ components co-rotate. Exactly how physicists use indexed quantities.

Equivalent description (use this in computations). Sections of $P \times_\rho V$ = equivariant functions $\phi: P \to V$, $\phi(p g) = \rho(g^{-1})\phi(p)$.

Why this is the linchpin. One principal bundle + one connection generates, for every representation $\rho$: a matter bundle and its Covariant Derivative. This is why a single photon field couples universally to every charged species — charge $n$ species = section of $P \times_{\rho_n} \mathbb{C}$, coupled via $\partial_\mu - ineA_\mu$. Woit's representation theory plugs in here.

Example. $TM = F(M) \times_{\rho} \mathbb{R}^n$ where $F(M)$ is the Frame Bundle and $\rho$ the defining rep of $GL(n)$.

Related: Principal Bundle · Group Representation · Covariant Derivative · Module 10 - Representations and Matter Fields

Sources: Sternberg Ch. VII; Woit (gauge theory chapters).