Associated Bundles and Matter Fields

A single principal bundle generates a whole family of vector bundles — one for each representation of the structure group. These associated bundles are where matter fields live: a field in representation $\rho$ of the gauge group is a section of the bundle associated via $\rho$. This page builds on principal-bundles.md and the representation theory of group-theory/reps-basics.md.

The associated bundle construction

Given a principal $G$-bundle $P\to M$ and a representation $\rho :G\to GL(V)$, the associated vector bundle is $$E=P{\times }_{\rho }V=(P\times V)/\sim ,(u\cdot g,v)\sim (u,\rho (g)v).$$ One quotients the product by the diagonal $G$-action, gluing $(u\cdot g,v)$ to $(u,\rho (g)v)$. The result is a vector bundle over $M$ with fiber $V$ and the same transition functions as $P$ but now acting through $\rho$: $g_{\alpha \beta }\mapsto \rho (g_{\alpha \beta })$. All associated bundles share the one principal bundle's topology, filtered through different representations.

Matter fields as sections

A matter field in representation $\rho$ is a section of $E=P{\times }_{\rho }V$. Equivalently — and this is the physicist's picture — it is a $V$-valued function $\psi$ on the principal bundle that is equivariant, $$\psi (u\cdot g)=\rho (g{)}^{-1}\psi (u),$$ which in a local gauge ${\sigma }_{\alpha }$ becomes the ordinary field ${\psi }_{\alpha }=\psi \circ {\sigma }_{\alpha }:U_{\alpha }\to V$. Under a change of gauge ${\sigma }_{\beta }={\sigma }_{\alpha }\cdot g_{\alpha \beta }$ it transforms as $${\psi }_{\beta }(x)=\rho (g_{\alpha \beta }(x){)}^{-1}{\psi }_{\alpha }(x)$$ — exactly the gauge-transformation law of a charged field. So the abstract "section of an associated bundle" is the transforming matter field of gauge theory. Differentiating sections requires a connection (a gauge potential); that is connections.md.

The adjoint bundle

The most important associated bundle uses the adjoint representation $\mathrm{Ad}:G\to GL(\mathfrak{g})$ on the Lie algebra (see group-theory/lie-algebras.md): $$\mathrm{ad}P=P{\times }_{\mathrm{Ad}}\mathfrak{g}.$$ Its sections are $\mathfrak{g}$-valued fields on $M$. This is where the gauge-theory data lives:

• the field strength $F_{\mu \nu }$ (curvature) is a $2$-form valued in $\mathrm{ad}P$ (curvature.md);
• the difference of two connections is an $\mathrm{ad}P$-valued $1$-form (the space of connections is an affine space modelled on these);
• infinitesimal gauge transformations are sections of $\mathrm{ad}P$;
• the gluon and other adjoint-representation matter (e.g. an adjoint Higgs) are its sections.

Structure-group reduction and symmetry breaking

The reduction of the structure group from $G$ to a subgroup $H\subset G$ — a choice of sub-principal-$H$-bundle $Q\subset P$ — encodes extra geometric or physical structure. Examples:

• Reducing the frame bundle from $GL(n)$ to $O(n)$ is exactly a Riemannian metric; to $SO(n)$, an orientation; to $U(n)$, a complex structure (see riemannian-bridge.md).
• In gauge theory, a Higgs field acquiring a vacuum value picks out a stabilizer $H\subset G$: spontaneous symmetry breaking $G\to H$ is a structure-group reduction, and the vacuum manifold $G/H$ is the fiber whose topology classifies the resulting defects via homotopy. This connects to symmetry-breaking/goldstone.md.

References

• Kobayashi & Nomizu, Foundations of Differential Geometry, Vol. 1, Ch. 1.
• Nakahara, Geometry, Topology and Physics, Ch. 9–10.
• Baez & Muniain, Gauge Fields, Knots and Gravity, Part II.