| fiber bundle | |
|---|---|
| Name | Fiber bundle |
| Field | Differential geometry; Algebraic topology |
| Introduced | 20th century |
| Related | Principal bundle, Vector bundle, Connection |
fiber bundle
A fiber bundle is a structure in Differential geometry and Algebraic topology consisting of a total space locally presented as a product of a base space and a fiber. In the context of Quantum Physics, fiber bundles provide the geometric language for gauge theory, quantum state spaces, and topological invariants underpinning phenomena such as the Berry phase and the Quantum Hall effect. They form the mathematical framework for expressing symmetrys, holonomy, and global obstruction classes that affect quantum dynamics.
A fiber bundle is a quadruple (E, B, π, F) where E is the total space, B the base space, π:E→B the projection map, and F the typical fiber, together with a covering of B by open sets Uα and homeomorphisms (local trivializations) φα:π^{-1}(Uα)→Uα×F satisfying transition function compatibility on overlaps Uα∩Uβ. The transition functions take values in the structure group G, a Lie group or topological group that acts on F. Standard examples include trivial bundles, Möbius strip (a nontrivial line bundle), and the tangent bundle TM of a manifold M. Characteristic classes such as Chern class, Stiefel–Whitney class, and Pontryagin class classify bundles up to isomorphism in many settings and are computed via cohomology theories like de Rham cohomology or Čech cohomology.
A Principal bundle P with structure group G underlies gauge fields in physics: physical gauge potentials are modeled as connections on P while matter fields live in associated bundles E=P×_G V for a representation V of G. In quantum field theory frameworks such as Yang–Mills theory, principal bundles over spacetime (often modeled as a smooth manifold with a Lorentz group structure) encode global gauge structure and topological sectors labelled by homotopy group invariants like π_n(G). Notable mathematical physics contexts include the Atiyah–Singer index theorem applications to anomalies, and the role of bundles in the classification of instantons on SU(2) or SU(N) principal bundles studied by Michael Atiyah and Isadore Singer.
Quantum states are sections of complex line bundles in single-particle contexts or of Hilbert bundles (infinite-dimensional vector bundles with fiber a Hilbert space) in families of systems. A complex line bundle L over parameter space B represents a family of quantum mechanical ground states up to phase; its first Chern class c1(L) controls quantized transport and degeneracies. In many-body and condensed matter physics, Bloch states define vector bundles over the Brillouin zone; their topological invariants (e.g., Chern numbers) determine conductance quantization in the Quantum Hall effect. Rigorous treatments relate these bundles to K-theory used by Michael Freed and Gregory Moore in topological phases, and to the Kitaev periodic table for topological insulators and superconductors.
A connection on a bundle gives a notion of parallel transport and covariant derivative; on principal bundles these are described by Lie-algebra valued 1-forms (gauge potentials). The curvature of a connection corresponds to field strength in physics (e.g., the electromagnetic tensor F in Maxwell's equations or the Yang–Mills field strength). Holonomy groups capture the result of parallel transport around closed loops and are linked to observable phases in quantum mechanics through the Wilczek–Zee non‑Abelian generalization. Mathematical tools include Ehresmann connection, Levi-Civita connection in Riemannian geometry, and the formalism of fiber bundles is central to modern treatments of gauge symmetry, BRST quantization, and anomaly inflow computations by teams at institutions such as Institute for Advanced Study and CERN.
Topological phases of matter are classified by global invariants of bundles: for two-dimensional systems, the integer-valued Chern number of the valence band bundle yields the Hall conductance via the TKNN formula (Thouless–Kohmoto–Nightingale–den Nijs). Time-reversal symmetric systems invoke Z2 invariants expressible using Stiefel–Whitney class or K-theory groups. The study of topological insulators and topological superconductors connects to bundle obstruction theory and to the work of Charles L. Kane and Eugene Mele as well as Alexei Kitaev's classification. Global anomalies and theta terms in quantum field theories are likewise controlled by characteristic classes like c1 and Pontryagin class.
The Berry phase arises from holonomy in a complex line bundle over the parameter manifold of Hamiltonians; the geometric phase is the integral of the connection 1-form (Berry connection) and its curvature is the Berry curvature. The quantum Hall effect connects to Chern numbers of bundles over the torus-shaped Brillouin zone; experimental confirmation of quantized conductance links topology with measurable transport coefficients. In quantum field theory, fiber bundles underpin Yang–Mills theory, the formulation of electroweak theory (based on SU(2)×U(1) principal bundles), and quantum chromodynamics (with SU(3) bundles), while topological quantum field theories (TQFT) and constructions like the Chern–Simons theory employ bundle invariants to compute knot and manifold invariants. Research on fiber bundles in quantum contexts involves collaborations across Princeton University, Harvard University, MIT, Perimeter Institute, and Max Planck Institute for Physics, and appears in foundational papers and texts by M. Nakahara, Michael Atiyah, Barry Simon, and others.
Category:Differential geometry Category:Quantum mechanics Category:Gauge theories