Bundle Gerbe¶
A geometric gerbe presentation built from a surjective submersion, a line bundle on paired fibers, and associative multiplication over triples.
Core Idea¶
A bundle gerbe is a concrete geometric object over a manifold M. Take a surjective submersion Y → M, place a complex line bundle on pairs Y[2] = Y ×_M Y, and give its pairwise lines an associative multiplication over triples. The line bundle is an internal part; the compatible triple product and submersion make the whole bundle gerbe.[ref-095805ed65e3][ref-2ef58842cbaa]
Its Dixmier–Douady class lies in H³(M, ℤ) and may be zero. A connection and curving can add differential information, but are optional. The class identifies stable-isomorphism classes in the cited theory, not every raw geometric presentation. A curvature three-form represents the class's real image and cannot by itself retain integral torsion.[ref-095805ed65e3][ref-db9e10f4ce9c]
Scope of Application¶
Use the term for a smooth geometric presentation whose line/C× bundle is over Y[2] and whose multiplication is associative on compatible triples and quadruples. Murray constructs a lifting bundle gerbe for a principal G-bundle and central extension. Murray and Stevenson construct a basic gerbe on specified unitary-group bases. The cited cases do not establish a universal model for every abstract stack gerbe or spacetime B-field.[ref-095805ed65e3][ref-2ef58842cbaa]
Clarity¶
Separate the presentation, its integral class, and optional connection/curving. A bare H³ element is not the full pair-line-bundle data. A curvature form is real-image information, not the integral class with torsion. Two presentations can have the same associated gerbe or class without being raw-isomorphic; stable isomorphism is the equivalence used for the H³ classification.[ref-095805ed65e3][ref-db9e10f4ce9c]
Manages Complexity¶
The definition reduces a global twisting question to three checks: what submersion supplies local data, what line bundle lives on Y[2], and whether multiplication over Y[3] is associative on Y[4]. In a lifting problem, those data turn a possible lift into a degree-three obstruction. In the unitary-group example, they organize spectral pair data and later curvature calculations.[ref-095805ed65e3][ref-2ef58842cbaa]
Abstract Reasoning¶
Identify M and Y → M. Verify that the line bundle is on the pair fiber product and show the triple multiplication and quadruple associativity. Then determine the Dixmier–Douady class and ask which equivalence notion the problem needs. If a differential statement is made, specify the connection and curving separately. Missing multiplication fails the bundle-gerbe definition; zero class or absent connection alone does not.[ref-095805ed65e3][ref-db9e10f4ce9c][^ref-2ef58842cbaa]
Knowledge Transfer¶
The same concrete roles apply to Murray's central-extension lifting gerbe and Murray–Stevenson's basic unitary-group gerbe, though their construction methods and questions differ. An abstract Gerbe is related: Murray associates one to a bundle gerbe, but the concrete presentation is not literally the stack. Live Line Bundle names the identity-bearing part inside this entry; the staged DAG uses composition/part_of with the parent inside the child.[ref-095805ed65e3][ref-2ef58842cbaa]
Example¶
Lifting bundle gerbe. A principal G-bundle Y → M and a central C× extension of G produce pairwise torsor/line data; the extension's multiplication composes them. Mapped roles: submersion → Y → M; line bundle → extension lifts of pairwise G differences; product → associative extension multiplication; class → obstruction to lifting the principal bundle; connection/curving → not needed for this bare construction.[^ref-095805ed65e3]
Basic unitary-group bundle gerbe. Murray and Stevenson construct a spectral-data submersion on specified unitary groups, a line bundle on paired choices, and coherent multiplication, then add connection and curving. Mapped roles: submersion → spectral-choice space over the chosen group; line bundle → paired spectral line data; product → Definition 2.1 coherence; class → integral twist whose real image is calculated; connection/curving → added differential data, not a universal requirement.[^ref-2ef58842cbaa]
Relationships to Other Abstractions¶
Current abstraction Bundle Gerbe Domain-specific
Parents (1) — more general patterns this builds on
-
Bundle Gerbe is part of Line Bundle Domain-specific
A complex line bundle on the pair fiber product is an identity-bearing internal constituent of every bundle gerbe.
Hierarchy path (1) — routes to 1 parentless root
- Bundle Gerbe → Line Bundle → Local-to-Global Aggregation
Neighborhood in Abstraction Space¶
Bundle Gerbe sits in a sparse region of the domain-specific corpus (84th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Vector Bundles & Classifying Constructions (10 abstractions)
Nearest neighbors
- Associated bundle — 0.82
- Holomorphic vector bundle — 0.82
- Bundle metric — 0.82
- Flat Vector Bundle — 0.82
- Dirac Structure — 0.81
Computed from structural-signature embeddings · 2026-10-08
Not to Be Confused With¶
A line bundle only on M, or even on Y[2] without associative multiplication, is not the whole bundle gerbe. A zero Dixmier–Douady class is allowed. The real curvature form does not contain all integral torsion. Raw bundle-gerbe isomorphism is finer than the stable relation classified by H³. The live Gerbe entry is stack-theoretic, not a second direct parent asserted here. Broader coherent local-to-global assembly outside smooth geometry is a future Prime question, not a proven cross-domain reach of this entry.[ref-095805ed65e3][ref-db9e10f4ce9c][^ref-2ef58842cbaa]
References¶
[^ref-095805ed65e3]: Michael K. Murray, Bundle Gerbes, original author preprint (25 July 1994), arXiv dg-ga/9407015, 19-page PDF. Especially Introduction PDF p. 3, §3 definition PDF pp. 7–8, §4 central-extension lifting PDF pp. 8–9, §5 Dixmier–Douady class, and §§6–8 connection/curvature. The later 1996 journal publication belongs to the same work lineage, not a second independent source. [^ref-db9e10f4ce9c]: Michael K. Murray and Daniel Stevenson, Bundle Gerbes, Stable Isomorphism and Local Theory, original author preprint (1999), arXiv math/9908135, §2 locally split maps and §3 Proposition 3.2 and Theorem 3.3, PDF pp. 4–5. Original title uses a colon after “Gerbes”; the comma in the link label preserves the full binder title. The H³ bijection is for stable-isomorphism classes. [^ref-2ef58842cbaa]: Michael K. Murray and Danny Stevenson, The Basic Bundle Gerbe on Unitary Groups, original author preprint (2008), arXiv 0804.3464. Especially Definition 2.1 PDF pp. 4–5, §3 construction, §4 connection, and §5 curving/curvature/class calculation. The group and normalization scope are those of the paper; connection and curving are additional data.