[Paper Review] The Geometry of Bundle Gerbes

Danny Stevenson|ArXiv.org|Apr 18, 2000

Homotopy and Cohomology in Algebraic TopologyMathematics18 references70 citations

TL;DR

This paper introduces the concept of bundle 2-gerbes as a higher-dimensional generalization of bundle gerbes, defining bundle 2-gerbe connections and 2-curvings. It establishes a canonical class in H⁴(M; ℤ) associated with any bundle 2-gerbe, proving that triviality of this class corresponds precisely to the bundle 2-gerbe being stably trivial, and conjecturing a bijection between stable 2-isomorphism classes of bundle 2-gerbes and H⁴(M; ℤ).

ABSTRACT

This thesis reviews the theory of bundle gerbes and then examines the higher dimensional notion of a bundle 2-gerbe. The notion of a bundle 2-gerbe connection and 2-curving are introduced and it is shown that there is a class in $H^{4}(M;\Z)$ associated to any bundle 2-gerbe.

Motivation & Objective

• To develop a geometric theory of bundle 2-gerbes as a higher-dimensional analog of bundle gerbes.
• To define bundle 2-gerbe connections and 2-curvings, extending the differential geometric structure of bundle gerbes.
• To associate a canonical characteristic class in H⁴(M; ℤ) to any bundle 2-gerbe and characterize its triviality.
• To establish a conjectural bijection between stable 2-isomorphism classes of bundle 2-gerbes and H⁴(M; ℤ).

Proposed method

• Uses simplicial techniques and geometric realization to model higher categorical structures.
• Defines bundle 2-gerbes via simplicial bundle gerbes and stable morphisms, generalizing the notion of bundle gerbes.
• Introduces bundle 2-gerbe connections and 2-curvings using differential forms and curvature data.
• Applies Čech, Deligne, and de Rham cohomology to compare characteristic classes associated with bundle 2-gerbes.
• Constructs classifying maps and relates bundle 2-gerbes to 2-gerbes bound by the sheaf C×_M.
• Uses duality and tensor products of stable bundle 2-gerbes to analyze their characteristic classes.

Experimental results

Research questions

• RQ1 What is the correct geometric and differential-geometric generalization of bundle gerbes to dimension four?
• RQ2 How can one define a connection and curvature (2-curving) for a bundle 2-gerbe?
• RQ3 What is the characteristic class in H⁴(M; ℤ) associated with a bundle 2-gerbe, and what does it classify?
• RQ4 When is a stable bundle 2-gerbe trivial, and how does this relate to its characteristic class?
• RQ5 Is there a bijection between stable 2-isomorphism classes of bundle 2-gerbes and H⁴(M; ℤ)?

Key findings

• A bundle 2-gerbe naturally gives rise to a class in H⁴(M; ℤ), which is the main characteristic invariant of the object.
• A stable bundle 2-gerbe is trivial if and only if its associated H⁴(M; ℤ) class vanishes, as proven in Corollary 13.1.
• The four-class of the dual bundle 2-gerbe Q* is the negative of the class of Q, and the class of the tensor product Q₁ ⊗ Q₂ is the sum of the classes of Q₁ and Q₂.
• The construction of a 2-gerbe from a stable bundle 2-gerbe shows that the class in H⁴(M; ℤ) is preserved under this correspondence.
• The paper provides a framework for comparing Čech, Deligne, and de Rham classes associated with a bundle 2-gerbe, showing their equivalence in the cohomological sense.
• The conjecture that stable 2-isomorphism classes of bundle 2-gerbes are in bijection with H⁴(M; ℤ) is supported by the fact that every class in H⁴(M; ℤ) arises as the class of some stable bundle 2-gerbe.