[Paper Review] From Sparks to Grundles--Differential Characters

Reese Harvey, H. Blaine Lawson|ArXiv.org|Jun 11, 2003

Homotopy and Cohomology in Algebraic Topology9 references3 citations

TL;DR

This paper introduces a unifying homological framework—(R, Z)-spark complexes—for differential characters, showing that diverse mathematical formulations, including de Rham-Federer currents, hypersparks, grundles (n-gerbes with connection), Cheeger-Simons cochains, and holonomy maps, yield canonically isomorphic groups of secondary invariants. The key contribution is a rigorous equivalence proof across disparate geometric, topological, and analytic approaches to differential characters via a 3×3 exact sequence grid and compatibility of spark complexes.

ABSTRACT

We introduce a new homological machine for the study of secondary geometric invariants. The objects, called spark complexes, occur in many areas of mathematics. The theory is applied here to establish the equivalence of a large family of spark complexes which appear naturally in geometry, topology and physics. These complexes are quite different. Some of them are purely analytic, some are simplicial, some are of Cech-type, and many are mixtures. However, the associated theories of secondary invariants are all shown to be canonically isomorphic. We also show that Differential characters factor to a much smaller, more geometric group, the set of holonomy maps. Numerous applications and examples are explored.

Motivation & Objective

• To establish a general homological framework for studying secondary geometric invariants such as differential characters and gerbes.
• To resolve the long-standing problem of showing canonical isomorphism between disparate formulations of differential characters—e.g., de Rham-Federer currents, hypersparks, grundles, Cheeger-Simons cochains.
• To introduce and formalize the concept of holonomy maps as a geometrically natural starting point for the (R,Z)-theory.
• To demonstrate that smooth hypersparks and grundles of degree k are equivalent via logarithmic/exponential duality in the lowest-degree component.
• To provide a systematic homological machine applicable to new secondary theories beyond differential characters.

Proposed method

• Introduces the concept of a homological spark complex and constructs a 3×3 grid of short exact sequences centered on spark classes.
• Defines compatible spark complexes whose associated spark class groups are canonically isomorphic, enabling cross-formulation comparison.
• Applies the framework to specific complexes: de Rham-Federer sparks, hypersparks, smooth hypersparks, grundles (n-gerbes with connection), Cheeger-Simons cochains, and holonomy maps.
• Uses the curvature condition dA = φ − [C] for Wess-Zumino sparks to link geometric constructions to differential characters.
• Establishes isomorphisms via the holonomy map H : eZk(X) → R/Z satisfying dH ≡ φ mod Z, showing DiffChark(X) ≅ Hol(X).
• Applies the Bockstein map and long exact sequences to relate torsion classes in bHk(X) to cohomology with Zn coefficients.

Experimental results

Research questions

• RQ1 Are the various formulations of differential characters—such as de Rham-Federer currents, hypersparks, grundles, and Cheeger-Simons cochains—truly isomorphic?
• RQ2 Can holonomy maps, defined as homomorphisms from current cycles to R/Z with curvature condition, provide a geometrically natural foundation for differential characters?
• RQ3 How are grundles (n-gerbes with connection) related to smooth hypersparks, and what is the precise duality between them?
• RQ4 What is the role of the Wess-Zumino term in constructing explicit L1_loc-spark representatives with curvature equal to the Cartan 3-form?
• RQ5 How do torsion classes in bHk(X) relate to cohomology with Zn coefficients, and can they be characterized via holonomy maps?

Key findings

• All (R,Z)-spark complexes—de Rham-Federer, hyperspark, smooth hyperspark, grundle, Cheeger-Simons cochain, current cochain, and holonomy map—yield canonically isomorphic groups of spark classes.
• The group of differential characters DiffChark(X) is isomorphic to the group of holonomy maps Hol(X), with h = H ∘ ρ, where ρ maps cycles to currents.
• Grundles of degree k and smooth hypersparks of degree k are equivalent via logarithmic/exponential duality in the lowest-degree component.
• Wess-Zumino sparks exist as AdG-invariant L1_loc-sparks A on compact simply-connected Lie groups satisfying dA = Φ − [C], where Φ is the Cartan 3-form and [C] is the current of the cut locus.
• Torsion classes in bHk(X) correspond bijectively to Hom(Hk(X,Z), Zn), and the holonomy map point of view makes this transparent via the curvature condition.
• The spark class [Sk+1(α)] of Zweck’s construction refines the kth Stiefel-Whitney class and equals the refined integer Stiefel-Whitney class cWk+1(F) ∈ Hk(X,S1) ⊂ bHk(X).