A functorial approach to differential characters

Abstract

We describe Cheeger-Simons differential characters in terms of a variant of Turaev's homotopy quantum field theories based on chains in a smooth manifold X .

TQFT’s and gerbes

“…The manifolds involved in the TQFT now come equipped with maps to a target manifold on which the bundle or c Geometry & Topology Publications gerbe lives. This brings us into the realm of Segal's category C [23] and string connection [22], as well as Turaev's Homotopy Quantum Field Theory (HQFT) [24] and a related construction by Brightwell and Turner [8], as well as subsequent developments by Rodrigues [19], Bunke, Turner and Willerton [10] and Turner [25]. Here we will base our approach on a general framework for TQFT and related constructions by Semião and the author [18], which defines a TQFT to be a certain type of monoidal functor, without using the cobordism approach.…”

Orientation Twisted Homotopy Field Theories and Twisted Unoriented Dijkgraaf–Witten Theory

“…Let T be a topological space and let λ ∈ Z n (T ; C × ). Independently, Turaev [47, I.2] and Turner [49] constructed an invertible oriented homotopy field theory P λ T : T -Cob or n,n−1 → Vect C valued in complex vector spaces. The construction of Turaev is direct while that of Turner is in terms of higher gerbes with connection.…”

Differential Characters and Geometric Chains

“…Chain field theories in the sense of [46] are a modification of topological field theories where the source category is replaced by a category with smooth cycles as objects and chains as morphisms. Chain field theories are closely related to differential characters.…”

“…By the smoothness condition (123), this yields a differential character in H n+1 (X; Z) with curvature ω. Moreover, chain field theories are classified up to equivalence by the differential characters obtained in this manner, see [46,Thm. 2.1].…”

“…By [13, p. 434], chain field theories in the sense of [46] generalize thin invariant field theories in the sense of [13]. By [13,Thm.…”