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A functorial approach to differential characters. (English) Zbl 1058.57024
This paper describes Cheeger-Simons differential characters [ J. Cheeger and J. Simons, Lect. Notes Math. 1167, 50–80 (1985; Zbl 0621.57010)] in terms of a variant of Turaev’s homotopy quantum field theories, based on chains in a smooth manifold \(X\).
To be more precise, an homotopy quantum field theory on \(X\) may be thought of as a bundle on the space of all \(n\)-manifolds equipped with a map to \(X\), together with a generalized connection where parallel transport is defined across \((n+1)\)-cobordism. This makes a remarkable difference with differential characters, which are defined in terms of homological information rather than bordism. In order to remedy this discrepancy, the author introduces an homological variant of Turaev’s construction: chain field theories.
Given a smooth manifold \(X\), the \((n+1)\)-dimensional chain category of \(X\) is the category \({\mathcal G}_{n+1}X\) whose objects are smooth \(n\)-cycles in \(X\) and whose morphisms are defined by \({\mathcal G}_{n+1}(\gamma_1,\gamma_2)=\{(n+1)\)-chains \(\sigma\) such that \(\partial\sigma=\gamma_2-\gamma_1\}\). Then, a chain field theory on \(X\) is defined as a functorial version of the definition of differential characters. Namely, an \((n+1)\)-dimensional chain field theory on \(X\) is defined as a symmetric monoidal functor \(E\) from \({\mathcal G}_{n+1}\) to the symmetric monoidal category of 1-dimensional vector spaces over \({\mathbb C}\), together with a closed differential \((n+2)\)-form \(c\) over \(X\) such that for any \((n+2)\)-chain \(\beta\) the following holds: \[ E(\partial\beta)(1)=\exp \left(2\pi i\int_\beta c\right). \] It can be thought of as a line bundle over the space of \(n\)-cycles of \(X\) with parallel transport defined across \((n+1)\)-chains. The tensor product in the category of 1-dimensional \({\mathbb C}\)-vector spaces induces a group operation on the set of isomorphism classes of \((n+1)\)-dimensional chain field theories. This group is denoted \(\operatorname{Ch}FT^{n+1}(X)\) and it is shown to be isomorphic (via holonomy) to the Cheeger-Simons group of \((n+1)\)-dimensional differential characters; in other words, the equivalence classes of \((n+1)\)-dimensional chain field theories are classified by the Cheeger-Simons group \(\widehat{H}^{n+1}(X)\).
Finally, a chain field theory \(E\) is said to be flat if the differential form \(c\) is zero. This is equivalent to \(E\) being invariant under chain deformations, i.e., \(E(\sigma)=E(\sigma+\partial \beta)\). Flat \((n+1)\)-dimensional chain field theories are classified by the group \(H^{n+1}(X,U(1))\).

MSC:
57R56 Topological quantum field theories (aspects of differential topology)
53C05 Connections (general theory)
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
58A10 Differential forms in global analysis
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