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Gerbes and homotopy quantum field theories. (English) Zbl 1054.57034

A line bundle over a base space \(X\) with connection can be regarded as a functor on the \(0+1\) dimensional cobordism category of \(X\) [G. Segal, Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 359, No. 1784, 1389–1398 (2001; Zbl 1041.81094)]. Its holonomy can be thought of as a multiplicative \(\mathbb{C}^\times\)-valued function \(f\) of closed \(1\)-dimensional manifolds in \(X\) together with a closed \(2\)-form \(c\) such that \(f(\partial S)=\exp (\int_S c)\) for all surfaces \(S\) in \(X\). Such pairs \((S,c)\) form the Cheeger-Simons group of differential characters. Degree 3 cohomology classes are realized by gerbes [J.-L. Brylinski, Loop spaces, characteristic classes and geometric quantization (Progress in Mathematics (Boston, Mass.) 107) (1993; Zbl 0823.55002)] in the same way: A gerbe with connection is a multiplicative \(\mathbb{C}^\times\)-valued function \(f\) of closed \(2\)-dimensional manifolds in \(X\) together with a closed \(3\)-form \(c\) for which the above formula holds. If \(c\) vanishes the gerbe is called flat.
The paper establishes an equivalence between the group of gerbes with connections and the group of thin-invariant field theories. Here, two manifolds are called thin cobordant if there is cobordism of \(0\)-volume in \(X\). Thin-invariant field theories are smooth symmetric monoidal functors from the thin-cobordism category to the category of one-dimensional vector spaces. Moreover, when restricted to the path category of the free loop space the authors prove that each theory gives a line bundle with connection on \(LX\). Flat gerbes correspond to normalized rank one homotopy quantum field theories, even if considered over arbitrary fields.
All correspondences are explicitly given and a number of examples for thin-invariant field theories are provided. The paper is very much self contained and recommended to everyone who wants to view a gerbe as a functor on a cobordism category.

MSC:

57R56 Topological quantum field theories (aspects of differential topology)
81T70 Quantization in field theory; cohomological methods
55P48 Loop space machines and operads in algebraic topology
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