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The geometry of loop spaces. II: Characteristic classes. (English) Zbl 1350.58011

The paper is a continuation of [the authors, Int. J. Math. 26, No. 4, Article ID 1540002, 26 p. (2015; Zbl 1327.58027)]. The two parts study the geometry of loop spaces by defining Chern-Weil type invariants using Wodzicki residues, a natural choice of trace on the Lie algebra of the structure group. For a bundle \(\mathcal{E}\to M\) of Sobolev spaces (on some auxiliary closed manifold \(N\)) with structure group \(\Psi\mathrm{DO}^*_0(N)\) (the pseudo-differential operators of order \(0\) on \(N\)), the authors define Wodzicki-Chern-Weil type and Wodzicki-Chern-Simons type invariants of connections on \(\mathcal{E}\). The Wodzicki-Chern-Weil invariants often vanish, e.g. if the structure group reduces to the gauge group. However, the authors find interesting cases where the Wodzicki-Chern-Simons invariants are non-zero. These examples are used to prove that \(\pi_1(\mathrm{Diff}(M))\) is infinite for a large class of \(5\)-manifolds \(M\).

MSC:

58J40 Pseudodifferential and Fourier integral operators on manifolds
58J28 Eta-invariants, Chern-Simons invariants
55Q52 Homotopy groups of special spaces

Citations:

Zbl 1327.58027
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References:

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