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Kotov, Alexei; Strobl, Thomas

Characteristic classes associated to \(Q\)-bundles. (English) Zbl 1311.58002

Int. J. Geom. Methods Mod. Phys. 12, No. 1, Article ID 1550006, 26 p. (2015).
Summary: A \(Q\)-manifold is a graded manifold endowed with a vector field of degree 1 squaring to zero. We consider the notion of a \(Q\)-bundle, that is, a fiber bundle in the category of \(Q\)-manifolds. To each homotopy class of “gauge fields” (sections in the category of graded manifolds) and each cohomology class of a certain subcomplex of forms on the fiber we associate a cohomology class on the base. As any principal bundle yields canonically a Q-bundle, this construction generalizes Chern-Weil classes. Novel examples include cohomology classes that are locally de Rham differential of the integrands of topological sigma models obtained by the AKSZ-formalism in arbitrary dimensions. For Hamiltonian Poisson fibrations one obtains a characteristic 3-class in this manner. We also relate the framework to equivariant cohomology and Lecomte’s characteristic classes of exact sequences of Lie algebras.

Cited in 55 Documents

MSC:

58A50 Supermanifolds and graded manifolds
55R10 Fiber bundles in algebraic topology
57R20 Characteristic classes and numbers in differential topology
81T13 Yang-Mills and other gauge theories in quantum field theory
81T45 Topological field theories in quantum mechanics

Keywords:

\(Q\)-manifolds; gauge theories; characteristic classes
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