2009/06/30 by S. L. Lyakhovich, E. A. Mosman, Elena A. Mosman +2
Mathematics · Physics and Astronomy · #Advanced Topics in Algebra #Algebra over a field #Black Holes and Theoretical Physics #Cohomology #Connection (principal bundle) #Covariant derivative #Covariant transformation #Differential geometry #Differential operator #Geometry #Homotopy and Cohomology in Algebraic Topology #Mathematics #Pure mathematics #hep-th #math-ph #math.DG #math.MP #msc:58Z05
paper · pdf · doi:10.1016/j.geomphys.2010.01.008
42 pages, journal version
openalex publication_date 2010/02/03 · arxiv created 2010/03/02 · arxiv updated 2015/05/13 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
A Q-manifold M is a supermanifold endowed with an odd vector field Q squaring to zero. The Lie derivative LQ along Q makes the algebra of smooth tensor fields on M into a differential algebra. In this paper, we define and study the invariants of Q-manifolds called characteristic classes. These take values in the cohomology of the operator LQ and, given an affine symmetric connection with curvature R, can be represented by universal tensor polynomials in the repeated covariant derivatives of Q and R up to some finite order. As usual, the characteristic classes are proved to be independent of the choice of the affine connection used to define them. The main result of the paper is a complete classification of the intrinsic characteristic classes, which, by definition, do not vanish identically on flat Q-manifolds. As an illustration of the general theory we interpret some of the intrinsic characteristic classes as anomalies in the BV and BFV-BRST quantization methods of gauge theories. An application to the theory of (singular) foliations is also discussed.