### Abstract

A Q-manifold M is a supermanifold endowed with an odd vector field Q squaring to zero. The Lie derivative L_{Q} 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 L_{Q} 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.

Original language | English |
---|---|

Pages (from-to) | 729-759 |

Number of pages | 31 |

Journal | Journal of Geometry and Physics |

Volume | 60 |

Issue number | 5 |

DOIs | |

Publication status | Published - May 2010 |

Externally published | Yes |

### Fingerprint

### Keywords

- Characteristic classes
- Gauge theories
- Q-manifolds

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Geometry and Topology

### Cite this

*Journal of Geometry and Physics*,

*60*(5), 729-759. https://doi.org/10.1016/j.geomphys.2010.01.008

**Characteristic classes of Q-manifolds : Classification and applications.** / Lyakhovich, S. L.; Mosman, E. A.; Sharapov, A. A.

Research output: Contribution to journal › Article

*Journal of Geometry and Physics*, vol. 60, no. 5, pp. 729-759. https://doi.org/10.1016/j.geomphys.2010.01.008

}

TY - JOUR

T1 - Characteristic classes of Q-manifolds

T2 - Classification and applications

AU - Lyakhovich, S. L.

AU - Mosman, E. A.

AU - Sharapov, A. A.

PY - 2010/5

Y1 - 2010/5

N2 - 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.

AB - 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.

KW - Characteristic classes

KW - Gauge theories

KW - Q-manifolds

UR - http://www.scopus.com/inward/record.url?scp=77949913711&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77949913711&partnerID=8YFLogxK

U2 - 10.1016/j.geomphys.2010.01.008

DO - 10.1016/j.geomphys.2010.01.008

M3 - Article

VL - 60

SP - 729

EP - 759

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

IS - 5

ER -