## Abstract

In this paper we describe a construction which produces classes in compactifications of the moduli space of curves. This construction extends a construction of Kontsevich which produces classes in the open moduli space from the initial data of a cyclic A_{∞}-algebra. The initial data for our construction are what we call a 'quantum A_{∞}-algebra', which arises as a type of deformation of a cyclic A_{∞}-algebra. The deformation theory for these structures is described explicitly. We construct a family of examples of quantum A_{∞}-algebras which extend a family of cyclic A_{∞}-algebras, introduced by Kontsevich, which are known to produce all the kappa classes using his construction.

Original language | English |
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Pages (from-to) | 115-130 |

Number of pages | 16 |

Journal | Letters in Mathematical Physics |

Volume | 89 |

Issue number | 2 |

DOIs | |

State | Published - Aug 2009 |

## Keywords

- A-algebra
- Deformation theory
- Maurer-Cartan set
- Moduli space of curves
- Noncommutative geometry