## Abstract

Given an odd symmetry acting on an associative algebra, I show that the summation over arbitrary ribbon graphs gives the construction of the solutions to the noncommutative Batalin-Vilkovisky equation, introduced in (Barannikov in IMRN, rnm075, 2007), and to the equivariant version of this equation. This generalizes the known construction of A_{∞}-algebra via summation over ribbon trees. I give also the generalizations to other types of algebras and graph complexes, including the stable ribbon graph complex. These solutions to the noncommutative Batalin-Vilkovisky equation and to its equivariant counterpart, provide naturally the supersymmetric matrix action functionals, which are the gl(N)-equivariantly closed differential forms on the matrix spaces, as in (Barannikov in Comptes Rendus Mathematique vol 348, pp. 359-362.

Original language | English |
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Pages (from-to) | 605-628 |

Number of pages | 24 |

Journal | Letters in Mathematical Physics |

Volume | 103 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jun 2013 |

Externally published | Yes |

## Keywords

- A-algebras
- B-model
- Batalin-Vilkovisky formalism
- cyclic homology
- Feynman diagrams
- matrix integrals
- mirror symmetry
- noncommutative algebraic geometry
- ribbon graphs