## Abstract

Semi-classically equivalent field theories are related by a quasi-isomorphism between their underlying

*L*_{∞}-algebras, but such a quasi-isomorphism is not necessarily a homotopy transfer. We demonstrate that all quasi-isomorphisms can be lifted to spans of*L*_{∞}-algebras in which the quasi-isomorphic*L*_{∞}-algebras are obtained from a correspondence*L*_{∞}-algebra by a homotopy transfer. Our construction is very useful: homotopy transfer is computationally tractable, and physically, it amounts to integrating out fields in a Feynman diagram expansion. Spans of*L*_{∞}-algebras allow for a clean definition of quasi-isomorphisms of cyclic L∞ -algebras. Furthermore, they appear naturally in many contexts within physics. As examples, we first consider scalar field theory with interaction vertices blown up in different ways. We then show that (non-Abelian) T-duality can be seen as a span of*L*_{∞}-algebras, and we provide full details in the case of the principal chiral model. We also present the relevant span of*L*_{∞}-algebras for the Penrose–Ward transform in the context of self-dual Yang–Mills theory and Bogomolny monopoles.Original language | English |
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Article number | 285208 |

Journal | Journal of Physics A: Mathematical and Theoretical |

Volume | 57 |

Issue number | 28 |

Early online date | 2 Jul 2024 |

DOIs | |

Publication status | Published - 12 Jul 2024 |

## Keywords

- T-duality
- semi-classical equivalence
- homotopy algebras
- homotopy transfer
- perturbative quantum field theory

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