Field Theory Equivalences as Spans of L∞-algebras

Mehran Jalali Farahani, Christian Saemann, Martin Wolf

Research output: Working paperPreprint


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 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 languageEnglish
Publication statusPublished - 9 May 2023


  • hep-th
  • math-ph
  • math.MP

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