Field Theory Equivalences as Spans of $L_{\mathrm{\infty }}$-algebras

Semi-classically equivalent field theories are related by a quasi-isomorphism between their underlying $L_{\mathrm{\infty }}$-algebras, but such a quasi-isomorphism is not necessarily a homotopy transfer. We demonstrate that all quasi-isomorphisms can be lifted to spans of $L_{\mathrm{\infty }}$-algebras in which the quasi-isomorphic $L_{\mathrm{\infty }}$-algebras are obtained from a correspondence $L_{\mathrm{\infty }}$-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 $L_{\mathrm{\infty }}$-algebras allow for a clean definition of quasi-isomorphisms of cyclic $L_{\mathrm{\infty }}$-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 $L_{\mathrm{\infty }}$-algebras, and we provide full details in the case of the principal chiral model. We also present the relevant span of $L_{\mathrm{\infty }}$-algebras for the Penrose-Ward transform in the context of self-dual Yang-Mills theory and Bogomolny monopoles.