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

Abstract

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

arXiv comment49 pages, v3: fixed a gap in the proof of Proposition 2.6, all other results unchanged

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