Field theory equivalences as spans of L∞ -algebras

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 57; no. 28; pp. 285208 - 285249
• Main Authors: Jalali Farahani, Mehran, Saemann, Christian, Wolf, Martin
• Format: Journal Article
• Language: English
• Published: IOP Publishing 12.07.2024
• Subjects: homotopy algebras; homotopy transfer; perturbative quantum field theory; semi-classical equivalence; T-duality
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/ad5521

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.