Field Theory Equivalences as Spans of $L_\infty$-algebras
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866917195210031104 |
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| author | Farahani, Mehran Jalali Saemann, Christian Wolf, Martin |
| author_facet | Farahani, Mehran Jalali Saemann, Christian Wolf, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_05473 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Field Theory Equivalences as Spans of $L_\infty$-algebras Farahani, Mehran Jalali Saemann, Christian Wolf, Martin High Energy Physics - Theory Mathematical Physics 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. |
| title | Field Theory Equivalences as Spans of $L_\infty$-algebras |
| topic | High Energy Physics - Theory Mathematical Physics |
| url | https://arxiv.org/abs/2305.05473 |