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

Mehran Jalali Farahani, Christian Saemann, Martin Wolf

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.

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

Abstract

Semi-classically equivalent field theories are related by a quasi-isomorphism between their underlying -algebras, but such a quasi-isomorphism is not necessarily a homotopy transfer. We demonstrate that all quasi-isomorphisms can be lifted to spans of -algebras in which the quasi-isomorphic -algebras are obtained from a correspondence -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 -algebras allow for a clean definition of quasi-isomorphisms of cyclic -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 -algebras, and we provide full details in the case of the principal chiral model. We also present the relevant span of -algebras for the Penrose-Ward transform in the context of self-dual Yang-Mills theory and Bogomolny monopoles.
Paper Structure (47 sections, 5 theorems, 137 equations)

This paper contains 47 sections, 5 theorems, 137 equations.

Table of Contents

  1. Introduction
  2. Quasi-isomorphisms and homotopy transfer
  3. Quasi-isomorphisms and homotopy transfer for Linfty-algebras
  4. $L_\infty$-algebras.
  5. Morphisms of $L_\infty$-algebras.
  6. Homotopy transfer.
  7. Quasi-isomorphisms not originating from homotopy transfer.
  8. Minimal model.
  9. Composition of homotopy transfers.
  10. Spans of Linfty-algebras
  11. Pullbacks of $L_\infty$-algebras.
  12. Spans of $L_\infty$-algebras.
  13. Lifting of homotopy transfers.
  14. Composition of spans of $L_\infty$-algebras.
  15. Quasi-isomorphisms of cyclic $L_\infty$-algebras.
  16. ...and 32 more sections

Key Result

proposition 1

Given $L_\infty$-algebras $\frL^{(1,2,3)}$ together with projections and embeddings of the underlying cochain complexes \begin{tikzcd} \big(\frL^{(1)},\mu_1^{(1)}\big)\arrow[r,shift left]{}{\sfp^{(21)}} & \big(\frL^{(2)},\mu^{(2)}_1\big)\arrow[l,shift left]{}{\sfe^{(21)}}\arrow[r and two special deformation retracts then there is a third special deformation retract \begin{tikz

Theorems & Definitions (12)

  • proposition 1
  • proof
  • corollary 1
  • proof
  • remark 1
  • theorem 1
  • proposition 2
  • proof
  • proposition 3
  • proof
  • ...and 2 more