Holography as Homotopy

TL;DR

This paper interprets holography and the AdS/CFT correspondence using homotopy algebras, showing how bulk theories relate to boundary theories through homotopy transfer of cyclic $L_{ olinebreak}_{ olinebreak}infty$ algebras, providing a new algebraic perspective.

Contribution

It introduces a novel algebraic framework for holography by connecting bulk and boundary theories via homotopy transfer of cyclic $L_{ olinebreak}_{ olinebreak}infty$ algebras, extending the dictionary to manifolds with boundaries.

Findings

Homotopy transfer relates bulk and boundary $L_{ olinebreak}_{ olinebreak}infty$ algebras.

Boundary actions correspond to on-shell bulk actions via homotopy equivalence.

New techniques for homotopy transfer of cyclic $L_{ olinebreak}_{ olinebreak}infty$ algebras are developed.

Abstract

We give an interpretation of holography in the form of the AdS/CFT correspondence in terms of homotopy algebras. A field theory such as a bulk gravity theory can be viewed as a homotopy Lie or $L_{\infty}$ algebra. We extend this dictionary to theories defined on manifolds with a boundary, including the conformal boundary of AdS, taking into account the cyclic structure needed to define an action with the correct boundary terms. Projecting fields to their boundary values then defines a homotopy retract, which in turn implies that the cyclic $L_{\infty}$ algebra of the bulk theory is equivalent, up to homotopy, to a cyclic $L_{\infty}$ algebra on the boundary. The resulting action is the `on-shell action' conventionally computed via Witten diagrams that, according to AdS/CFT, yields the generating functional for the correlation functions of the dual CFT. These results are established with the help of new techniques regarding the homotopy transfer of cyclic $L_{\infty}$ algebras.