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Operadic control of OPE via an L-infinity algebra and d-algebra structures

Establish that (a) the complex Ω^d(H)=H⊗∧^d(R^d) carries a natural L-infinity algebra structure; (b) there exists an L-infinity morphism from Ω^d(H) to the L-infinity algebra controlling formal deformations of H as an algebra over the cooperad built from Fulton–Macpherson compactifications; and (c) the graded complex Ω^(•)(H) has a natural d-algebra structure compatible with these operations.

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Background

The OPE near all strata of Fulton–Macpherson compactifications suggests an operadic framework. The cooperad built from asymptotics on FM(Rd) is expected to govern OPE deformations.

The conjecture posits precise homotopy-algebraic structures: an L∞-algebra on Ωd(H), a controlling L∞-morphism, and a d-algebra structure on Ω•(H), aligning deformation theory of QFTs with operadic methods.

References

Conjecture \label{conjecture about controlling algebra} Let $$ be the $L_{\infty}$-algebra which controls formal deformations of $H$ as an ${\cal A}$-algebra. Then:

a) There is a structure of $L_{\infty}$-algebra on $\Omegad(H)=H\otimes\wedged({}d).$

b) There exists an $L_{\infty}$-morphism $\Omegad(H)\to.$

c) There is a structure of a $d$-algebra (see [KoSo2]) on $\Omega{\bullet}(H)$.

— Moduli space of Conformal Field Theories and non-commutative Riemannian geometry (2506.00896 - Soibelman, 1 Jun 2025) in Section 6.4, The operad controlling the OPE (Conjecture)