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Relationship between r‑twisted BD algebras and BD_d algebras

Determine the precise relationship, in general, between the r‑twisted Beilinson–Drinfeld algebras introduced in this paper and the BD_d algebras of Calaque–Pantev–Toën–Vaquié–Vezzosi (CPTVV17), beyond the special case r = 1 where they coincide with BD_0.

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Background

The paper works with a variant of Beilinson–Drinfeld algebras (called r‑twisted BD algebras) tailored to the contexts of shifted Poisson and BV structures arising from graph complexes and (quantized) cyclic A∞/L∞ settings.

CPTVV17 introduces BD_d algebras in a different framework. The authors explicitly state that, aside from the case r=1 (which matches BD_0), they do not know the general relationship between their notion and that of CPTVV17.

References

In , section 3.5.1, the notion of a $BD_d$ algebra is introduced. It is not clear to the author what is the relationship of this notion and ours, in general. However, for $r=1$ an $r$-twisted BD algebra is the same as a $BD_0$ algebra.

— Kontsevich's Cocycle Construction and Quantization of the Loday-Quillen-Tsygan Theorem (2506.15210 - Ulmer, 18 Jun 2025) in Section 2 (Notation and Preliminaries), remark after the definition of r‑twisted BD algebras