Heisenberg Actions on Hochschild–Serre Cohomology

Definitions and mathematical statement

For an object $S$ for which the relevant products and quotient stacks are defined, the symmetric group $\mathfrak S_n$ acts on $S^n$ by permuting the factors. The corresponding symmetric quotient stack is \[ \mathcal X_n=[S^n/\mathfrak S_n]. \] This is a conventional interpretation of "symmetric quotient stacks"; the supplied record does not specify the permitted objects $S$, the base field, or the range of $n$.

An ordinary Heisenberg Lie algebra associated with a vector space $V$ over a field $K$ and an alternating bilinear form $\omega$ is the central extension \[ \mathfrak h(V,\omega)=V\oplus Kc,\qquad [v,w]=\omega(v,w)c,\qquad [c,v]=0. \] An action on a vector space $M$ means a Lie algebra homomorphism $\mathfrak h(V,\omega)\to\operatorname{End}_K(M)$, where the target has the commutator bracket. These definitions explain ordinary Heisenberg terminology only: the source does not identify $K$, $V$, $\omega$, the central action, or whether a graded or super version is intended.

Source question. Is there a Heisenberg algebra controlling the Hochschild--Serre cohomology of symmetric quotient stacks?

Unresolved formulation. The supplied record does not define "Hochschild--Serre cohomology" in this setting or the requirement expressed by "controlling." In particular, it does not specify whether the intended structure is an action on the cohomology of each $\mathcal X_n$, an action on a sum over different $n$, or a stronger description by a Fock-space representation. Consequently, the question above is the preserved research target, but the evidence does not determine a fully quantified existence statement or prescribed commutation relations.

Retained contextual qualifications. The appended source excerpt concerns a space denoted $S$, its Hilbert schemes written $\operatorname{Hilb}^n S$, negative Hochschild--Serre cohomology of $S$, and a calculation of $\mathrm{HH}^{2}(\operatorname{Hilb}^n S)$. It says that, to prove an unavailable assertion "(2)," one takes $k=0$, uses all negative Hochschild--Serre cohomology of $S$, and subsequently takes $\ast=2$. It further describes extracting geometric deformations in a Hochschild--Kostant--Rosenberg decomposition by bootstrapping from earlier results and cancelling contributions in the Hochschild--Serre calculation. However, the component notation is damaged: it contains V2, H2 (Hilbn S, OHilbn S ), and H2 (Hilbn S, THilbn S ). The definitions of $k$, the grading denoted by $\ast$, assertion (2), the earlier results, the component notation, and the hypotheses on $S$ are absent. These fragments therefore supply context without establishing additional hypotheses or a precise auxiliary assertion.

Short English statement

The question asks whether a suitable Heisenberg algebra governs Hochschild--Serre cohomology for symmetric quotient stacks. The supplied evidence leaves the cohomology theory, geometric hypotheses, representation space, and meaning of "controlling" unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005909$, title Heisenberg Actions on Hochschild--Serre Cohomology, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record supplies the year 2024 and no named proposer. Its status is labelled open; its statement status is labelled corrected_verified, with the supplied qualification that only formula delimiters, notation, grammar, and typography were repaired and that mathematical content and scope were preserved. These are registry labels, not independent verification here.
• Supplied report citation: Hochschild (Co)Homology and Applications (2024), Oberwolfach Reports, pp. 1097--1162, DOI 10.4171/owr/2024/20. https://doi.org/10.4171/owr/2024/20. The record describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. No precise page, speaker attribution, or complete surrounding mathematical setup is supplied for this question.
• Belmans, Fu and Krug, Hochschild cohomology of Hilbert schemes. Supplied literature reference: https://arxiv.org/abs/2309.06244.
• Anno, Baranovsky and Logvinenko, Hochschild homology. Supplied literature reference: https://arxiv.org/abs/2512.25039.
• Supplied literature assessment, checked on exactly 2026-08-22: the record labels the problem open and states that Heisenberg/Fock-space structures are known for related Grothendieck and Hochschild-homology constructions, while the requested structure for Hochschild--Serre cohomology is not established. This is supplied dated curation derived from the OWR curation, not an independently verified literature assessment. The record explicitly calls for independent verification of time-sensitive status and cited claims; no source visit or such verification is claimed here.