Higgs Branches as Associated Varieties of Vertex Algebras

Definitions and mathematical statement

Let $\mathcal T$ denote a four-dimensional $\mathcal N=2$ Argyres--Douglas theory, let $\mathcal H(\mathcal T)$ denote its Higgs branch, and let $V(\mathcal T)$ denote the vertex algebra assigned to it by the four-dimensional/two-dimensional correspondence. The supplied record does not specify the class of theories, a mathematical construction of their Higgs branches, or the precise correspondence.

For a vertex algebra $V$ over ${\mathbb{C}}$, write its fields as $Y(a,z)=\sum_{n\in{\mathbb{Z}}}a_{(n)}z^{-n-1}$, and define \[ C_2(V)=\operatorname{span}_{{\mathbb{C}}}\{a_{(-2)}b:a,b\in V\},\qquad R_V=V/C_2(V). \] The quotient has the commutative product $[a][b]=[a_{(-1)}b]$ and Poisson bracket $\{[a],[b]\}=[a_{(0)}b]$. Under the reduced affine convention, its associated variety is \[ X_V=\operatorname{Spec}\left(R_V/\sqrt{(0)}\right). \] This convention uses complex reduced affine spaces; finite generation is needed to regard them as affine algebraic varieties in the usual finite-type sense. The source does not specify reduced versus scheme-theoretic conventions, finite-generation hypotheses, or the structure to be preserved by the proposed isomorphism.

Intended conjectural comparison, subject to unresolved qualifications. For every theory $\mathcal T$ in the intended, unspecified class of four-dimensional $\mathcal N=2$ Argyres--Douglas theories, the proposed correspondence is \[ \mathcal H(\mathcal T)\cong X_{V(\mathcal T)}. \] This is a formulation template under the conventions above, not a fully specified universally quantified conjecture: the range of $\mathcal T$, the construction of both sides, and the category of isomorphism require clarification.

The record gives typical corresponding vertex algebras of the forms \[ L_k(\mathfrak g),\qquad W_k(\mathfrak g,f),\qquad k=-h^\vee+\frac{h^\vee}{q}, \] where $\mathfrak g$ has type $A$, $D$, or $E$, and $h^\vee$ denotes its dual Coxeter number. In the conventional complex Lie-algebra interpretation, $\mathfrak g$ is finite-dimensional and simple, $L_k(\mathfrak g)$ is the simple affine vertex algebra at level $k$, and a quantum Drinfeld--Sokolov $W$-algebra uses a nilpotent element $f\in\mathfrak g$. These conventions do not supply the missing hypotheses: the source gives neither the domain and restrictions on $q$ (the displayed expression requires $q\ne0$), nor the allowed nilpotent orbits of $f$, nor the simple-versus-universal convention for $W_k(\mathfrak g,f)$, nor an assignment of these algebras to particular theories. The examples therefore do not assert the comparison for every choice of these parameters.

The supplied literature assessment expressly describes the extraction as an incomplete fragment ending at a conjecture and says that no precise mathematical claim could be verified. That qualification remains unresolved despite the registry's separate corrected_verified statement label.

Short English statement

The intended conjecture identifies the Higgs branch of a four-dimensional $\mathcal N=2$ Argyres--Douglas theory with the associated variety of its corresponding vertex algebra. The supplied fragment leaves the family of theories, parameter restrictions, and meaning of isomorphism unspecified, so this is a qualified statement-only draft requiring mathematical review. The registry labels the problem open; that label is supplied curation, not an independently verified status.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json; permanent ID $30004020$, title Higgs Branches as Associated Varieties of Vertex Algebras, exact source code catalogue entry. The supplied record belongs to the Oberwolfach Reports Open Problems collection. No individual proposer is supplied; the proposed year is recorded as 2018.
• Supplied report citation: Enveloping Algebras and Geometric Representation Theory (2019), Oberwolfach Reports, pp. 3081--3146, DOI 10.4171/owr/2018/52. Supplied source URL and literature-source link, labeled "Oberwolfach report 52": https://doi.org/10.4171/owr/2018/52. This is report-level provenance; no precise conjecture page, speaker, or passage is supplied.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open, describes the extracted text as an incomplete fragment ending at a conjecture, and reports that no authoritative proof or counterexample for the exact fragment was identified. This is supplied dated curation, not independently verified literature review or source consultation. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports that mathematical content was preserved while notation, typography, duplicate clauses, or omitted definitions were repaired. This supplied editorial assertion does not resolve the accompanying warning that the fragment lacks a verifiable precise claim.