arXiv subjects

Thomas Creutzig

Publications and source records attributed to Thomas Creutzig.

At least 19 recordsLinked to original sources

arXiv2026

Feigin-Semikhatov duality at the critical level

The Feigin-Semikhatov duality asserts that the Heisenberg cosets of the subregular $W$-algebra of $\mathfrak{sl}_n$ at level $k$ and the one of the principal $W$-superalgebra of $\mathfrak{sl}_{n|1}$ at level $\ell$ coincide when the levels satisfy the Feigin-Frenkel relation $(k+n)(\ell+n-1)=1$. A similar duality holds between the subregular $W$-algebra of $\mathfrak{so}_{2n+1}$ and the principal $W$-superalgebra of $\mathfrak{osp}_{2|2n}$. We study these dualities in the critical/large level limit. We describe the centerless subregular $W$-algebra at the critical level as an orbifold of the large level limit of the principal $W$-superalgebra times a lattice VOA. Our construction yields a functor between certain categories of the two involved vertex algebras. We show that in this set-up one in fact gets block-wise equivalences of categories. Studying the principal block of the large level limit of the principal $W$-superalgebra then gives us the structure of the principal blocks of the subregular $W$-algebras in the category of weight modules (which is much larger than the more common category of lower bounded modules).

math.RT↗

arXiv2026

Universal $2$-parameter $\mathcal{N}=2$ supersymmetric $\mathcal{W}_{\infty}$-algebra

The universal $2$-parameter vertex algebra $\mathcal{W}_{\infty}$ of type $\mathcal{W}(2,3,\dots)$ is a classifying object for vertex algebras of type $\mathcal{W}(2,3,\dots,N)$ for some $N$; under mild hypotheses, all such vertex algebras arise as quotients of $\mathcal{W}_{\infty}$. In 2017, Gaiotto and Rap\v{c}\'ak introduced a family of such vertex algebras called $Y$-algebras, and conjectured that they fall into groups of three that are mutually isomorphic. This is a common generalization of both Feigin-Frenkel duality and the coset realization of principal $\mathcal{W}$-algebras in type $A$, and was proven in 2021 for the simple $Y$-algebras (i.e., one label is zero) by the first and third authors. In this paper, we extend this entire story to the $\mathcal{N}=2$ superconformal setting. First, we prove the 2013 conjecture of Gaberdiel and Candu that there exists a universal $2$-parameter vertex algebra $\mathcal{W}^{\mathcal{N}=2}_{\infty}$ which is an extension of the $\mathcal{N}=2$ superconformal algebra, and has four additional generators in weights $i, i + \frac{1}{2}, i + \frac{1}{2}, i+1$, for each integer $i > 1$. This admits many $1$-parameter quotients which we call $\mathcal{N}=2$ supersymmetric $Y$-algebras, and we prove the dualities among these algebras which were conjectured in 2018 by Prochazka and Rap\v{c}\'ak. A special case is the coset realization of the principal $\mathcal{W}$-algebra $\mathcal{W}^k(\mathfrak{sl}_{n+1|n})$ which was conjectured in 1992 by Ito. As a corollary, we obtain the strong rationality of $\mathcal{W}_k(\mathfrak{sl}_{n+1|n})$ for $k = -1 + \frac{1}{n+a+1}$ for all positive integers $n,a$, and we describe its module category. This generalizes Adamovi\'c's 1999 result on $\mathcal{N}=2$ minimal models, which is the case $n=1$.

math.RT↗

arXiv2026

W-algebras of the Deligne-Cvitanovi\'{c} Exceptional series and the minimal 3d ${\mathcal N}=4$ SCFT

We propose a three-dimensional field theory construction that realizes the vertex algebras associated with the intermediate Lie algebras and the related $C_2$-cofinite minimal $W$-algebras of the Deligne-Cvitanovi\'c (DC) series as boundary algebras. The construction is based on the minimal three-dimensional ${\mathcal N}=4$ superconformal field theory coupled to a topological field theory. For a Neumann-type boundary condition compatible with the topological $A$-twist, the algebra of boundary local operators realizes the minimal $W$-algebra $W_{-h^\vee/6}(\mathfrak{g},f_{\text{min}})$. While this boundary condition is not deformable to the $B$-twist, we argue that a holomorphic-topological ($HT^B$) twist instead realizes the level-one affine algebras of the intermediate Lie algebras, providing a uniform three-dimensional origin for these vertex algebra structures.

hep-th↗

arXiv2026

The Kazhdan-Lusztig category of W-algebras of simply-laced Lie algebras at irrational levels

Let $\mathfrak{g}$ be a simple, simply-laced Lie algebra and $f \in \mathfrak{g}$ nilpotent. The Kazhdan-Lusztig category of the W-algebra $W^\kappa(\mathfrak{g},f)$ associated with $(\mathfrak{g},f)$ at level $\kappa \in \mathbb{C}$ is obtained from the Kazhdan-Lusztig category of the affine vertex algebra $V^\kappa(\mathfrak{g})$ via the quantum Hamiltonian reduction associated with $f$. We show that this is a braided tensor equivalence for any $f$ and any irrational level $\kappa\in \mathbb{C} \backslash \mathbb{Q}$.

math.RT↗

arXiv2025

New universal vertex algebras as glueings of the basic ones

There are three universal $2$-parameter vertex algebras $\mathcal{W}_{\infty}$, $\mathcal{W}^{\text{ev}}_{\infty}$, and $\mathcal{W}^{\mathfrak{sp}}_{\infty}$ which are freely generated of types $\mathcal{W}(2,3,4,\dots)$, $\mathcal{W}(2,4,6,\dots)$, and $\mathcal{W}(1^3, 2, 3^3, 4,\dots)$, respectively. They serve as classifying objects for vertex algebras with these generating types satisfying mild hypotheses. Their $1$-parameter quotients are expected to be the building blocks of all $\mathcal{W}$-algebras of classical Lie types. Furthermore, such $\mathcal{W}$-algebras are expected to be organized into families that are governed by new universal $2$-parameter vertex algebras, which are themselves glueings of copies of $\mathcal{W}_{\infty}$ in type $A$ (together with a Heisenberg algebra), and copies of $\mathcal{W}^{\text{ev}}_{\infty}$ and $\mathcal{W}^{\mathfrak{sp}}_{\infty}$ in types $B$, $C$, and $D$. We denote these universal objects by $\mathcal{W}^{X,S,M}_{\infty}$, where $X$ denotes the Lie type (either $A$, $C$, or $BD$ since types $B$ and $D$ can be treated uniformly), and $S$, $M$ are sets of positive integers that determine certain families of partitions. More precisely, for a partition $P = (n_0^{m_0}, n_1^{m_1},\dots, n_{t}^{m_t})$ of $N = \sum_{i=0}^t n_i m_i$ consisting of $m_i$ parts of size $n_i$, where $n_0> n_1 > \cdots > n_t \geq 2$, $M = \{m_0,\dots, m_t\}$ is the set of multiplicities, and $S = \{d_1,\dots, d_t\}$ is the set of height differences $d_{i+1} = n_i - n_{i+1}$. After introducing this general conjectural picture, we will construct the first nontrivial example $\mathcal{W}^{\mathfrak{so}_2}_{\infty}:=\mathcal{W}^{BD, \emptyset, \{2\}}_{\infty}$, which is a glueing of two copies of $\mathcal{W}^{\text{ev}}_{\infty}$.

math.QA↗

arXiv2025

W-algebras as conformal extensions of affine VOAs

We provide a criterion for a vertex operator superalgebra homomorphism from an affine vertex algebra to another vertex superalgebra to be conformal, and an additional criterion that guarantees that this homomorphism is surjective. This situation is applied to W-algebras and W-superalgebras and we list all cases where our criterion applies. This gives many new examples of W-algebras that collapse to affine vertex algebras or are conformal extensions. In particular, we provide many examples of simple W-algebras at non-admissible levels that collapse to admissible level affine vertex algebras.

math.QA↗

arXiv2025

Minimal W-algebras of $\mathfrak{so}_N$ at level minus one

For $N \in\mathbb Z_{\geq 7}$ we show that the simple minimal $\mathcal{W}$-algebra of $\mathfrak{so}_N$ at level minus one is isomorphic to the even subalgebra of the tensor product of the simple affine vertex superalgebra of $\mathfrak{osp}_{1|2}$ at level $\frac{N-6}{2}$ with $N-4$ free fermions. In particular when $N$ is even this minimal $\mathcal{W}$-algebra is strongly rational as conjectured by Arakawa-Moreau.

math.RT↗

arXiv2025

A Conjecture of Warnaar-Zudilin from Deformations of Lie Superalgebras

We prove a collection of $q$-series identities conjectured by Warnaar and Zudilin and appearing in recent work with H. Kim in the context of superconformal field theory. Our proof utilizes a deformation of the simple affine vertex operator superalgebra $L_k(\mathfrak{osp}_{1|2n})$ into the principal subsuperspace of $L_k(\mathfrak{sl}_{1|2n+1})$ in a manner analogous to earlier work of Feigin-Stoyanovsky. This result fills a gap left by Stoyanovsky, showing that for all positive integers $N$, $k$ the character of the principal subspace of type $A_N$ at level $k$ can be identified with the (super)character of a simple affine vertex operator (super)algebra at the same level.

math.QA↗

arXiv2024

$N=1$ super Virasoro tensor categories

We show that the category of $C_1$-cofinite modules for the universal $N=1$ super Virasoro vertex operator superalgebra $\mathcal{S}(c,0)$ at any central charge $c$ is locally finite and admits the vertex algebraic braided tensor category structure of Huang-Lepowsky-Zhang. For central charges $c^{\mathfrak{ns}}(t)=\frac{15}{2}-3(t+t^{-1})$ with $t\notin\mathbb{Q}$, we show that this tensor category is semisimple, rigid, and slightly degenerate, and we determine its fusion rules. For central charge $c^{\mathfrak{ns}}(1)=\frac{3}{2}$, we show that this tensor category is rigid and that its simple modules have the same fusion rules as $\mathrm{Rep}\,\mathfrak{osp}(1\vert 2)$, in agreement with earlier fusion rule calculations of Milas. Finally, for the remaining central charges $c^{\mathfrak{ns}}(t)$ with $t\in \mathbb{Q}^\times$, we show that the simple $\mathcal{S}(c^{\mathfrak{ns}}(t),0)$-module $\mathcal{S}_{2,2}$ of lowest conformal weight $h^{\mathfrak{ns}}_{2,2}(t)=\frac{3(t-1)^2}{8t}$ is rigid and self-dual, except possibly when $t^{\pm 1}$ is a negative integer or when $c^{\mathfrak{ns}}(t)$ is the central charge of a rational $N=1$ superconformal minimal model. As $\mathcal{S}_{2,2}$ is expected to generate the category of $C_1$-cofinite $\mathcal{S}(c^{\mathfrak{ns}}(t),0)$-modules under fusion, rigidity of $\mathcal{S}_{2,2}$ is the first key step to proving rigidity of this category for general $t\in\mathbb{Q}^\times$.

math.QA↗

arXiv2024

Resolving Verlinde's formula of logarithmic CFT

Verlinde's formula for rational vertex operator algebras computes the fusion rules from the modular transformations of characters. In the non semisimple and non finite case, a logarithmic Verlinde formula has been proposed together with David Ridout. In this formula one replaces simple modules by their resolutions by standard modules. Here and under certain natural assumptions this conjecture is proven in generality. The result is illustrated in the examples of the singlet algebras and of the affine vertex algebra of $\mathfrak{sl}_2$ at any admissible level, i.e. in particular the Verlinde conjectures in these cases are true. In the latter case it is also explained how to compute the actual fusion rules from knowledge of the Grothendieck ring.

math.QA↗

arXiv2024

Ribbon categories of weight modules for affine $\mathfrak{sl}_2$ at admissible levels

We show that the braided tensor category of finitely-generated weight modules for the simple affine vertex operator algebra $L_k(\mathfrak{sl}_2)$ of $\mathfrak{sl}_2$ at any admissible level $k$ is rigid and hence a braided ribbon category. The proof uses a recent result of the first two authors with Shimizu and Yadav on embedding a braided Grothendieck-Verdier category $\mathcal{C}$ into the Drinfeld center of the category of modules for a suitable commutative algebra $A$ in $\mathcal{C}$, in situations where the braided tensor category of local $A$-modules is rigid. Here, the commutative algebra $A$ is Adamovi\'{c}'s inverse quantum Hamiltonian reduction of $L_k(\mathfrak{sl}_2)$, which is the simple rational Virasoro vertex operator algebra at central charge $1-\frac{6(k+1)^2}{k+2}$ tensored with a half-lattice conformal vertex algebra. As a corollary, we also show that the category of finitely-generated weight modules for the $N = 2$ super Virasoro vertex operator superalgebra at central charge $-6\ell-3$ is rigid for $\ell$ such that $(\ell+1)(k+2) = 1$.

math.QA↗

arXiv2024

Commutative algebras in Grothendieck-Verdier categories, rigidity, and vertex operator algebras

Let $A$ be a commutative algebra in a braided monoidal category $\mathcal{C}$; e.g., $A$ could be an extension of a vertex operator algebra (VOA) $V$ in a category $\mathcal{C}$ of $V$-modules. We study when the category $\mathcal{C}_A$ of $A$-modules in $\mathcal{C}$ and its subcategory $\mathcal{C}_A^{\text{loc}}$ of local modules inherit rigidity from $\mathcal{C}$, and then we find conditions for $\mathcal{C}$ and $\mathcal{C}_A$ to inherit rigidity from $\mathcal{C}_A^{\text{loc}}$. First, we assume $\mathcal{C}$ is a braided finite tensor category and prove rigidity of $\mathcal{C}_A$ and $\mathcal{C}_A^{\text{loc}}$ under conditions based on criteria of Etingof-Ostrik for $A$ to be an exact algebra in $\mathcal{C}$. As a corollary, we show that if $A$ is a simple $\mathbb{Z}_{\geq 0}$-graded VOA with a strongly rational vertex operator subalgebra $V$, then $A$ is strongly rational, without requiring the categorical dimension of $A$ as a $V$-module to be non-zero. Next, we assume $\mathcal{C}$ is a Grothendieck-Verdier category, i.e., $\mathcal{C}$ admits a weaker duality structure than rigidity. We first prove $\mathcal{C}_A$ is also a Grothendieck-Verdier category. Using this, we prove that if $\mathcal{C}_A^{\text{loc}}$ is rigid, then so is $\mathcal{C}$ under conditions such as a mild non-degeneracy assumption on $\mathcal{C}$, an assumption that every simple object of $\mathcal{C}_A$ is local, and that induction from $\mathcal{C}$ to $\mathcal{C}_A$ commutes with duality. These conditions are motivated by free field-like VOA extensions $V\subseteq A$ where $A$ is often an indecomposable $V$-module, so our result will make it more feasible to prove rigidity for many vertex algebraic monoidal categories. In a follow-up work, our result will be used to prove rigidity of the category of weight modules for the simple affine VOA of $\mathfrak{sl}_2$ at any admissible level.

math.QA↗

arXiv2024

Building blocks for $W$-algebras of classical types

The universal $2$-parameter vertex algebra $W_{\infty}$ of type $W(2,3,4,\dots)$ serves as a classifying object for vertex algebras of type $W(2,3,\dots,N)$ for some $N$ in the sense that under mild hypothesis, all such vertex algebras arise as quotients of $W_{\infty}$. There is an $\mathbb{N} \times \mathbb{N}$ family of such $1$-parameter vertex algebras which, after tensoring with a Heisenberg algebra, are known as $Y$-algebras. They were introduced by Gaiotto and Rap\v{c}\'ak and are expected to be the building blocks for all $W$-algebras in type $A$, i.e., every $W$-(super) algebra in type $A$ is an extension of a tensor product of finitely many $Y$-algebras. Similarly, the orthosymplectic $Y$-algebras are $1$-parameter quotients of a universal $2$-parameter vertex algebra $W^{\text{ev}}_{\infty}$ of type $W(2,4,6,\dots)$, which is a classifying object for vertex algebras of type $W(2,4,\dots, 2N)$ for some $N$. Unlike type $A$, these algebras are not all the building blocks for $W$-algebras of types $B$, $C$, and $D$. In this paper, we construct a new universal $2$-parameter vertex algebra of type $W(1^3, 2, 3^3, 4, 5^3,6,\dots)$ which we denote by $W^{\mathfrak{sp}}_{\infty}$ since it contains a copy of the affine vertex algebra $V^k(\mathfrak{sp}_2)$. We identify $8$ infinite families of $1$-parameter quotients of $W^{\mathfrak{sp}}_{\infty}$ which are analogues of the $Y$-algebras. We regard $W^{\mathfrak{sp}}_{\infty}$ as a fundamental object on equal footing with $W_{\infty}$ and $W^{\text{ev}}_{\infty}$, and we give some heuristic reasons for why we expect the $1$-parameter quotients of these three objects to be the building blocks for all $W$-algebras of classical types. Finally, we prove that $W^{\mathfrak{sp}}_{\infty}$ has many quotients which are strongly rational. This yields new examples of strongly rational $W$-superalgebras.

math.RT↗

arXiv2024

On lisse non-admissible minimal and principal W-algebras

We discuss a possible generalization of a result by the third-named author on the rationality of non-admissible minimal W-algebras. We then apply this generalization to finding rational non-admissible principal W-algebras.

math.RT↗

arXiv2024

Mirror Symmetry and Level-rank Duality for 3d $\mathcal{N} = 4$ Rank 0 SCFTs

We introduce a family of 3d $\mathcal{N} = 4$ superconformal field theories that have zero-dimensional Coulomb and Higgs branches and propose that the rational vertex operator algebras $W^{\text{min}}_{k - \scriptstyle{\frac{1}{2}}}(\mathfrak{sp}_{2N})$ and $L_{k}(\mathfrak{osp}_{1|2N})$ model the modular tensor categories of line operators in their topological $A$ and $B$ twists, respectively. Our analysis indicates that the action of 3d mirror symmetry on this family of theories is related to a novel level-rank duality and leads to several conjectural $q$-series identities of independent interest.

hep-th↗

arXiv2024

On the structure of W-algebras in type A

We formulate and prove examples of a conjecture which describes the W-algebras in type A as successive quantum Hamiltonian reductions of affine vertex algebras associated with several hook-type nilpotent orbits. This implies that the affine coset subalgebras of hook-type W-algebras are building blocks of the W-algebras in type A. In the rational case, it turns out that the building blocks for the simple quotients are provided by the minimal series of the regular W-algebras. In contrast, they are provided by singlet-type extensions of W-algebras at collapsing levels which are irrational. In the latter case, several new sporadic isomorphisms between different W-algebras are established.

math.RT↗

arXiv2024

Kazhdan-Lusztig Correspondence for Vertex Operator Superalgebras from Abelian Gauge Theories

We prove the Kazhdan-Lusztig correspondence for a class of vertex operator superalgebras which, via the work of Costello-Gaiotto, arise as boundary VOAs of topological B twist of 3d $\mathcal{N}=4$ abelian gauge theories. This means that we show equivalences of braided tensor categories of modules of certain affine vertex superalgebras and corresponding quantum supergroups. We build on the work of Creutzig-Lentner-Rupert to this large class of VOAs and extend it since in our case the categories don't have projective objectives and objects can have arbitrary Jordan H\"older length. Our correspondence significantly improves the understanding of the braided tensor category of line defects associated to this class of TQFT, by realizing line defects as modules of a Hopf algebra. In the process, we prove a case of the conjecture of Semikhatov-Tipunin, relating logarithmic CFTs to Nichols algebras of screening operators.

hep-th↗

arXiv2023

Whittaker vectors for $\mathcal{W}$-algebras from topological recursion

We identify Whittaker vectors for $\mathcal{W}_k(\mathfrak{g})$-modules with partition functions of higher Airy structures. This implies that Gaiotto vectors, describing the fundamental class in the equivariant cohomology of a suitable compactification of the moduli space of $G$-bundles over $\mathbb{P}^2$ for $G$ a complex simple Lie group, can be computed by a non-commutative version of the Chekhov-Eynard-Orantin topological recursion. We formulate the connection to higher Airy structures for Gaiotto vectors of type A, B, C, and D, and explicitly construct the topological recursion for type A (at arbitrary level) and type B (at self-dual level). On the physics side, it means that the Nekrasov partition function for pure $\mathcal{N} = 2$ four-dimensional supersymmetric gauge theories can be accessed by topological recursion methods.

math-ph↗