arXiv subjects

Sven Möller

Publications and source records attributed to Sven Möller.

17 recordsLinked to original sources

arXiv2026

Character Identities Between Affine and Virasoro Vertex Operator Algebra Modules

The affine vertex operator algebras for $\mathfrak{sl}_2$ and the Virasoro minimal models are related by Drinfeld-Sokolov reduction and by the Goddard-Kent-Olive coset construction. In this work, we propose another connection based on certain character identities between these vertex operator algebras and their modules. This relates the simple affine vertex operator algebras $L_k(\mathfrak{sl}_2)$ at admissible levels $k=-2+q/p$ to the rational $(q,3p)$-minimal models $L_\mathrm{Vir}(c_{q,3p},0)$, and also extends to the nonadmissible levels with $q=1$. Several special cases are particularly interesting. In the nonadmissible case $q=1$, the character identities extend to certain abelian intertwining algebras, specifically $\mathcal{V}^{(p)}$ and the doublet $\mathcal{A}^{(3p)}$. Specialising further to $p=2$, where $\mathcal{V}^{(2)}$ is the simple small $\mathcal{N}=4$ superconformal algebra of central charge $c=-9$, this recovers, via the 4d/2d-correspondence, a known identity between the Schur indices of the 4d $\mathcal{N}=4$ supersymmetric Yang-Mills theory for $\mathrm{SU}(2)$ and the 4d $\mathcal{N}=2$ $(3,2)$ Argyres-Douglas theory. In the boundary admissible case $q=2$, in a similar vein, we obtain an identity between the Schur indices of 4d $\mathcal{N}=2$ Argyres-Douglas theories of types $(A_1,D_{2n+1})$ and $(A_1,A_{6n})$. On the other hand, for integral levels, $p=1$, where both involved vertex operator algebras are strongly rational, our character identity induces a Galois conjugation between the representation categories $\mathrm{Rep}(L_{-2+q}(\mathfrak{sl}_2))$ and $\mathrm{Rep}(L_\mathrm{Vir}(c_{q,3},0))$; and for small values of $q$, the characters are related by the action of certain Hecke operators. Finally, we also sketch how to extend the results of this paper to relaxed highest-weight and Whittaker modules.

math.QA↗

arXiv2026

Vertex Superalgebras for Hypertoric Varieties and 3d Abelian Gauge Theories

Hypertoric (or toric hyperkähler) varieties are a class of symplectic singularities and their resolutions, obtained as Hamiltonian reductions of a symplectic vector space acted on by a torus. In physics, they appear as Higgs (and Coulomb) branches of 3d $\mathcal{N}=4$ supersymmetric quantum field theories with abelian gauge group. In this work, we construct an $\hbar$-adic (in the sense of microlocalisation) sheaf of vertex operator superalgebras over a given smooth hypertoric variety. Its global sections give the $A$-twisted boundary of the corresponding 3d gauge theory. We use this to prove that the associated affine variety of this hypertoric vertex operator superalgebra recovers the singular hypertoric variety. This proves the 3d Higgs branch conjecture for a large class of boundary vertex operator superalgebras. In particular, these vertex operator superalgebras are quasi-lisse. This is in contrast to the (purely even) hypertoric vertex operator superalgebras (and their $\hbar$-adic localisations) constructed previously by Kuwabara as global sections of sheaves on families of universal Poisson deformations of the hypertoric varieties. These are generally not quasi-lisse. We show that the vertex operator superalgebras defined in this paper are (fermionic) simple-current extensions of those defined by Kuwabara, and investigate the consequences for symplectic duality and characters. We observe that the latter are upgraded from partial (or false) theta functions to quasimodular forms.

math.QA↗

arXiv2026

Modular $\mathbb{Z}_2$-Crossed Tambara-Yamagami-like Categories for Even Groups

We explicitly construct nondegenerate braided $\mathbb{Z}_2$-crossed tensor categories of the form $\operatorname{Vect}_Γ\oplus\operatorname{Vect}_{Γ/2Γ}$. They are $\mathbb{Z}_2$-crossed extensions, in the sense of arXiv:0909.3140, of the braided tensor category $\operatorname{Vect}_Γ$ with $\mathbb{Z}_2$-action given by $-\mathrm{id}$ on the finite, abelian group $Γ$. Thus, we obtain generalisations of the Tambara-Yamagami categories, where now the abelian group $Γ$ may have even order and the nontrivial sector $\operatorname{Vect}_{Γ/2Γ}$ more than one simple object. The idea for this construction comes from a physically motivated approach in arXiv:2409.16357 to construct $\mathbb{Z}_2$-crossed extensions of $\operatorname{Vect}_Γ$ for any $Γ$ from an infinite Tambara-Yamagami category $\operatorname{Vect}_{\mathbb{R}^d}\oplus\operatorname{Vect}$, which itself is not fully rigorously defined, and then using condensation from $\operatorname{Vect}_{\mathbb{R}^d}$ to $\operatorname{Vect}_Γ$, which we prove commutes with crossed extensions. The $\mathbb{Z}_2$-equivariantisation of $\operatorname{Vect}_Γ\oplus\operatorname{Vect}_{Γ/2Γ}$ yields new modular tensor categories, which correspond to the orbifold of an arbitrary lattice vertex operator algebra under a lift of $-\mathrm{id}$, as discussed in arXiv:2409.16357.

math.QA↗

arXiv2026

Systematic Orbifold Constructions of Schellekens' Vertex Operator Algebras from Niemeier Lattices

We present a systematic, rigorous construction of all 70 strongly rational, holomorphic vertex operator algebras $V$ of central charge 24 with non-zero weight-one space $V_1$ as cyclic orbifold constructions associated with the 24 Niemeier lattice vertex operator algebras $V_N$ and certain 226 short automorphisms in $\operatorname{Aut}(V_N)$. We show that up to algebraic conjugacy these automorphisms are exactly the generalised deep holes, as introduced in arXiv:1910.04947, of the Niemeier lattice vertex operator algebras with the additional property that their orders are equal to those of the corresponding outer automorphisms. Together with the constructions in arXiv:1708.05990 and arXiv:1910.04947 this gives three different uniform constructions of these vertex operator algebras, which are related through 11 algebraic conjugacy classes in $\operatorname{Co}_0$. Finally, by considering the inverse orbifold constructions associated with the 226 short automorphisms, we give the first systematic proof of the result that each strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with non-zero weight-one space $V_1$ is uniquely determined by the Lie algebra structure of $V_1$.

math.QA↗

arXiv2025

Equivalence Relations on Vertex Operator Algebras, II: Witt Equivalence and Orbifolds

When can two strongly rational vertex operator algebras or 1+1d rational conformal field theories (RCFTs) be related by topological manipulations? For vertex operator algebras, the term "topological manipulations" refers to operations like passing to a conformal extension or restricting to a conformal subalgebra; for RCFTs, topological manipulations include operations like gauging (or orbifolding) a finite subpart of a generalized global symmetry or interpolating to a new theory via a topological line interface of finite quantum dimension. Inspired by results in the theory of even lattices and tensor categories, we say that two strongly rational vertex operator algebras are Witt equivalent if their central charges agree and if their modular tensor categories are Witt equivalent. Two RCFTs are said to be Witt equivalent if their central charges agree and if their associated 2+1d topological field theories can be separated by a topological surface. We argue that Witt equivalence is necessary for two theories to be related by topological manipulations. We conjecture that it is also sufficient, and give proofs in various special cases. We relate this circle of ideas to the problem of classifying RCFTs, and to lore concerning deformation classes of quantum field theories. We use the notion of Witt equivalence to argue, assuming the conjectural classification of unitary, $c=1$ RCFTs, that all of the finite symmetries of the $SU(2)_1$ Wess-Zumino-Witten model are invertible. We also sketch a "quantum Galois theory" for chiral CFTs, which generalizes prior mathematical literature by incorporating non-invertible symmetries; we illustrate this non-invertible Galois theory in the context of the monster CFT, for which we produce two Fibonacci lines. Finally, we discuss $p$-neighborhood of vertex operator algebras, which is a special topological manipulation related to $\mathbb{Z}_p$-orbifolding.

hep-th↗

arXiv2024

Computing $G$-Crossed Extensions and Orbifolds of Vertex Operator Algebras

In this article, we develop tools for computing $G$-crossed extensions of braided tensor categories. Their equivariantisations appear as categories of modules of fixed-point subalgebras (or orbifolds) of vertex operator algebras and are often difficult to determine. As the first tool, we show how the seminal work of Etingof, Nikshych and Ostrik on the uniqueness of $G$-crossed extensions can be used to determine the category of modules of orbifold vertex operator algebras. As an application, we determine the modular tensor category of the orbifold of a lattice vertex operator algebra under a lift of $-\mathrm{id}$ for a lattice with odd-order discriminant form. In that case, the de-equivariantisation is of Tambara-Yamagami type. As the second tool, we describe how $G$-crossed extensions and condensations by commutative algebras commute in a suitable sense. This leads to an effective approach to compute new $G$-crossed extensions. As one application, we produce the coherence data that are then used in arXiv:2411.12251 to define a generalisation of the Tambara-Yamagami categories with more than one simple object in the twisted sector. This also yields the modular tensor category of the orbifold of an arbitrary lattice vertex operator algebra under a lift of $-\mathrm{id}$. Finally, we sketch how to categorically approach the general problem of lattice orbifolds under lifts of arbitrary lattice involutions.

math.QA↗

arXiv2024

A Geometric Classification of the Holomorphic Vertex Operator Algebras of Central Charge 24

We associate with a generalised deep hole of the Leech lattice vertex operator algebra a generalised hole diagram. We show that this Dynkin diagram determines the generalised deep hole up to conjugacy and that there are exactly 70 such diagrams. In an earlier work we proved a bijection between the generalised deep holes and the strongly rational, holomorphic vertex operator algebras of central charge 24 with non-trivial weight-1 space. Hence, we obtain a new, geometric classification of these vertex operator algebras, generalising the classification of the Niemeier lattices by their hole diagrams.

math.QA↗

arXiv2024

Equivalence Relations on Vertex Operator Algebras, I: Genus

In this first of a series of two papers, we investigate two different equivalence relations obtained by generalizing the notion of genus of even lattices to the setting of vertex operator algebras (or two-dimensional chiral algebras). The bulk genus equivalence relation was defined in arXiv:math/0209333 and groups (suitably regular) vertex operator algebras according to their modular tensor category and central charge. Hyperbolic genus arXiv:2004.01441 tests isomorphy after tensoring with a hyperbolic plane vertex algebra. Physically, two rational chiral algebras are said to belong to the same bulk genus if they live on the boundary of the same 2+1d topological quantum field theory; they belong to the same hyperbolic genus if they can be related by current-current exactly marginal deformations after tensoring a non-chiral compact boson. As one main result, we prove the conjecture that the hyperbolic genus defines a finer equivalence relation than the bulk genus. This is based on a new, equivalent characterization of the hyperbolic genus that uses the maximal lattice inside a vertex operator algebra and its commutant (or coset). We discuss the implications of these constructions for the classification of rational conformal field theory. In particular, we propose a program for (partially) classifying $c=32$, holomorphic vertex operator algebras (or chiral conformal field theories), and obtain novel lower bounds, via a generalization of the Smith-Minkowski-Siegel mass formula, on the number of vertex operator algebras at higher central charges. Finally, we conjecture a Siegel-Weil identity which computes the "average" torus partition function of an ensemble of chiral conformal field theories defined by any hyperbolic genus, and interpret this formula physically in terms of disorder-averaged holography.

hep-th↗

arXiv2024

Classification of Self-Dual Vertex Operator Superalgebras of Central Charge at Most 24

We classify the self-dual (or holomorphic) vertex operator superalgebras of central charge 24, or in physics parlance the purely left-moving, fermionic 2-dimensional conformal field theories with just one primary field. There are exactly 969 such vertex operator superalgebras under suitable regularity assumptions (essentially strong rationality) and the assumption that the shorter moonshine module $V\!B^\natural$ is the unique self-dual vertex operator superalgebra of central charge 23.5 whose weight-1/2 and weight-1 spaces vanish. Additionally, there might be self-dual vertex operator superalgebras arising as fake copies of $V\!B^\natural$ tensored with a free fermion $F$. We construct and classify the self-dual vertex operator superalgebras by determining the 2-neighbourhood graph of the self-dual vertex operator algebras of central charge 24 and also by realising them as simple-current extensions of a dual pair containing a certain maximal lattice vertex operator algebra. We show that all vertex operator superalgebras besides $V\!B^\natural\otimes F$ and potential fake copies thereof stem from elements of the Conway group $\mathrm{Co}_0$, the automorphism group of the Leech lattice $Λ$. By splitting off free fermions $F$, if possible, we obtain the classification for all central charges less than or equal to 24.

math.QA↗

arXiv2023

Hilbert Schemes of Points in the Plane and Quasi-Lisse Vertex Algebras with $\mathcal{N}=4$ Symmetry

To each complex reflection group $Γ$ one can attach a canonical symplectic singularity $\mathcal{M}_Γ$ arXiv:math/9903070. Motivated by the 4D/2D duality arXiv:1312.5344, arXiv:1707.07679, Bonetti, Meneghelli and Rastelli arXiv:1810.03612 conjectured the existence of a supersymmetric vertex operator superalgebra $\mathsf{W}_Γ$ whose associated variety is isomorphic to $\mathcal{M}_Γ$. We prove this conjecture when the complex reflection group $Γ$ is the symmetric group $S_N$ by constructing a sheaf of $\hbar$-adic vertex operator superalgebras on the Hilbert scheme of $N$ points in the plane. For that case, we also show the free-field realisation of $\mathsf{W}_Γ$ in terms of $\operatorname{rk}(Γ)$ many $βγbc$-systems proposed in arXiv:1810.03612, and identify the character of $\mathsf{W}_Γ$ as a certain quasimodular form of mixed weight and multiple $q$-zeta value. In physical terms, the vertex operator superalgebra $\mathsf{W}_{S_N}$ constructed in this article corresponds via the 4D/2D duality arXiv:1312.5344 to the four-dimensional $\mathcal{N}=4$ supersymmetric Yang-Mills theory with gauge group $\operatorname{SL}_N$.

math.RT↗

arXiv2022

Dimension Formulae and Generalised Deep Holes of the Leech Lattice Vertex Operator Algebra

We prove a dimension formula for the weight-1 subspace of a vertex operator algebra $V^{\operatorname{orb}(g)}$ obtained by orbifolding a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with a finite-order automorphism $g$. Based on an upper bound derived from this formula we introduce the notion of a generalised deep hole in $\operatorname{Aut}(V)$. Then we show that the orbifold construction defines a bijection between the generalised deep holes of the Leech lattice vertex operator algebra $V_Λ$ with non-trivial fixed-point Lie subalgebra and the strongly rational, holomorphic vertex operator algebras of central charge 24 with non-vanishing weight-1 space. This provides the first uniform construction of these vertex operator algebras and naturally generalises the correspondence between the deep holes of the Leech lattice $Λ$ and the 23 Niemeier lattices with non-vanishing root system found by Conway, Parker and Sloane.

math.QA↗

arXiv2021

Natural Construction of Ten Borcherds-Kac-Moody Algebras Associated with Elements in $M_{23}$

Borcherds-Kac-Moody algebras generalise finite-dimensional, simple Lie algebras. Scheithauer showed that there are exactly ten Borcherds-Kac-Moody algebras whose denominator identities are completely reflective automorphic products of singular weight on lattices of square-free level. These belong to a larger class of Borcherds-Kac-Moody (super)algebras Borcherds obtained by twisting the denominator identity of the Fake Monster Lie algebra. Borcherds asked whether these Lie (super)algebras admit natural constructions. For the ten Lie algebras from the classification we give a positive answer to this question, i.e. we prove that they can be realised uniformly as the BRST cohomology of suitable vertex algebras.

math.QA↗

arXiv2021

Schellekens' List and the Very Strange Formula

In 1993 Schellekens proved that the weight-one space $V_1$ of a strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 must be one of 71 Lie algebras. During the following three decades, in a combined effort by many authors, it was proved that each of these Lie algebras is realised by such a vertex operator algebra and that, except for $V_1=\{0\}$, this vertex operator algebra is uniquely determined by $V_1$. In this paper we give a fundamentally different, simpler proof of Schellekens' list of 71 Lie algebras. Using the dimension formula in arXiv:1910.04947 and Kac's "very strange formula" we show that every strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 with $V_1\neq\{0\}$ can be obtained by an orbifold construction from the Leech lattice vertex operator algebra $V_Λ$. This suffices to restrict the possible Lie algebras that can occur as weight-one space of $V$ to the 71 of Schellekens. Moreover, the fact that each strongly rational, holomorphic vertex operator algebra $V$ of central charge 24 comes from the Leech lattice $Λ$ can be used to classify these vertex operator algebras by studying properties of the Leech lattice. We demonstrate this for 43 of the 70 non-zero Lie algebras on Schellekens' list, omitting those cases that are too computationally expensive.

math.QA↗

arXiv2018

Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras

We prove a dimension formula for orbifold vertex operator algebras of central charge 24 by automorphisms of order $n$ such that $Γ_0(n)$ is a genus zero group. We then use this formula together with the inverse orbifold construction for automorphisms of orders 2, 4, 5, 6 and 8 to establish that each of the following fifteen Lie algebras is the weight-one space $V_1$ of exactly one holomorphic, $C_2$-cofinite vertex operator algebra $V$ of CFT-type of central charge 24: $A_5C_5E_{6,2}$, $A_3A_{7,2}C_3^2$, $A_{8,2}F_{4,2}$, $B_8E_{8,2}$, $A_2^2A_{5,2}^2B_2$, $C_8F_4^2$, $A_{4,2}^2C_{4,2}$, $A_{2,2}^4D_{4,4}$, $B_5E_{7,2}F_4$, $B_4C_6^2$, $A_{4,5}^2$, $A_4A_{9,2}B_3$, $B_6C_{10}$, $A_1C_{5,3}G_{2,2}$ and $A_{1,2}A_{3,4}^3$.

math.QA↗

arXiv2018

Orbifold Vertex Operator Algebras and the Positivity Condition

In this note we show that the irreducible twisted modules of a holomorphic, $C_2$-cofinite vertex operator algebra $V$ have $L_0$-weights at least as large as the smallest $L_0$-weight of $V$. Hence, if $V$ is of CFT-type, then the twisted $V$-modules are almost strictly positively graded. This in turn implies that the fixed-point vertex operator subalgebra $V^G$ for a finite, solvable group of automorphisms of $V$ almost satisfies the positivity condition. These and some further results are obtained by a careful analysis of Dong, Li and Mason's twisted modular invariance.

math.QA↗

arXiv2017

Construction and Classification of Holomorphic Vertex Operator Algebras

We develop an orbifold theory for finite, cyclic groups acting on holomorphic vertex operator algebras. Then we show that Schellekens' classification of $V_1$-structures of meromorphic conformal field theories of central charge 24 is a theorem on vertex operator algebras. Finally we use these results to construct some new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds.

math.RT↗

arXiv2017

A Cyclic Orbifold Theory for Holomorphic Vertex Operator Algebras and Applications

In this thesis we develop an orbifold theory for a finite, cyclic group $G$ acting on a suitably regular, holomorphic vertex operator algebra $V$. To this end we describe the fusion algebra of the fixed-point vertex operator subalgebra $V^G$ and show that $V^G$ has group-like fusion. Then we solve the extension problem for vertex operator algebras with group-like fusion. We use these results to construct five new holomorphic vertex operator algebras of central charge 24 as lattice orbifolds, contributing to the classification of the $V_1$-structures of suitably regular, holomorphic vertex operator algebras of central charge 24. As another application we present the BRST construction of ten Borcherds-Kac-Moody algebras whose denominator identities are completely reflective automorphic products of singular weight.

math.QA↗