CyberSec Research

Vertex algebras that arise from four-dimensional, $\mathcal{N}=2$ superconformal field theories inherit a collection of novel structural properties from their four-dimensional ancestors. Crucially, when the parent SCFT is unitary, the corresponding vertex algebra is not unitary in the conventional sense. In this paper, we motivate and define a generalized notion of unitarity for vertex algebras that we call \emph{graded unitarity}, and which captures the consequences of four-dimensional unitarity under this correspondence. We also take the first steps towards a classification program for graded-unitary vertex algebras whose underlying vertex algebras are Virasoro or affine Kac--Moody vertex algebras. Remarkably, under certain natural assumptions about the $\mathfrak{R}$-filtration for these vertex algebras, we show that only the $(2,p)$ central charges for Virasoro VOAs and boundary admissible levels for $\mathfrak{sl}_2$ and $\mathfrak{sl}_3$ Kac--Moody vertex algebras can possibly be compatible with graded unitarity. These are precisely the cases of these vertex algebras that are known to arise from four dimensions.

This paper addresses the interactions between three properties that a group algebra or more generally a pointed Hopf algebra may possess: being noetherian, having finite Gelfand-Kirillov dimension, and satisfying the Dixmier-Moeglin equivalence. First it is shown that the second and third of these properties are equivalent for group algebras $kG$ of polycyclic-by-finite groups, and are, in turn, equivalent to $G$ being nilpotent-by-finite. In characteristic 0, this enables us to extend this equivalence to certain cocommutative Hopf algebras. In the second and third parts of the paper finiteness conditions for group algebras are studied. In the second section we examine when a group algebra satisfies the Goldie conditions, while in the final section we discuss what can be said about a minimal counterexample to the conjecture that if $kG$ is noetherian then G is polycyclic-by-finite.

The purpose of this note is to prove a conjecture of Shvartsman relating a complex projective reflection group with the quotient of a suitable complex braid group by its center. Shvartsman originally proved this result in the case of real projective reflection groups, and we extend it to all complex projective reflection groups. Our study also allows us to correct a result of Brou\'e, Malle, Rouquier on projective reflection groups.

We explicitly compute the McKay quivers of small finite subgroups of $GL(2,\mathbb{C})$ relative to the natural representation, using character theory and the McKay quivers of finite subgroups of $SU(2)$. We present examples that shows the rich symmetry and combinatorial structure of these quivers. We compare our results with the MacKay quivers computed by Auslander and Reiten.

In this article we investigate the regularity properties of linear degenerations of flag varieties. We classify the linear degenerations of (partial) flag varieties that are smooth. Furthermore, we study the singular locus of irreducible degenerations and provide estimates for its dimension. We also introduce a new stratification of the total space of representations. Within each stratum, we identify the loci corresponding to flat and flat irreducible degenerations. As a consequence of our results, we show that irreducible linear degenerations are normal varieties.

For logarithmic conformal field theories whose monodromy data is given by a not necessarily semisimple modular category, we solve the problem of constructing and classifying the consistent systems of correlators. The correlator construction given in this article generalizes the well-known one for rational conformal field theories given by Fuchs-Runkel-Schweigert roughly twenty years ago and solves conjectures of Fuchs, Gannon, Schaumann and Schweigert. The strategy is, even in the rational special case, entirely different. The correlators are constructed using the extension procedures that can be devised by means of the modular microcosm principle. It is shown that, as in the rational case, the correlators admit a holographic description, with the main difference that the holographic principle is phrased in terms of factorization homology. The latter description is used to prove that the coefficients of the torus partition function are non-negative integers. Moreover, we show that the derived algebra of local operators associated to a consistent system of correlators carries a Batalin-Vilkovisky structure. We prove that it is equivalent to the Batalin-Vilkovisky structure on the Hochschild cohomology of the pivotal module category of boundary conditions, for the notion of pivotality due to Schaumann and Shimizu. This proves several expectations formulated by Kapustin-Rozansky and Fuchs-Schweigert for general conformal field theories.

It is well-known that ADE Dynkin diagrams classify both the simply-laced simple Lie algebras and simple singularities. We introduce a polygonal wheel in a plane for each case of ADE, called the Coxeter wheel. We show that equivalence classes of edges and spokes of a Coxeter wheel form a geometric root system isomorphic to the classical root system of the corresponding type. This wheel is in fact derived from the Milnor fiber of corresponding simple singularities of two variables, and the bilinear form on the geometric root system is the negative of its symmetrized Seifert form. Furthermore, we give a completely geometric definition of simple Lie algebras using arcs, Seifert form and variation operator of the singularity theory.

We consider the centre of the affine vertex algebra at the critical level associated with the orthosymplectic Lie superalgebra. It is well-known that the centre is a commutative superalgebra, and we construct a family of its elements in an explicit form. In particular, this gives a new proof of the formulas for the central elements for the orthogonal and symplectic Lie algebras. Our arguments rely on the properties of a new extended Brauer-type algebra.

A brief proof of Lie's classification of finite dimensional subalgebras of vector fields on the complex plane that have a proper Levi decomposition is given. The proof uses basic representation theory of sl(2, C). This, combined with \cite{ABF2} and \cite{ABF3} completes the classification of finite dimensional subalgebras of vector fields on the complex plane.

Let \( \pi \) be a polarized, regular algebraic, cuspidal automorphic representation of \( \GL_n(\bb{A}_F) \) where \( F \) is totally real or imaginary CM, and let \( (\rho_\lambda)_\lambda \) be its associated compatible system of Galois representations. We prove that if \( 7\nmid n \) and \( 4 \nmid n \) then there is a Dirichlet density \( 1 \) set of rational primes \( \mc{L} \) such that whenever \( \lambda\mid \ell \) for some \( \ell\in \mc{L} \), then \( \rho_\lambda \) is irreducible.

We prove that every supersymmetric Schur polynomial has a saturated Newton polytope (SNP). Our approach begins with a tableau-theoretic description of the suport, which we encode as a polyhedron with a totally unimodular constraint matrix. The integrality of this polyhedron follows from the Hoffman-Kruskal criterion, thereby establishing the SNP property. To our knowledge, this is the first application of total unimodularity to the SNP problem.

Several recent problems in the representation theory of finite groups require determining whether certain characters of almost simple groups belong to the principal block. Since the values of these characters are not yet known, we employ alternative group-theoretical techniques to address the "going down" case. This approach enables us to reduce the block version of well-known results by the third and fourth authors to a question about almost simple groups. Moreover, this suggests a Galois analogue of the height-zero-equal-degree conjecture of Malle and Navarro, which we formulate. However, the "going up" case of irreducible extensions of principal block characters remains unresolved.

Let $\mathcal{A}$ be the group algebra $\mathbf{k}[S_n]$ of the $n$-th symmetric group $S_n$ over a commutative ring $\mathbf{k}$. For any two subsets $A$ and $B$ of $[n]$, we define the elements \[ \nabla_{B,A}:=\sum_{\substack{w\in S_n;\\w\left( A\right) =B}} w \qquad \text{and} \qquad \widetilde{\nabla}_{B,A}:=\sum_{\substack{w\in S_n;\\w\left( A\right) \subseteq B}}w \] of $\mathcal{A}$. We study these elements, showing in particular that their minimal polynomials factor into linear factors (with integer coefficients). We express the product $\nabla_{D,C}\nabla_{B,A}$ as a $\mathbb{Z}$-linear combination of $\nabla_{U,V}$'s. More generally, for any two set compositions (i.e., ordered set partitions) $\mathbf{A}$ and $\mathbf{B}$ of $\left\{ 1,2,\ldots,n\right\} $, we define $\nabla_{\mathbf{B},\mathbf{A}}\in\mathcal{A}$ to be the sum of all permutations $w\in S_n$ that send each block of $\mathbf{A}$ to the corresponding block of $\mathbf{B}$. This generalizes $\nabla_{B,A}$. The factorization property of minimal polynomials does not extend to the $\nabla_{\mathbf{B},\mathbf{A}}$, but we describe the ideal spanned by the $\nabla_{\mathbf{B},\mathbf{A}}$ and a further ideal complementary to it. These two ideals have a "mutually annihilative" relationship, are free as $\mathbf{k}$-modules, and appear as annihilators of tensor product $S_n$-representations; they are also closely related to Murphy's cellular bases, Specht modules, pattern-avoiding permutations and even some algebras appearing in quantum information theory.

We prove that, up to scaling, there exist only finitely many isometry classes of Hermitian lattices over $O_E$ of signature $(1,n)$ that admit ball quotients of non-general type, where $n>12$ is even and $E=\mathbb{Q}(\sqrt{-D})$ for prime discriminant $-D<-3$. Furthermore, we show that even-dimensional ball quotients, associated with arithmetic subgroups of $\operatorname{U}(1,n)$ defined over $E$, are always of general type if $n > 207$, or $n>12$ and $D>2557$. To establish these results, we construct a nontrivial full-level cusp form of weight $n$ on the $n$-dimensional complex ball. A key ingredient in our proof is the use of Arthur's multiplicity formula from the theory of automorphic representations.

We show that bounded type implies finite type for a constructible subcategory of the module category of a finitely generated algebra over a field, which is a variant of the first Brauer-Thrall conjecture. A full subcategory is constructible if it consists of all modules that vanish on a finitely presented functor. Our approach makes use of the Ziegler spectrum of a ring and a connection, established in this work, with the scheme of finite dimensional modules. We also discuss a variant of the second Brauer-Thrall conjecture in this setting. It is shown to be true under additional assumptions on the algebra, but wrong in general. Lastly, it is proven that exact structures and matrix reductions give rise to constructible subcategories. Based on this, a translation from matrix reductions to reductions of exact structures is provided.

We classify the irreducible representations of a family of finite-dimensional pointed liftings $H_\lambda$ of the Nichols algebra associated with the diagram $A_2$ with parameter $q=-1$. We show that these algebras have infinite representation type and construct an indecomposable $H_\lambda$-module of dimension $n$ for each $n\in\mathbb{N}$. Finally, we study a semisimple category $\underline{\operatorname{Rep}} H_\lambda$ arising as a quotient of $\operatorname{Rep} H_\lambda$.

This paper is devoted to constructing simple modules of the planar Galilean conformal algebra. We study the tensor products of finitely many simple $\mathcal{U}(\mathcal{H})$-free modules with an arbitrary simple restricted module, where $\mathcal{H}$ is the Cartan subalgebra. We establish necessary and sufficient conditions for simplicity and determine the corresponding isomorphism classes.

This paper investigates simple modules of the semi-direct product algebra $\mathcal{W}\ltimes\widehat{H_4}$, where $\mathcal{W}$ is the Witt algebra and $\widehat{H_4}$ is the loop Diamond algebra. We first use simple modules over the Weyl algebra to construct a family of simple $\mathcal{W}\ltimes\widehat{H_4}$-modules. Then, we classify simple $\mathcal{W}\ltimes\widehat{H_4}$-modules that are free $U(\mathbb{C}L_0\oplus\mathbb{C} a_0)$-modules of rank 1. Finally, we give a necessary and sufficient condition for finitely many simple $U(\mathbb{C}L_0\oplus\mathbb{C}a_0)$-free modules to be simple, and then determine their isomorphism classes.