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.

We study the Fuss--Catalan algebras, which are generalizations of the Temperley--Lieb algebra and act on generalized Dyck paths, through non-crossing partitions. First, the Temperley--Lieb algebra is defined on non-crossing partitions, and a bijection between a Dyck path and a non-crossing partition is shown to be compatible with the Temperley--Lieb algebra on Dyck paths, or equivalently chord diagrams. We show that the Kreweras endomorphism on non-crossing partitions is equivalent to the rotation of chord diagrams under the bijection. Secondly, by considering an increasing $r$-chain in the graded lattice of non-crossing partitions, we define the Fuss--Catalan algebras on increasing $r$-chains. Through a bijection between an increasing $r$-chain and a generalized Dyck path, one naturally obtains the Fuss--Catalan algebra on generalized Dyck paths. As generalizations of the Fuss--Catalan algebra, we introduce the one- and two-boundary Fuss--Catalan algebras. Increasing $r$-chains of symmetric non-crossing partitions give symmetric generalized Dyck paths by the bijection, and the boundary Fuss--Catalan algebras naturally act on them. We show that these representations are compatible with the diagrammatic representations of the algebras by use of generalized chord diagrams. Thirdly, we discuss the integrability of the Fuss--Catalan algebras. For the Fuss--Catalan algebras with boundaries, we obtain a new solution of the reflection equation in the case of $r=2$.

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.

Monoidal product, braiding, balancing and weak duality are pieces of algebraic information that are well-known to have their origin in oriented genus zero surfaces and their mapping classes. More precisely, each of them correspond to operations of the cyclic framed $E_2$-operad. We extend this correspondence to include another algebraic piece of data, namely the modified trace, by showing that it amounts to a homotopy fixed point structure with respect to the homotopy involution that reverses the orientation of surfaces and dualizes the state spaces. We call such a homotopy fixed point structure reflection equivariance. As an application, we describe the effect of orientation reversal on spaces of conformal blocks and skein modules in the non-semisimple setting, throughout relying on their factorization homology description. This has important consequences: For a modular functor that is reflection equivariant relative to a rigid duality, i) the circle category is modular, and the resulting mapping class group representations are automatically the ones built by Lyubashenko, and ii) the resulting internal skein algebras have one simple representation, carrying a unique projective mapping class group representation making the action equivariant. While i) is a new topological characterization of not necessarily semisimple modular categories, ii) generalizes the implicit description of spaces of conformal blocks purely through the representation theory of moduli algebras given by Alekseev-Grosse-Schomerus from rational conformal field theories admitting a Hopf algebra description to finite rigid logarithmic conformal field theories. This also generalizes several results of Faitg from ribbon factorizable Hopf algebras to arbitrary modular categories.

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 $\operatorname{Rep} H_\lambda$.

We present a quantum algorithm for solving perfect mazes by casting the pathfinding task as a structured search problem. Building on Grover's amplitude amplification, the algorithm encodes all candidate paths in superposition and evaluates their proximity to the goal using a reversible fitness operator based on quantum arithmetic. A Grover-compatible oracle marks high-fitness states, and an adaptive cutoff strategy refines the search iteratively. We provide formal definitions, unitary constructions, and convergence guarantees, along with a resource analysis showing efficient scaling with maze size and path length. The framework serves as a foundation for quantum-hybrid pathfinding and planning. The full algorithmic pipeline is specified from encoding to amplification, including oracle design and fitness evaluation. The approach is readily extensible to other search domains, including navigation over tree-like or acyclic graphs.

If Kaluza-Klein ideas were correct as an explanation of Yang-Mills and General Relativity on spacetime, the extra fibre geometry would have to be a sphere of constant size of the order of 10 Planck lengths, hence subject to quantum gravity corrections. Conversely, it was shown in previous work that modelling such corrections by noncommutative coordinates indeed forces the Kaluza-Klein cylinder ansatz form of the metric, and we now propose that the remaining restrictions needed come from quantum gravity on the fibre. Working with a fuzzy sphere fibre, we find that the expected value of the metric is indeed spherical and we propose that it can be taken as of constant size due to freedom in the renormalisation of divergences. In this way, we outline a mechanism whereby the observed structure of gravity plus Yang-Mills can emerge at low energies as a consequence of quantum gravity effects.

We present a construction of the chiral de Rham complex over an algebraic surface with at most rational singularities of $A_n$-type. An explicit formula for the character of the chiral structure sheaf is also provided.

Unitary vertex operator algebras (VOAs) and conformal nets are the two most prominent mathematical axiomatizations of two-dimensional unitary chiral conformal field theories. They are conjectured to be equivalent, but a rigorous comparison has proven challenging. We resolve one direction of the conjecture by showing that every conformal net has an associated unitary VOA.

The aim of the paper is to define noncommutative cluster structure on several algebras ${\mathcal A}$ related to marked surfaces possibly with orbifold points of various orders, which includes noncommutative clusters, i.e., embeddings of a given group $G$ into the multiplicative monoid ${\mathcal A}^\times$ and an action of a certain braid-like group $Br_{\mathcal A}$ by automorphisms of each cluster group in a compatible way. For punctured surfaces we construct new symmetries, noncommutative tagged clusters and establish a noncommutative Laurent Phenomenon.

In this paper, we present two Hamiltonian simulation algorithms for multiscale linear transport equations, combining the Schr\"odingerization method [S. Jin, N. Liu and Y. Yu, Phys. Rev. Lett, 133 (2024), 230602][S. Jin, N. Liu and Y. Yu, Phys. Rev. A, 108 (2023), 032603] and exponential integrator while incorporating incoming boundary conditions. These two algorithms each have advantages in terms of design easiness and scalability, and the query complexity of both algorithms, $\mathcal{O}(N_vN_x^2\log N_x)$, outperforms existing quantum and classical algorithms for solving this equation. In terms of the theoretical framework, these are the first quantum Hamiltonian simulation algorithms for multiscale linear transport equation to combine the Schr\"odingerization method with an effective asymptotic-preserving schemes, which are efficient for handling multiscale problems with stiff terms.

Gaetz, Pechenik, Pfannerer, Striker, and Swanson introduced the concept of hourglass plabic graphs and provided a method for computing web diagrams and invariants corresponding to $4\times n$ Young tableaux, while Elkin, Musiker, and Wright applied Lam's method to explicitly compute the webs compatible with cluster variables in Gr(3,n) and their twists, namely, the preimages of the immanant map introduced by Fraser, Lam, and Le. In this paper, we use these two methods to compute both the web diagrams and the dual webs corresponding to quadratic and cubic cluster variables in the Grassmannian cluster algebra C[Gr(4,8)].

We develop an obstruction theory for the extension of truncated minimal $A$-infinity bimodule structures over truncated minimal $A$-infinity algebras. Obstructions live in far-away pages of a (truncated) fringed spectral sequence of Bousfield--Kan type. The second page of this spectral sequence is mostly given by a new cohomology theory associated to a pair consisting of a graded algebra and a graded bimodule over it. This new cohomology theory fits in a long exact sequence involving the Hochschild cohomology of the algebra and the self-extensions of the bimodule. We show that the second differential of this spectral sequence is given by the Gerstenhaber bracket with a bimodule analogue of the universal Massey product of a minimal $A$-infinity algebra. We also develop a closely-related obstruction theory for truncated minimal $A$-infinity bimodule structures over (the truncation of) a fixed minimal $A$-infinity algebra; the second page of the corresponding spectral sequence is now mostly given by the vector spaces of self-extensions of the underlying graded bimodule and the second differential is described analogously to the previous one. We also establish variants of the above for graded algebras and graded bimodules that are $d$-sparse, that is they are concentrated in degrees that are multiples of a fixed integer $d\geq1$. These obstruction theories are used to establish intrinsic formality and almost formality theorems for differential graded bimodules over differential graded algebras. Our results hold, more generally, in the context of graded operads with multiplication equipped with an associative operadic ideal, examples of which are the endomorphism operad of a graded algebra and the linear endomorphism operad of a pair consisting of a graded algebra and a graded bimodule over it.

We define the notion of a Lie superalgebra over a field $k$ of characteristic $2$ which unifies the two pre-existing ones - $\mathbb{Z}/2$-graded Lie algebras with a squaring map and Lie algebras in the Verlinde category ${\rm Ver}_4^+(k)$, and prove the PBW theorem for this notion. We also do the same for the restricted version. Finally, discuss mixed characteristic deformation theory of such Lie superalgebras (for perfect $k$), introducing and studying a natural lift of our notion of Lie superalgebra to characteristic zero - the notion of a mixed Lie superalgebra over a ramified quadratic extension $R$ of the ring of Witt vectors $W(k)$.

We show that a vector space valued TQFT constructed in work of De Renzi et al. [DGGPR23] extends naturally to a topological field theory which takes values in the symmetric monoidal category of linear cochains. Specifically, we consider a bordism category whose objects are surfaces with markings from the category of cochains Ch(A) over a given modular tensor category (such as the category of small quantum group representations), and whose morphisms are 3-dimensional bordisms with embedded ribbon graphs traveling between such marked surfaces. We construct a symmetric monoidal functor from the aforementioned ribbon bordism category to the category of linear cochains. The values of this theory on surfaces are identified with Hom complexes for Ch(A), and the 3-manifold invariants are alternating sums of the renormalized Lyubashenko invariant from [DGGPR23]. We show that our cochain valued TQFT furthermore preserves homotopies, and hence localizes to a theory which takes values in the derived $\infty$-category of dg vector spaces. The domain for this $\infty$-categorical theory is, up to some approximation, an $\infty$-category of ribbon bordism with labels in the homotopy $\infty$-category K(A). We suggest our localized theory as a starting point for the construction of a "derived TQFT" for the $\infty$-category of derived quantum group representations.

These notes are based on the three lectures that one of the authors gave at Tsinghua University in the summer of 2023 as part of the workshop on Geometric Representation Theory and Applications. They contain an introduction to the evaluation of $\mathsf{SL}(3)$ foams and the associated topological theory of trivalent planar graphs and foam cobordisms between them. A categorification of the Kuperberg quantum $\mathfrak{sl}_3$ web and link invariant and the Robert-Wagner $\mathsf{SL}(N)$ foam evaluation are reviewed as well.

We examine higher-form symmetries of quantum lattice gauge theories through the lens of homotopy theory and operator algebras. We show that in the operator-algebraic approach both higher-form symmetries and 't Hooft anomalies arise from considering restrictions of symmetry transformations to spatial regions. The data of these restrictions are naturally packaged into a higher group. For example, for gauge theories in two spatial dimensions, this information is encoded in a crossed square of groups, which is an algebraic model of a 3-group. In general, we propose that higher groups appear in lattice models and QFT as crossed n-cubes of groups via a nonabelian version of the Cech construction.