CyberSec Research

We produce graded monoidal categorifications of the quantum boson algebras in any symmetrizable Kac-Moody type. Our categories are defined in terms of diagrammatic generators and relations and have a faithful 2-representation on Khovanov-Lauda and Rouquier's categorification of the corresponding positive part quantum group. We use our construction to produce interesting bases of the quantum boson algebras and quantum bosonic extensions.

We compare the notions of metric-compatibility and torsion of a connection in the frameworks of Beggs-Majid and Mesland-Rennie. It follows that for $\ast$-preserving connections, compatibility with a real metric in the sense of Beggs-Majid corresponds to Hermitian connections in the sense of Mesland-Rennie. If the calculus is quasi-tame, the torsion zero conditions are equivalent. A combination of these results proves the existence and uniqueness of Levi-Civita connections in the sense of Mesland-Rennie for unitary cocycle deformations of a large class of Riemannian manifolds as well as the Heckenberger-Kolb calculi on all quantized irreducible flag manifolds.

Let $\mathcal{A}_{q}$ be an arbitrary quantum cluster algebra with principal coefficients. We give the fundamental relations between the quantum cluster variables arising from one-step mutations from the initial cluster in $\mathcal{A}_{q}$. Immediately and directly, we obtain an algebra homomorphism from the corresponding (untwisted) quantum group to $\mathcal{A}_{q}$.

We study skein modules using $Sp(2n)$ webs. We define multivariable analogues of Chebyshev polynomials in the Type $C$ setting and use them to construct transparent elements in the skein module at roots of unity. Our arguments are diagrammatic and make use of an explicit braiding formula for $1$ and $k$ labeled strands and an analogue of Kuperberg's tetravalent vertex in the annular setting.

It is natural to consider extending the typical construction of relative Poisson algebras from commutative differential algebras to the context of bialgebras. The known bialgebra structures for relative Poisson algebras, namely relative Poisson bialgebras, are equivalent to Manin triples of relative Poisson algebras with respect to the symmetric bilinear forms which are invariant on both the commutative associative and Lie algebras. However, they are not consistent with commutative and cocommutative differential antisymmetric infinitesimal (ASI) bialgebras as the bialgebra structures for commutative differential algebras. Alternatively, with the invariance replaced by the commutative $2$-cocycles on the Lie algebras, the corresponding Manin triples of relative Poisson algebras are proposed, which are shown to be equivalent to certain bialgebra structures, namely relative PCA bialgebras. They serve as another approach to the bialgebra theory for relative Poisson algebras, which can be naturally constructed from commutative and cocommutative differential ASI bialgebras. The notion of the relative PCA Yang-Baxter equation (RPCA-YBE) in a relative PCA algebra is introduced, whose antisymmetric solutions give coboundary relative PCA bialgebras. The notions of $\mathcal{O}$-operators of relative PCA algebras and relative pre-PCA algebras are also introduced to give antisymmetric solutions of the RPCA-YBE.

We give a cabling formula for the Links--Gould polynomial of knots colored with a $4n$-dimensional irreducible representation of $\mathrm{U}^H_q\mathfrak{sl}(2|1)$ and identify them with the $V_n$-polynomial of knots for $n=2$. Using the cabling formula, we obtain genus bounds and a specialization to the Alexander polynomial for the colored Links--Gould polynomial that is independent of $n$, which implies corresponding properties of the $V_n$-polynomial for $n=2$ conjectured in previous work of two of the authors, and extends the work done for $n=1$. Combined with work of one of the authors arXiv:2409.03557, our genus bound for $\mathrm{LG}^{(2)}=V_2$ is sharp for all knots with up to $16$ crossings.

We show that isomorphism classes $[\mathcal{A}]$ of flat $q\times q$ matrix bundles $\mathcal{A}$ (or projectively flat rank-$q$ complex vector bundles $\mathcal{E}$) on a pro-torus $\mathbb{T}$ are in bijective correspondence with the \v{C}ech cohomology group $H^2(\mathbb{T},\mu_q:=\text{$q^{th}$ roots of unity})$ (respectively $H^2(\mathbb{Z})$) via the image of $[A]\in H^1(\mathbb{T},PGL(q,\mathcal{C}_{\mathbb{T}}))$ through $H^1(\mathbb{T},PGL(q,\mathcal{C}_{\mathbb{T}}))\xrightarrow{\quad}H^2(\mathbb{T},\mu(q,\mathcal{C}_{\mathbb{T}}))$ (respectively the first Chern class $c_1(\mathcal{E})$). This is in the spirit of Auslander-Szczarba's result identifying real flat bundles on the torus with their first two Stiefel-Whitney classes, and contrasts with classifying spaces $B\Gamma$ of compact Lie groups $\Gamma$ (as opposed to $\mathbb{T}^n\cong B\mathbb{Z}^n$), on which flat non-trivial vector bundles abound. The discussion both recovers the Disney-Elliott-Kumjian-Raeburn classification of rational non-commutative tori $\mathbb{T}^n_{\theta}$ with a different, bundle-theoretic proof, and sheds some light on the connection between topological invariants associated to $\mathbb{T}^2_{\theta}$, $\theta\in\mathbb{Q}$ by Rieffel and respectively H{\o}egh-Krohn-Skjelbred.

These lecture notes are the product of a week-long learning workshop on the work of Johnson-Freyd and Reutter on the problem of the existence of minimal nondegenerate extensions of braided fusion categories (arXiv:2105.15167). They recount the mathematical arguments of the original paper from an expository angle, with background material covering the algebra and homotopy theory required to understand the statement and follow the proof. The notes are aimed at newcomers to the field of (braided) fusion 1- and 2-categories.

We use a 2-categorical version of (de-)equivariantization to classify (3+1)d topological orders with a finite $G$-symmetry. In particular, we argue that (3+1)d fermionic topological order with $G$-symmetry correspond to $\mathbf{2SVect}$-enriched $G$-crossed braided fusion 2-categories. We then show that the categorical data necessary to define these theories agrees with that arising from a fermionic generalization of the Wang-Wen-Witten construction of bosonic topological theories with $G$-symmetry saturating an anomaly. More generally, we also explain how 2-categorical (de-) equivariantization yields a classification of all braided fusion 2-categories.

Recently, the first author with A. Ardehali, M. Lemos, and L. Rastelli introduced the notion of graded unitarity for vertex algebras. This generalization of unitarity is motivated by the SCFT/VOA correspondence and introduces a novel Hilbert space structure on the state space of a large class of vertex algebras that are not unitary in the conventional sense. In this paper, we study the relative semi-infinite cohomology of graded-unitary vertex algebras that admit a chiral quantum moment map for an affine current algebra at twice the critical level. We show that the relative semi-infinite chain complex for such a graded-unitary vertex algebra has a structure analogous to that of differential forms on a compact K\"ahler manifold, generalizing a strong form of the classic construction of Banks--Peskin and Frenkel--Garland--Zuckerman. We deduce that the relative semi-infinite cohomology is itself graded-unitary, which establishes graded unitarity for a large class of vertex operator algebras arising from three- and four-dimensional supersymmetric quantum field theories. We further establish an outer USp$(2)$ action on the semi-infinite cohomology (which does not respect cohomological grading), analogous to the Lefschetz $\mathfrak{sl}(2)$ in K\"ahler geometry. We also show that the semi-infinite chain complex is quasi-isomorphic as a differential graded vertex algebra to its cohomology, in analogy to the formality result of Deligne--Griffiths--Morgan--Sullivan for the de Rham cohomology of compact K\"ahler manifolds. We conclude by observing consequences of these results to the associated Poisson vertex algebras and related finite-type derived Poisson reductions.

The adjunction between coalgebras and Hopf algebras, first described by Takeuchi, allows one to prove that the semi-abelian category of cocommutative Hopf algebras has enough $\mathcal E$-projective objects with respect to the class $\mathcal{E}$ of cleft extensions. One then proves that, for any cocommutative Hopf algebra, there exists a weak $\mathcal{E}$-universal normal (=central) extension. This fact allows one to apply the methods of categorical Galois theory to classify normal $\mathcal{E}$-extensions and to provide an explicit description of the fundamental group of a cocommutative Hopf algebra in terms of a generalized Hopf formula. Moreover, with any cleft extension, we associate a 5-term exact sequence in homology that can be seen as a Hopf-theoretic analogue of the classical Stallings-Stammbach exact sequence in group theory.

It is well-known that an averaging operator on a commutative associative algebra gives rise to a perm algebra. This paper lifts this process to the level of bialgebras. For this purpose, we first give an infinitesimal bialgebra structure for averaging commutative associative algebras and characterize it by double constructions of averaging Frobenius commutative algebras. To find the bialgebra counterpart of perm algebras that is induced by such averaging bialgebras, we need a new two-part splitting of the multiplication in a perm algebra, which differs from the usual splitting of the perm algebra (into the pre-perm algebra) by the characterized representation. This gives rise to the notion of an averaging-pre-perm algebra, or simply an apre-perm algebra. Furthermore, the notion of special apre-perm algebras which are apre-perm algebras with the second multiplications being commutative is introduced as the underlying algebra structure of perm algebras with nondegenerate symmetric left-invariant bilinear forms. The latter are also the induced structures of symmetric Frobenius commutative algebras with averaging operators. Consequently, a double construction of averaging Frobenius commutative algebra gives rise to a Manin triple of special apre-perm algebras. In terms of bialgebra structures, this means that an averaging commutative and cocommutative infinitesimal bialgebra gives rise to a special apre-perm bialgebra.

We investigate the annihilator condition $(a.c.)$ for skew Poincar\'e-Birkhoff-Witt extensions. We prove that some results about the annihilator condition $(a.c.)$ for skew polynomial rings also hold for skew PBW extensions. We also demonstrate that the behavior of annihilators is determined by the defining relations of the extension. Our results extend those corresponding presented for skew polynomial rings

This paper studies the connection between the quantum trace map -- which maps the $\mathfrak{sl}_2$-skein module to the quantum Teichm\"uller space for surfaces and to the quantum gluing module for 3-manifolds -- and the quantum UV-IR map -- which maps the $\mathfrak{gl}_2$-skein module to the $\mathfrak{gl}_1$-skein module of the branched double cover. We show that the two maps are compatible in a precise sense, and that the compatibility map is natural under changes of triangulation; for surfaces, this resolves a conjecture of Neitzke and Yan. As a corollary, under a mild hypothesis on the 3-manifold, the quantum trace map can be recovered from the quantum UV-IR map, hence providing yet another construction of the recently introduced 3d quantum trace map.

We study the q-characters and modular data of exceptional W-algebras and give several examples and applications. We establish equality of q-characters and modular data between certain boundary W-algebras, leading in particular to a largely complete determination of fusion rules of exceptional W-algebras in type A.

This survey reviews recent advances connecting link homology theories to invariants of smooth 4-manifolds and extended topological quantum field theories. Starting from joint work with Morrison and Walker, I explain how functorial link homologies that satisfy additional invariance conditions become diagram-independent, give rise to braided monoidal 2-categories, extend naturally to links in the 3-sphere, and globalize to skein modules for 4-manifolds. Later developments show that these skein lasagna modules furnish invariants of embedded and immersed surfaces and admit computation via handle decompositions. I then survey structural properties, explicit computations, and applications to exotic phenomena in 4-manifold topology, and place link homology and skein lasagna modules within the framework of extended topological quantum field theories.

We introduce a modified version of the necklace Lie bialgebra associated to a quiver, in which the bracket and cobracket insert (rather than remove) pairs of arrows in involution. This structure is then related to canonical quartic Poisson/Batalin-Vilkovisky structures on suitable representation varieties of the quiver. Constructions on the representation side take place in symmetric monoidal $\Pi$-categories, which prompts a discussion of graded differential operators on commutative monoids in any such category. The generality of the categorical approach allows us to fully recover necklace structures, showing how the modified and classical necklace operations are related via dualisability.

Let $\mathbb V$ be an $\mathbb N$-graded, $C_2$-cofinite vertex operator algebra (VOA) admitting a non-lowest generated module in $\mathrm{Mod}(\mathbb V)$ (e.g., the triplet algebras $\mathcal{W}_p$ for $p\in \mathbb{Z}_{\geq 2}$ or the even symplectic fermion VOAs $SF_d^+$ for $d\in \mathbb{Z}_+$). We prove that, unlike in the rational case, the spaces of conformal blocks associated to certain $\mathbb V$-modules do not form a vector bundle on $\overline{\mathcal{M}}_{0,N}$ for $N\geq 4$ by showing that their dimensions differ between nodal and smooth curves. Consequently, the sheaf of coinvariants associated to these $\mathbb V$-modules on $\overline{\mathcal{M}}_{0,N}$ is not locally free for $N\geq 4$. It also follows that, unlike in the rational case, the mode transition algebra $\mathfrak A$ introduced by Damiolini-Gibney-Krashen is not isomorphic to the end $\mathbb E=\int_{\mathbb X\in \mathrm{Mod}(\mathbb X)}\mathbb X\otimes \mathbb{X}'$ as an object of $\mathrm{Mod}(\mathbb{V}^{\otimes 2})$.