CyberSec Research

We establish an identification between the spaces of $\alpha$-fusion trees in non-semisimple topological quantum computation (NSS TQC) and a family of homological representations of the braid group known as the Lawrence representations specialized at roots of unity. Leveraging this connection, we provide a new proof of Ito's colored Alexander invariant formula using graphical calculus. Inspired by Anghel's topological model, we derive a formula involving the Hermitian pairing of fusion trees. This formula verifies that non-semisimple quantum knot invariants can be explicitly encoded via the language of fusion trees in the NSS TQC mathematical architecture.

In a series of papers we have considered a non-stationary difference equation which was originally discovered for the deformed Virasoro conformal block. The equation involves mass parameters and, when they are tuned appropriately, the equation is regarded as a quantum KZ equation for $U_q(A_{1}^{(1)})$. We introduce a $\widehat{\mathfrak{gl}}_N$ generalization of the non-stationary difference equation. The Hamiltonian is expressed in terms of $q$-commuting variables and allows both factorized forms and a normal ordered form. By specializing the mass parameters appropriately, the Hamiltonian can be identified with the $R$-matrix of the symmetric tensor representation of $U_q(A_{N-1}^{(1)})$, which in turn comes from the 3D (tetrahedron) $R$-matrix. We conjecture that the affine Laumon partition function of type $A_{N-1}^{(1)}$ gives a solution to our $\widehat{\mathfrak{gl}}_N$ non-stationary difference equation. As a check of our conjecture, we work out the four dimensional limit and find that the non-stationary difference equation reduces to the Fuji-Suzuki-Tsuda system.

Given a vertex operator algebra $ V $ with a general automorphism $ g $ of $ V $, we introduce a notion of $ C_n $-cofiniteness for weak $ g $-twisted $ V $-modules. When $ V $ is $ C_2 $-cofinite and of CFT type, we show that all finitely-generated weak $ g $-twisted $ V $-modules are $ C_n $-cofinite for all $ n \in \mathbb{Z}_{>0} $.

We explore 4-Legendrian rack structures and the effectiveness of 4-Legendrian racks to distinguish Legendrian knots. We prove that permutation racks with a 4-Legendrain rack structure cannot distinguish sets of Legendrian knots with the same knot type, Thurston-Bennequin number, and rotation number.

We appeal to the theory of Jouanolou torsors to model the coherent cohomology of configuration spaces of points in d-dimensional affine space. Using this model, we develop the operadic notion of chiral operations, thus generalizing the notion of chiral algebras of Beilinson and Drinfeld to higher dimensions. To produce examples, we use a higher-dimensional conceptualization of the residue which is inspired by Feynman graph integrals.

We prove the statement in the title, solving in this way a conjecture stated by Ginot for manifolds with corners. Along the way, we establish a derived Swiss-cheese additivity theorem and an alternative proof for the hyperdescent of factorization algebras over those manifolds.

The goal of this paper is twofold: (i) define a symplectic Khovanov type homology for a transverse link in a fibered closed $3$-manifold $M$ (with an auxiliary choice of a homotopy class of loops that intersect each fiber once) and (ii) give conjectural combinatorial dga descriptions of surface categories that appear in (i). These dgas are higher-dimensional analogs of the strands algebras in bordered Heegaard Floer homology, due to Lipshitz-Ozsv\'ath-Thurston \cite{LOT}.

This paper has two main objectives. The first one is to show that the Connes formulation of Dirac theory can be applied in the framework of quantum principal bundles for any n dimensional spectral triple, any quantum group, any quantum principal connection and any finite dimensional corepresentation of the quantum group. The second objective is to demonstrate that, under certain conditions, one can define a Yang Mills functional that measures the squared norm of the curvature of a quantum principal connection, in contrast to the Yang Mills functional proposed by Connes, which measures the squared norm of the curvature of a compatible quantum linear connection. An illustrative example based on the noncommutative n torus is presented, highlighting the differences and similarities between the two functionals.

According to the Hall algebras of quivers with automorphisms under Lusztig's construction, the polynominal forms of several structure coefficients for quantum groups of all finite types are presented in this note. We first provide a geometric realization of the coefficients between PBW basis and the canonical basis via standard sheaves on quiver moduli spaces with admissible automorphisms. This realization is constructed through Lusztig sheaves equipped with periodic functors and their modified Grothendieck groups. Second, within this geometric framework, we present an alternative proof for the existence of Hall polynomials originally due to Ringel. Finally, we give a slight generalization of the Reineke-Caldero expression for the bar involution of PBW basis elements in symmetrizable cases. When the periodic functor $\mathbf{a}^*$ is taken $\operatorname{id}$, our results are the same as Lusztig's and Caldero-Reineke's.

We study the representation theory of various convolution algebras attached to the $q$-deformation of $\mathrm{SL}(2,\mathbb{R})$ from an algebraic perspective and beyond the unitary case. We show that many aspects of the classical representation theory of real semisimple groups can be transposed to this context. In particular, we prove an analogue of the Harish-Chandra isomorphism and we introduce an analogue of parabolic induction. We use these tools to to classify the (non-unitary) irreducible admissible representations of $q$-deformed $\mathrm{SL}(2,\mathbb{R})$. Moreover, we explicitly show how these irreducible representations converge to the classical admissible dual of $\mathrm{SL}(2,\mathbb{R})$. For that purpose, we define a version of the quantized universal enveloping algebra defined over the ring of analytic functions on $\mathbb{R}_+^*$, which specializes at $q = 1$ to the enveloping $\ast$-algebra of $\mathfrak{sl}(2,\mathbb{R})$ .

Behrend, Liao, and Xu showed that differential graded (DG) manifolds of positive amplitude forms a category of fibrant objects. In particular, this ensures that notion of derived intersection -- more generally, homotopy fibre product -- is well-defined up to weak equivalences. We prove that the Atiyah and Todd classes of DG manifolds of positive amplitude are invariant under the weak equivalences. As an application, we study Hochschild cohomology of DG manifolds of positive amplitude defined using poly-differential operators, which is compatible with Kontsevich formality theorem and Duflo--Kontsevich-type theorem established by Liao, Sti\'enon and Xu. We prove that this Hochschild cohomology is invariant under weak equivalences.

Let $U$ be a set of positive roots of type $ADE$, and let $\Omega_U$ be the set of all maximum cardinality orthogonal subsets of $U$. For each element $R \in \Omega_U$, we define a generalized Rothe diagram whose cardinality we call the level, $\rho(R)$, of $R$. We define the generalized quantum Hafnian of $U$ to be the generating function of $\rho$, regarded as a $q$-polynomial in $U$. Several widely studied algebraic and combinatorial objects arise as special cases of these constructions, and in many cases, $\Omega_U$ has the structure of a graded partially ordered set with rank function $\rho$. A motivating example of the construction involves a certain set of $k^2$ roots in type $D_{2k}$, where the elements of $\Omega_U$ correspond to permutations in $S_k$, the generalized Rothe diagrams are the traditional Rothe diagrams associated to permutations, the level of a permutation is its length, the generalized quantum Hafnian is the $q$-permanent, and the partial order is the Bruhat order. We exhibit many other natural examples of this construction, including one involving perfect matchings, two involving labelled Fano planes, and one involving the invariant cubic form in type $E_6$.

We construct a crystal base of the negative half of a quantum orthosymplectic superalgebra. It can be viewed as a limit of the crystal bases of $q$-deformed irreducible oscillator representations. We also give a combinatorial description of the embedding from the crystal of a $q$-oscillator representation to that of the negative half subalgebra given in terms of a PBW type basis. It is given as a composition of embeddings into the crystals of intermediate parabolic Verma modules, where the most non-trivial one is from an oscillator module to a maximally parabolic Verma module with respect to a quantum subsuperalgebra for $\mathfrak{gl}_{m|n}$. A new crystal theoretic realization of Burge correspondence of orthosymplectic type plays an important role for the description of this embedding.

In arXiv:2503.19532 new examples of ribbon Hopf algebras based on the construction due to Nenciu were presented. This piece serves as a sequel where we study the representation theory of these new examples of ribbon Hopf algebras. We classify indecomposable projective and simple modules, find the Krull-Schmidt decomposition of the adjoint representation of Nenciu algebras, and prove fusion rules between its components. We also comment on the properties of M\"uger centres of their representation categories, in particular that they can be non-semisimple. Finally, we consider a new family of ribbon Hopf algebras over fields of prime characteristic $p>2$ in the context of 4-dimensional TQFTs presented in arXiv:2306.03225 that constitute an improvement over examples given therein, although still seemingly falling short of producing powerful invariants of 4-manifolds.

We calculate the Roger-Yang skein algebra of the annulus with two interior punctures, $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, and show there is a surjective homomorphism from this algebra to the Kauffman bracket skein algebra of the closed torus. Using this homomorphism, we characterize the irreducible, finite-dimensional representations of $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, showing that they can be described by certain complex data and that the correspondence is unique if certain polynomial conditions are satisfied. We also use the relationship with the skein algebra of the torus to compute structural constants for a bracelets basis for $ \mathcal S^{RY}(\Sigma_{0, 2, 2})$, giving evidence for positivity.

We show a variation of the usual homological freeness criterion for operadic modules over a Koszul operad. We then apply this result to decorated partition posets for some operads, showing that their augmentation is Cohen-Macaulay and computing its homology. This work answers several open questions asked by B\'er\'enice Delcroix-Oger and Cl\'ement Dupont in a recent article.

We propose a new construction of vertex operators of the elliptic quantum toroidal algebra $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$ by combining representations of the algebra and formulas of the elliptic stable envelopes for the $A^{(1)}_{N-1}$ quiver variety ${\cal M}(v,w)$. Compositions of the vertex operators turn out consistent to the shuffle product formula of the elliptic stable envelopes. Their highest to highest expectation values provide K-theoretic vertex functions for ${\cal M}(v,w)$. We also derive exchange relation of the vertex operators and construct a $L$-operator satisfying the $RLL=LLR^*$ relation with $R$ and $R^*$ being elliptic dynamical $R$-matrices defined as transition matrices of the elliptic stable envelopes. Assuming a universal form of $L$ and defining a comultiplication $\Delta$ in terms of it, we show that our vertex operators are intertwining operators of the $U_{t_1,t_2,p}(\mathfrak{gl}_{N,tor})$-modules w.r.t $\Delta$.

We investigate models of algebraic theories in the category of cocommutative coalgebras over a field. We establish some of their categorical properties, similar to those of algebraic varieties. We introduce a class of categories of coalgebraic models of algebraic theories endowed with an underlying structure of cocommutative Hopf algebra, and show that these categories are semi-abelian. We call them ``categories of Omega-Hopf algebras'', since it is possible to characterize them as coalgebraic models of algebraic theories of Omega-groups.