Papers

We prove the Kazhdan-Lusztig correspondence for the Lie superalgebra $\mathfrak{osp}_{1|2n}$ at irrational levels, that is, we show the category $\mathrm{KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ of finite-length even ordinary modules for the affine vertex operator superalgebra of $\mathfrak{osp}_{1|2n}$ at level $k \in \mathbb{C} \setminus \mathbb{Q}$ is braided tensor equivalent to the category of finite-dimensional even weight modules for the quantum group of $\mathfrak{osp}_{1|2n}$ at parameter $q = e^{\pi i/(2k+2n+1)}$. We also prove that ${\rm KL}_k^{\rm ev}(\mathfrak{osp}_{1|2n})$ is braided tensor equivalent to the category ${\rm KL}_\ell^{\rm ns}(\mathfrak{so}_{2n+1})$ of finite-length ordinary modules with non-spinorial top level for the affine vertex operator algebra of $\mathfrak{so}_{2n+1}$ at level $\ell$ such that $ \frac{1}{\ell+ 2n-1} = \frac{1}{2k+2n+1} + 1 \ \ ({\rm mod}\ 2\mathbb Z).$ Consequently, by gluing vertex operator (super)algebras via tensor categories, we construct a few new families of simple conformal vertex (super)algebras, including the mixed kernel VOAs that were the missing ingredient for proving certain Feigin-Frenkel type dualities in previous work of the first-named author with Linshaw, Nakatsuka, and Sato.

We study the quantized six-vertex (Q6V) model on arbitrary admissible diagrams at $q=\varepsilon$, where $\varepsilon$ is a primitive $N$th root of unity with $N$ odd. For the commuting layer transfer matrices $\mathbb{T}(y)$ with mixed boundary conditions, we establish the quantum Frobenius property \[ \mathbb{T}(y)\mathbb{T}(\varepsilon y)\cdots \mathbb{T}(\varepsilon^{N-1}y) = \mathscr{P}_N(y^N)\mathbb{I}, \] where $\mathscr{P}_N$ is a scalar Laurent polynomial obtained directly from the monomial expansion of $\mathbb{T}(y)$. Our proof exploits the underlying three-dimensional integrability, in particular the tetrahedron equations and their consequences. The quantum Frobenius relation essentially determines the joint eigenvalues of the associated Hamiltonians up to multiplicities and shows that they take a free-parafermion form, thereby extending such structures from previously known one-dimensional examples to a broad class of genuinely two-dimensional quantized six-vertex models. The construction also admits independent local gauge choices and recovers the long-range-interacting $\tau_{\; 2}$ model, which includes the original free-parafermion model as a special case. We also apply the same framework to the relativistic quantum Toda chain at odd roots of unity and obtain the free-parafermion spectra for the relevant Q6V transfer matrices with fixed boundary states.

We provide a generators and relations presentation of pivotal pointed fusion categories, $Vec(G,\omega,\pi)$. Unlike the well-known skeletal model, our presentation is strict and allows multiple isomorphic objects. Our main tool is skein theory, which allows us to apply topological tools to understand the relations of morphisms in the category.

Let $k$ be an algebraically closed field of characteristic zero and $G$ a finite group. We realize the based Grothendieck ring $R_G$ of ${\mathcal{R}\mathit{ep}}(D(G))$-module categories as the degree-two cocycle-decorated double Burnside ring and derive an explicit Clifford formula for multiplication and for the action of $R_G$ on the Grothendieck group of ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-module categories. We determine the extremal based ideals, study factorization through smaller groups, and prove a Mackey theorem for standard two-sided subgroup inductions. We prove that $\mathbb C\otimes_{\mathbb Z}R_G$ is semisimple exactly when $G$ is cyclic. We study the Brauer–Picard action on indecomposable ${\operatorname{\operatorname{\mathsf{Vec}}}}_G$-module categories and show that it is transitive exactly when $G$ is abelian of square-free exponent. For abelian $G$, we determine the possible abstract group types of Lagrangian subgroups of $G\oplus\widehat G$, apply the known orthogonal classification in the homocyclic case, and exhibit same-type nonconjugate Lagrangians for mixed exponents.

We investigate the maximal total quantum dimension at a bounded categorical rank cutoff $R$ in three-dimensional Dijkgraaf–Witten theories. This optimization problem physically governs the maximal state-optimized torus topological entanglement entropy at a fixed torus ground-state degeneracy, while providing a quantitative constraint on the search space required to classify these topological order phases. We analyze how topological twists (3-cocycles) and gauge-group structures independently govern this trade-off. For elementary-Abelian gauge groups at a fixed prime $p$, we prove that introducing cocycle twists elevates the maximum total quantum dimension from square-root growth $\Theta_p(R^{1/2})$ to linear growth $\Theta_p(R)$ in the rank cutoff, driven by the geometric protection of cyclic holonomies. For untwisted theories, we establish sharp envelopes for symmetric and alternating gauge groups, and derive group-theoretic bounds across almost-simple, solvable, and radical-free families. In particular, almost-simple groups and their direct products obey a quadratic-logarithmic upper envelope: $\log\mathcal{D} = O((\log R)^2)$. Finally, we compare these scales against WZW models and discuss the status of the general quadratic-logarithmic envelope conjecture.

Let $\mathcal{C}$ be a differential $\mathbb{Z}/2$-graded category over $\mathbb{C}$. Its periodic cyclic homology $HH^{per}_*(\mathcal{C})$, when viewed as a vector bundle over the formal punctured disk, is equipped with a canonical connection $\nabla^{\mathcal{C}}_{\partial_t}$ called the Getzler-Gauss-Manin connection in the $t$-direction (or the categorical $t$-connection). Our main result is that when $\mathcal{C}$ is smooth and proper, this connection has a regular singularity at $t=0$ and quasi-unipotent monodromy, affirming a conjecture of Katzarkov-Kontsevich-Pantev \cite{KKP}. Our proof follows a reduction mod $p$ argument using a spreading out technique of Toën \cite{To} and a regularity criterion of Katz \cite{Ka1}. The main novelty is the proof of a multiplicative property of the $p$-curvature of $\nabla^{\mathcal{C}}_{\partial_t}$ through an interpretation in terms of the two-colored Kontsevich-Soibelman operad. We then explore two applications of the main result. First, we give an explicit description (under additional assumptions) of the non-commutative Hodge filtration on the periodic cyclic homology of a smooth proper d$(\mathbb{Z}/2)$g category, following a construction of Shklyarov \cite{Shk}. The second application, which is special to our particular method or proof, is an upper bound on the sizes of Jordan blocks of the monodromy of $\nabla^{\mathcal{C}}_{\partial_t}$, which simultaneously generalizes Scherk's local monodromy theorem for isolated hypersurface singularities \cite{Sche} and (partially) a recent result of Pomerleano-Seidel on the quantum connection of a closed monotone symplectic manifold \cite{PS2}. As a specialization, we show that the sizes of these Jordan blocks are bounded above by the diagonal dimension of $\mathcal{C}$ plus one.

In this note we investigate the construction of near-group fusion categories via building explicit representations of them in certain graph planar algebras. To achieve this we first give a presentation for a cyclic near-group fusion category. Our presentation is “subfactor-centric” in the sense that we use a Q-system as one of our generating morphisms. Using this presentation we then obtain a rational system of equations, a solution of which corresponds to a faithful embedding of a near-group fusion category into a certain 2-coloured graph planar algebra. We end the note by providing solutions to these systems of equations for the odd cyclic groups up to order 13. This gives an alternate construction to the Evans-Gannon construction of the corresponding near-group fusion categories.

We introduce and develop an integral diagrammatic framework for two new monoidal categories, the finite and affine B-web categories. They arise naturally from representation categories of type BCD Lie algebras, and can be viewed as a Brauer type generalization of the affine web category which we introduced earlier. Integral bases consisting of bubbled elementary diagrams are established for the finite and affine B-webs. A simplified presentation of these categories over a field of characteristic zero is also obtained.

We investigate the associated vertex operator algebras and Higgs branches of two closely related families of generalised Argyres–Douglas theories, arising from the twisted $A_{2N}$ and twisted $D_{N+1}$ series, respectively. We propose that the $A$-series theories are realised as nilpotent Higgsings of the $D$-series, and we assemble a variety of pieces of evidence for this proposal. A consequence is that the associated vertex operator algebras of the $A$-series are finite extensions of affine Kac–Moody vertex algebras at (non-boundary) admissible levels, rather than those Kac–Moody algebras themselves, and that their Higgs branches are in many cases double covers of nilpotent orbit closures. The $\mathbb{Z}_2$ gaugings of the $A$-series theories then furnish examples of unitary SCFTs whose associated vertex operator algebras are precisely affine Kac–Moody algebras at non-boundary admissible levels. We propose that these vertex algebras are equipped with a non-standard $\mathfrak{R}$-filtration, and comment on compatibility with graded unitarity.

For every odd $N$, we construct a Haagerup-Izumi (HI) category for $\mathbb Z/N$, using Barnes double-sine functions. Equivariantization gives near-group categories of type $((\mathbb Z/N)^2,N^2)$ for every odd $N$. We classify connected separable algebras up to Morita equivalence in such HI categories and describe their dual categories. These duals are often HI for noncyclic groups with nontrivial pointed associators. We prove the Evans-Gannon conjecture that the modular data of the center is always a smashed sum involving a finite metric group of order $N^2+4$ and recover this group and its quadratic form directly from the data of the HI category. We also give the HI equations and reconstruction in arbitrary characteristic and prove a Frobenius symmetry in characteristic two. Using this, we obtain the complete list of (untwisted, subfactor type) HI categories over $\mathbb{C}$ for odd $N\le 43$. Other applications include finding three pairwise non-Morita-equivalent $\mathbb Z/15$ categories whose centers have the same modular data. We show that the center of a $\mathbb Z/5$ HI category cannot be defined (i.e., admits no split ribbon form) over a cyclotomic field, disproving a conjecture of Davidovich-Hagge-Wang.

The Lie algebra of multiple zeta values is realized as the space $\operatorname{ARI}^{\operatorname{pol}}_{\underline{al}\ast \underline{il}}$ of alternal moulds whose swap is alternil up to a constant mould, equipped with the $ari$ bracket. In this paper, we study the larger space $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$ of alternil, swap-invariant bimoulds, which is conjecturally the Lie algebra for multiple $q$-zeta values. We propose an explicit formula for a Lie bracket $uri$ on this space. Moreover, we prove that the Lie algebra $\operatorname{ARI}^{\operatorname{pol}}_{\underline{al}\ast \underline{il}}$ embeds into $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$, with the $ari$ bracket translating directly into the $uri$ bracket. Finally, we examine the associated-depth graded of $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$, extending the known depth-graded setup for $\operatorname{ARI}_{\underline{al}\ast \underline{il}}^{\operatorname{pol}}$.

We introduce an analogue of Gröbner bases, or equivalently, Bergman's diamond lemma, for linear (super)monoidal categories. It gives a systematic way to study monoidal ideals and to prove basis theorems. The central notion is that of a monoidal Gröbner system and is based on higher linear rewriting theory. We then apply the theory to categories of affine Brauer type: linear (super)monoidal categories that have the same hom-basis as the affine Brauer category, but possibly distinct composition and tensor product. Our main result is a criterion for a category to be of affine Brauer type, reducing the question to an explicit list of local computations, which we partially implement in the computer algebra system FORM. This gives new combinatorial proofs of the basis theorems for the affine Brauer category, the nil-Brauer category and the affine VW supercategory, and yields new examples. In particular, we construct the odd nil-Brauer category, a conjectural supercategorification of the split $\imath$quantum group of rank one, and the quantized affine VW supercategory, whose non-affine part is the quantized periplectic Brauer category and which we expect to satisfy a higher quantum Schur–Weyl duality. Finally, we classify the categories of affine Brauer type admitting a sufficiently simple presentation.

In this paper we propose a universal way to define multivariate orthogonal polynomials for the positive branch of WLZZ matrix models. This allows us to generalize from the Gaussian case, previously considered in the literature, the strict superintegrability property and the Miwa deformation formulas.

In this paper, we investigate the relation between the category of 3-crossed modules and the category of Gray-type 4-groups. The notion of a 3-crossed module was first introduced by Arvasi et al., motivated by the question of what kind of algebraic structure completely encodes a homotopy 4-type. On the other hand, from the point of view that higher groups are equivalent to algebraic realizations of higher categories – as exemplified by the relationship between 2-crossed modules and Gray 3-groups established by Sarikaya–Ulualan – it had not been clear how the 3-crossed modules of Arvasi et al. relate to any higher category. In our previous paper, we proposed a new definition of a 3-crossed module and observed that it admits a natural interpretation in terms of higher categories. In this paper, we make this interpretation precise: we introduce a 4-category, which reduces to a semistrict braided monoidal 2-category when restricted to a single object and a single 1-morphism, and prove that the category of our 3-crossed modules is equivalent to the category of Gray 4-groups, defined as single-object versions of this 4-category in which all morphisms are invertible. Furthermore, while Gray 4-groups provide a conceptual framework for 4-dimensional higher structures, they are often computationally intractable. Our equivalence establishes 3-crossed modules as a concrete, group-theoretic calculus for Gray 4-groups, providing a powerful computational tool for studying surface knots, state-sum invariants, and higher gauge theories.

We construct a polynomial invariant of planar curves via skein relations, which generalizes Arnold's numerical invariant $\operatorname{St}$.

In this paper, we study strictly weakly integral fusion categories of rank 6 and give a complete classification of their Grothendieck rings. We obtain twelve candidate Grothendieck rings. Eight of them can be realized by fusion categories, while the remaining four are ruled out by necessary obstructions to categorification. Together with previously known results, this completes the classification of Grothendieck rings of strictly weakly integral fusion categories of rank at most 6.

The notion of extremal subsets was introduced by Assaf, Dranowski, and González as a natural generalization of Demazure crystals. In this paper, we show that the crystal basis of a finite-dimensional irreducible module over a quantum group of finite type has a unique maximal proper extremal subset, whose complement is the set of elements that generate the whole crystal basis as an extremal subset. We conjecture that this complement consists of all nonzero elements obtained by successively applying the lowering Kashiwara operators to a distinguished element, which is constructed from the lowest element of the crystal basis. We prove this conjecture in type A.

For braces, associativity of the $\star$-product is equivalent to two-sidedness. We extend this result to skew braces by proving that any two of the following conditions imply the third: two-sidedness, associativity of the $\star$-product, and centrality of $A\star A$ in the additive group. We then study two-sided skew braces with associative $\star$-product through the radical ring associated with their additive abelianization. This yields a canonical fibre-product description modulo an annihilator ideal, exact sequences relating higher $\star$-products to the powers of the radical ring, coincidence of the left, right and strong $\star$-series, and sharp bounds on their nilpotency indices. We also obtain an explicit description of the lower central series, classify the structures whose additive abelianization is infinite cyclic, and characterize when they are bi-skew.

We study regular $\mathbb{N}$-graded vertex operator algebras $V=\bigoplus_{n\geq 0}V_n$ whose weight-zero algebra $V_0$ is a nontrivial finite-dimensional local Gorenstein algebra, asking which structural features of strongly rational vertex operator algebras persist without the CFT-type condition $V_0=\mathbb{C}{\bf 1}$. For semisimple Lie subalgebras of the left Leibniz algebra $V_1$, assuming $\ker\left(\left.L(-1)\right|_{V_0}\right)=\mathbb{C}\mathbf{1}$, the product $u_1v$ induces an invariant symmetric bilinear form. Under $C_2$-cofiniteness and a nondegeneracy condition, each simple component with nonzero form generates an affine vertex operator algebra at positive integral level and acts integrably on $V$. When $V_1$ is solvable, the Frobenius structure of $V_0$ yields a distinguished nondegenerate subspace $M\subset V_1$. Under a quasi-primary condition, $M$ is abelian and generates a Heisenberg vertex operator algebra. With additional semisimplicity, full-rank integrality, and cocycle compatibility assumptions, $V$ contains a conformally embedded lattice vertex operator algebra $V_K$, where $K$ is positive-definite and even, with rank $\dim_{\mathbb{C}}M$ and minimum norm at least $4$. Conformally shifted lattice vertex operator algebras provide explicit models illustrating the distinction among weight-one Lie, mode-generated Lie, and lattice structures. They also show that regularity alone does not ensure semisimplicity of arbitrary Heisenberg zero-mode actions, so the additional lattice-theorem hypotheses represent genuine structural obstructions.

We construct explicit examples of skew braces $A$ for which the third term $A^3$ in the left series is not generated by the right-normed star products $a_1*(a_2*a_3)$ of length $3$ as a sub-skew brace.

We prove that Stark–Shintani ray class invariants (Stark units) associated to real quadratic fields are algebraic numbers. These invariants are given by special values of Faddeev's modular quantum dilogarithm, introduced by Garoufalidis–Kashaev–Zagier. Our main discovery is that special values of the modular quantum dilogarithm satisfy an explicit overdetermined system of polynomial equations, matching a variation on the defining equations of Andersen–Kashaev's notion of a quantum dilogarithm on a product of two cyclic groups. We give two and a half proofs that this system of equations defines a zero-dimensional variety. The simplest follow from an uncertainty principle for finite Fourier transform and $2$-adic valuation bounds. The last proof is more involved and shows finite quanatum dilogarithms can be used to categorify fusion rings introduced by Izumi, and the algebraicity of the special values then follows by Ocneanu's rigidity theorem. As a byproduct, we obtain an explicit infinite family of irrational near-group fusion categories. As a further application, we prove a family of quadratic relations for Stark units recently conjectured by Appleby, Flammia, and Kopp motivated by Zauner's conjecture about SIC-POVMs (complex equiangular lines).

The main purpose of this paper is to provide a cohomology theory of restricted Lie triple systems in low degrees and their algebraic interpretations. We define restricted cochains up to degree 5, together with the corresponding coboundary operators. Moreover, we provide interpretations of the low-degree restricted cohomology groups in terms of restricted derivations, extensions of restricted modules, formal deformations, and extensions of restricted Lie triple systems.

Recent work has constructed Abelian Chern-Simons theories as categorical TQFTs, allowing us to naturally incorporate categorical defects and construct defect extensions of Abelian Chern-Simons TQFTs. We first identify the Turaev-Viro realizations of Abelian Chern-Simons theory in the center and doubled pointed modular cases, clarifying the distinction between single bulk realizations and canonical doubled ones. Then we obtain Turaev-Viro partition functions in terms of Reidemeister-torsion densities. Alternatively, the Alterfold construction supplies the associated topological boundaries, domain walls, and condensation sectors, establishing an explicit Alterfold/Chern-Simons dictionary. We show that the finite quadratic module is the invariant controlling the bulk theory, its topological symmetries, orientation-reversal invariance, and defects. We further show that multicomponent Abelian BF theory arises as the extended TQFT of an off-diagonal Abelian Chern-Simons theory, placing it naturally within the same extended framework. Finally, we demonstrate that recently proposed Abelian Chern-Simons dualities do not define a genuine TQFT duality. These results provide a concrete model for defects in Abelian topological orders and suggest a route toward the non-Abelian case.

We study the quantum $K$-theory of the flag variety $G/B$. For each minuscule fundamental weight $\varpi$, we construct an explicit relation in the torus-equivariant quantum $K$-theory $QK_T(G/B)$. The relation can be regarded as a quantum deformation of the character of the irreducible representation with highest weight $\varpi$.

Let $V$ be a graded vertex operator coalgebra and $\mathcal{C}$ the category of admissible $V$-comodules. We first give the construction of contragredient comodule of an admissible $V$-comodule. We then introduce cointertwining operators among admissible $V$-comodules. Using cointertwining operators, we define the cotensor decomposition of an admissible $V$-comodule by a universal property. Finally, we prove that the complexified Grothendieck group $\mathbb{K}[\mathcal{C}]=\mathbb{C}\otimes_{\mathbb{Z}}K[\mathcal{C}]$ is a cocommutative coassociative coalgebra. Moreover, this coalgebra is naturally dual to the fusion algebra of the category of admissible modules over the dual vertex operator algebra $V'$.

The article arXiv:2410.21454 constructed a braided $\mathrm{W^*}$-tensor category of superselection sectors associated to a net of von Neumann algebras on a suitably geometric poset using localized and transportable endomorphisms. In this article, we construct an equivalent tensor product and braiding using the Connes fusion tensor product without localizing, using ideas from arXiv:1812.04470 for conformal nets. As an application, we prove that the category of superselection sectors associated to the 2D braided quantum spin system constructed from a unitary braided fusion category $\mathcal{B}$ in arXiv:2506.19969 is equivalent to $\mathsf{Hilb}(\mathcal{B})^{\rm op}$, the opposite of the unitary ind-completion of $\mathcal{B}$, as a braided $\mathrm{W^*}$-tensor category.

In 2020 M. Mucciconi and L. Petrov introduced a long-range deformation of the quantum open non-relativistic Toda system. We prove the integrability of the deformed Toda system by constructing a $2 \times 2$ Lax operator, which produces the commutative family of differential operators containing the Hamiltonian of the deformed Toda system. Moreover, we show that the same integrable deformation exists on both classical and quantum levels and can be applied to both non-relativistic and relativistic Toda systems. For the open non-relativistic deformed Toda systems we also present an $n \times n$ Lax matrix and prove that it produces the same family of Hamiltonians. We also show how to obtain the van Diejen-type deformed Toda system. Lastly, we show that on the quantum level the algebraic Bethe ansatz technique can be applied to the deformed Toda system.

We compute the groups of minimal nondegenerate extensions of the real symmetric fusion categories $\mathrm{sVec}_{\mathbb R}$ and $\mathrm{sVec}_{\mathbb H}$. We find that they are both isomorphic to the Klein-four group $(\mathbb{Z}/2\mathbb{Z})^2$. Along the way, we classify all nondegenerately braided fusion categories over $\mathbb R$ that have Frobenius-Perron dimension 4, and we determine exactly which of these categories is a Drinfeld center. We end the paper by discussing a homotopy-theoretic conjecture that organizes all these structures.

We introduce the Hopf image of a right coaction of a Hopf algebra on an algebra as the smallest Hopf subalgebra through which the coaction factors. We establish its basic properties and show that the induced coaction on the Hopf image is inner-faithful. For coactions on Nichols algebras arising from Yetter–Drinfeld modules, we prove that the Hopf image is the smallest Hopf subalgebra detected by the underlying coaction on the Yetter–Drinfeld module. We give explicit examples of proper non-trivial Hopf images arising from bosonizations.

Rump asked in 2016 whether every finite quasi-linear cycle set is retractible. We reinterpret this question in terms of automorphisms of certain pointed magmas, and propose a conjecture on the automorphisms of these magmas which would imply an affirmative answer to Rump's question. We provide some theoretical and computational evidence in support of our conjecture.