Papers

For a vertex operator algebra $V$ and a certain category of its modules, we propose a construction for spaces of conformal blocks organized into an open-closed modular functor with singularities. This is inspired by the idea of implementing directly from the start the principle of holomorphic factorization. More precisely, using the strategy of modular extension introduced by Costello and developed further in our previous work, we build for each surface $\Sigma$ with at least one boundary component per path component and specified boundary labels attached to marked intervals or boundary circles a representation $\Omega_V(\Sigma;-)$ of the mapping class group of $\Sigma$. The construction can be described explicitly on generating Dehn twists. This approach is a priori independent from other constructions based on algebraic geometry or topological techniques involving e.g. surgery, but we include an overview over the available comparisons. In the special case in which the module category of $V$ is a not necessarily semisimple modular category $\mathcal{A}$, the spaces $\Omega_V(\Sigma)$ are equivalent to the string-net spaces for $\mathcal{A}$ and hence to the modular functor for the Drinfeld center $Z(\mathcal{A})\simeq \bar{\mathcal{A}}\boxtimes\mathcal{A}$. However, the construction of $\Omega_V$ in this paper has the advantage of being available beyond rationality, rigidity, self-contragredience and finiteness. Moreover, we prove that $\Omega_V$ satisfies excision, is finite-dimensional in the $C_2$-cofinite case and produces in genus one a generalization of the elliptic double of Brochier-Jordan. We prove for the triplet $\mathcal{W}_{2,3}$ with non-exact fusion product that the boundary conditions introduced by Gaberdiel-Runkel-Wood produce, as expected by these authors, correlation functions, provided that one uses the notion of a modular functor with singularities that we develop.

In this paper, we introduce a generalized framework for star products that preserve associativity while not necessarily obeying the canonical commutation relations. Within this framework, we formulate the augmented star product and investigate the associated augmented star exponentials. We further demonstrate several applications of these constructions to special functions and to problems arising in quantum physics.

We introduce the differential perm bialgebras by generalizing the study of perm bialgebras to the context of differential perm algebras. They are characterized by the Manin triple of differential perm algebras as well as the matched pairs of differential perm algebras. The quasi-triangular (resp. triangular, factorizable) differential perm bialgebra are introduced. In particular, symmetric solutions of an analogue of perm Yang-Baxter equation in differential perm algebras provide triangular differential perm bialgebras, whereas in turn the notions of $\mathcal{O}$-operators of differential perm algebras. As an application, the relationship between differential perm bialgebras and related bialgebra structures is discussed in detail. We show that there is a natural differential infinitesimal bialgebra structure on the tensor product of a differential perm bialgebra and a quadratic Zinbiel algebra, and this differential infinitesimal bialgebra structure is quasi-triangular (resp. triangular, factorizable) if the original differential perm bialgebra is quasi-triangular (resp. triangular, factorizable). Starting from a differential perm bialgebras with certain conditions, we can construct Novikov bialgebras, and these Novikov bialgebras can also be implemented by using the tensor product of diNovikov bialgebras and quadratic Zinbiel algebras.

We introduce the Quantum Information Space at Infinity (Quinfinity) $\mathcal{Q}_\infty$ as the inverse limit of the symbolic Quantum $N$-Spaces $\mathcal{Q}_N$, identified with the complete local real algebra of formal power series under the adic topology. The pure state pro-variety is formalized as the inverse limit of $N$-level pure state varieties, driven by the algebraic colimit of a direct system of real radical ideals. This real architecture internalizes the imaginary unit, collapsing the metric landscapes onto an invariant Riemannian isometry, which identifies the pro-variety directly with the classical Fisher-Rao statistical manifold. Within this framework, the Heisenberg uncertainty principle undergoes a structural regularization at infinity being intrinsically manifested as a localized truncation obstruction governed by adic algebraic derivations. Concurrently, the continuous Liouville-von Neumann equation operates as an internal derivation tangent to the pro-variety, while the open Gorini-Kossakowski-Sudarshan-Lindblad asymptotic master equation is intrinsically obtained via a non-associative symmetric Jordan product. This dissipative flow acts as a contracting radial vector field that dampens higher-order jet configurations, driving the state trajectories down the hierarchical tree toward the absolute zero element as a non-singular universal attractor.

A web is a plane graph encoding an invariant vector in a tensor product of fundamental representations of a quantum group. A stranding of an $\mathfrak{sl}_n$ web is a system of colored oriented curves recording one monomial of the vector it encodes. This article focuses on identifying and constructing leading term strandings, those recording the leading term of a web's vector with respect to a lexicographic order on monomials. We show that every open strand of a leading term stranding is clockwise, so the boundary data of such a stranding is recorded by a row-strict tableau. From a row-strict tableau, we construct a web with a prescribed leading term, and the resulting webs form a web basis. This non recursive construction recovers Fontaine's bases and produces others. For $\mathfrak{sl}_3$ webs with no flat vertices, we identify a leading term stranding using the depths of the faces of the web. Finally, we ask whether any two strandings of a web are connected by a sequence of local moves called strand reversals. We answer this question affirmatively for $\mathfrak{sl}_3$, extending a known result for closed webs to webs with boundary; in particular, a leading term stranding of any $\mathfrak{sl}_3$ web is always reachable from an arbitrary stranding.

Weak bialgebras are natural generalizations of bialgebras that play a fundamental role in quantum groups, tensor categories, and representation theory. In this paper, we systematically classify all (weak) bialgebra structures on the finite-dimensional semisimple algebra \(\Bbbk^{\oplus n}\), with a distinguished basis of primitive orthogonal idempotents. For the bialgebra case, we prove that such structures are in bijection with finite monoids, and that bialgebra isomorphisms correspond exactly to monoid isomorphisms. Extending the result to weak bialgebras, we show that every weak bialgebra structure uniquely determines a finite small category.

In this paper, we mainly provide some methods for constructing antisymmetric infinitesimal (ASI) bialgebras and Lie bialgebras using Zinbiel bialgebras and diassociative bialgebras. We first show that there is an ASI bialgebra structure on the tensor product of a diassociative bialgebra and a quadratic Zinbiel algebra, while there is an infinite-dimensional ASI bialgebra structure on the tensor product of a Zinbiel bialgebra and a quadratic $\bz$-graded diassociative algebra. For a special quadratic $\bz$-graded Leibniz algebra, the property that its tensor product with a Zinbiel bialgebra forms an ASI bialgebra characterizes the Zinbiel bialgebra. By examining the relationship between solutions of the (classical) Yang-Baxter equation in a Zinbiel algebra and the induced associative algebra, we prove that the induced ASI bialgebra is quasi-triangular (resp. triangular, factorizable) whenever the original Zinbiel bialgebra is quasi-triangular (resp. triangular, factorizable). These conclusions enable us to provide a method for constructing Lie bialgebras from Zinbiel bialgebras and two approaches for constructing Lie bialgebras from diassociative bialgebras. We also provide specific descriptions of the connections between the solutions of the Yang-Baxter equations and the connections between the $O$-operators corresponding to these constructions.

We establish a grading bound for colored $\mathfrak{gl}_N$ link homology that can be read off from any link diagram. In the context of contact geometry, this gives a bound on the Thurston–Bennequin number of a Legendrian link from $\mathfrak{gl}_N$ link homology, in the spirit of Ng.

We classify all characters of quantum symmetric pair coideal subalgebras for generalised Satake diagrams of finite and affine type.

We first present an introduction to the differential geometry of noncommutative algebras that carry a representation of a triangular Hopf algebra. Their noncommutativity is encoded in the universal R-matrix. The differential geometry is canonically constructed from these data. We then develop a new approach to the Bianchi identities for curvature and torsion of arbitrary connections (not necessarily bimodule connections). Using the Cartan calculus for connections, we prove the equivalence of their global formulations in terms of exterior forms and tensor fields. In particular, we obtain the noncommutative analogues of the first and second Bianchi identities familiar from general relativity.

We study the center of the quantum Schur superalgebra $\mathcal{S}_v(m|n,r)$ associated with the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$. Using the super Schur–Weyl duality due to Mitsuhashi between the quantum supergroup $U_v(\mathfrak{gl}_{m|n})$ and the Hecke algebra $\mathcal{H}_v(\mathfrak{S}_r)$, we transfer two known bases of the center of $\mathcal{H}_v(\mathfrak{S}_r)$, namely the Geck–Rouquier basis and the Jones basis, to the center of $\mathcal{S}_v(m|n,r)$. This yields two distinct bases for $\mathscr{Z}(\mathcal{S}_v(m|n,r))$, indexed respectively by the symmetrized hook set $H^{\vee}(m|n,r):=H(\min(m,n)\mid \max(m,n),r)$ and by the full $(m|n)$-hook set $H(m|n,r)$. Our approach relies on a detailed analysis of the ring $\Lambda_{m|n}$ of doubly symmetric polynomials satisfying $f|_{x_m=t=-y_n}$ independent of $t$, and of its power-sum bases.

Let $\mathbb{k}$ be a finite field and $\mathcal{A}, \mathcal{B}$ be $\mathbb{k}$-linear $\mathsf{Ext}$-finite hereditary abelian categories. A theorem of Cramer asserts that, under suitable assumptions, a derived equivalence $\mathcal{D}^b(\mathcal{A}) \!\longrightarrow\! \mathcal{D}^b(\mathcal{B})$ between two such categories induces an algebra isomorphism of the corresponding double Hall algebras $\mathsf{DH}_\mathcal{A} \!\longrightarrow\! \mathsf{DH}_\mathcal{B}$. It turns out that a counting formula for certain distinguished triangles in $\mathcal{D}^b(\mathcal{A})$, on which Cramer's proof relies, is incorrect in general. We give a corrected proof of Cramer's theorem which preserves the overall strategy of his approach.

For an oriented link diagram $D$, the Khovanov cochain complex carries a canonical Hermitian inner product that defines a combinatorial Hodge Laplacian. Its kernel is naturally isomorphic to rational Khovanov cohomology, while its positive spectrum depends on the chosen diagram. Building on the numerical work of Jones and Wei, we develop a structural study of this diagram-dependent higher spectrum. In the minimal and maximal cube degrees, we identify the Khovanov Laplacian with signless Laplacians of explicit weighted graphs, up to diagonal sign conjugation in maximal degree. The graph model describes rational Khovanov classes and harmonic representatives through its bipartite components and gives inverse-polynomial gap bounds in fixed-width bands near the extremal $q$-degrees when the corresponding rational cohomology vanishes. It also yields exact $\Theta(N^{-2})$ bidegree gaps for twisted unknots, odd $(2,N)$-torus knots, and even twist knots. The exact gap calculations are motivated in part by the spectral-resolution requirement in a recent quantum algorithm for Khovanov homology. We also prove a finite-dimensional analytic–combinatorial torsion correspondence for the Khovanov complex showing that an alternating product of nonzero Laplacian pseudodeterminants recovers integral Khovanov torsion in rationally acyclic $q$-degrees and differs from it by an explicit regulator in general. Finally, we transfer Lee's deformation to a filtered differential on the harmonic Khovanov subspace, giving a canonical harmonic model for the Lee spectral sequence.

We formulate split Galois descent for fusion categories through framed equivalences, reducing the existence of a split form to a group-theoretic splitting problem. The associated degree-two obstruction combines categorical coherence with stable descent of the simple objects, while fusion spaces impose index and parity restrictions. For quantum groups at roots of unity, we use the quasi-$R$-matrix to make the bar involution tensor-compatible and prove that the underlying fusion category $\mathcal C(\mathfrak g,k)$, associated with the simply connected root datum, has a split form over the maximal totally real subfield of the cyclotomic field generated by the quantum parameter. The same holds for all its cyclotomic Galois conjugates; in particular, every $\mathcal C(\mathfrak g,k)$ has a split real form. By contrast, no split real form exists for the standard braiding except on $\mathcal C(E_8,1)$ and $\mathcal C(D_{4m},1)$, $m \geq 1$. We further show that associative zesting can destroy real descent of $\mathcal C(\mathfrak g,k)$, while every braided zesting of $\mathcal C(\mathfrak g,k)$ still admits a split real form as an underlying fusion category. Applications to pointed and finite-group representation categories give criteria for real descent in terms of cohomology and Frobenius–Schur indicators.

The $F$-symbols of an anyon theory encode the associativity of fusion and constitute some of the theory's most fundamental and, simultaneously, subtle data. Much of the subtlety lies in the gauge-dependence of the $F$-symbols. Despite their generic complexity, many braided anyon theories admit gauges in which all $F$-symbols are real, a phenomenon for which no general organising principle has been known. We identify a physical mechanism underlying this reality. For a unitary ribbon fusion category admitting an appropriate braided charge-conjugation symmetry, we show that the complex-conjugated $F$-symbols are related to the original ones by a gauge transformation. Finding a real gauge is thereby reduced to a condition on these transformations. When the charge-conjugation symmetry is suitably "flat,” or equivalently when the associated "twisted” Frobenius-Schur (or "generalized” Kawanaka-Matsuyama) data respects a grading, these local transformations can be trivialised and a real gauge exists. This framework unifies a broad range of previously disparate examples, including families of Chern-Simons theories whose $F$-symbols are difficult to directly compute. Finally, we exhibit a unitary ribbon category that realizes a novel obstruction to the existence of a real gauge and therefore has inherently complex $F$-symbols.

We calculate the R-matrix for the exceptional Lie superalgebra $\mathfrak{d}(2,1;\kappa)$ in the smallest representation of its quantum supergroup, using a method of Hasse diagrams.

We report a Lean 4 formal audit accompanying the paper "Completing the Arakawa–Moreau Conjecture on Maximal Ideals of Affine Vertex Algebras" (arXiv:2607.25249). In a single Lean source, the development kernel-checks the paper-local deductive architecture used for the new maximal-ideal cases: affine-kernel detection and lifting, extremal PBW-line deductions, finite Ramond–Casimir arithmetic and case enumerations, reduced-simplicity contradiction schemes, and the case-level affine conclusions for the level -1 D_l family and the positive-parameter cases of D4, E6, E7, and E8. It also audits the paper's alternative D_l level -2 argument and the separate rank-reduction application of Section 8. Previously published exceptional n=0 cases are not claimed as newly formalized results of this bundle. The released project builds successfully with the pinned Lean/Mathlib environment and contains live #print axioms queries for the principal final wrappers. The development does not reconstruct affine vertex algebras, minimal W-algebras, BRST/DS reduction, Ramond Zhu theory, or Li spectral flow from foundational definitions in Mathlib. Those ingredients, together with the concrete realization of case-specific representation-theoretic objects, are exposed as theorem parameters and semantic interfaces. Accordingly, the precise verification claim is a kernel-checked deduction of the paper-local proof from explicitly stated VOA/BRST/DS boundary inputs, rather than a from-scratch formalization of the ambient representation theory.

We classify Satake diagrams for general linear and orthosymplectic non-twisted affine Lie superalgebras and prove that each of them generates a family of proper spherical subalgebras. Furthermore, we demonstrate that every such family contains subalgebras with matrix invariants, which are viewed as classical analogs of K-matrices solving supersymmetric Reflection equation.

We construct Wakimoto-type free field realizations of $N=1$ SUSY affine vertex algebras associated with basic classical Lie superalgebras. Using the superfield formalism, we obtain a SUSY analogue of the Wakimoto realization in terms of $\beta\gamma$–$bc$ systems and boson–fermion systems (or SUSY Heisenberg vertex algebras). We also construct screening operators for the free field realizations. A distinctive feature of the SUSY setting is that, unlike the ordinary Wakimoto realization, the injectivity of the free field homomorphism depends on the level. We prove that the homomorphism is injective for nondegenerate $\kappa$, whereas at $\kappa=0$ it fails to be injective. Thus, the SUSY Wakimoto realization exhibits a degeneration at $\kappa=0$ that has no direct counterpart in the ordinary affine case.

In 1987, Woronowicz asked whether the linear span of irreducible characters is norm dense in the cocommutative part of the ambient C-star algebra of a compact quantum group. After thirty-nine calendar years, we give a negative answer to the universal form of this question, even for compact matrix quantum groups of Kac type. Starting from an infinite finitely generated simple group with property T and a finite bicharacter twist, we construct a compact quantum group whose universal C-star algebra is a full group C-star algebra. A Kazhdan projection is shown to remain cocommutative under the twisted coproduct, while a vector state arising from an induced representation separates this projection from every algebraic cocommutative element. Quantitatively, the distance from the Kazhdan projection to the closed linear span of irreducible characters is at least one half. Moreover, the projection lies in the kernel of the reducing morphism. The obstruction is therefore genuinely universal and disappears after passage to the reduced compact quantum group, where the known character-density theorem remains valid. The construction identifies a sharp boundary between universal and reduced character theory and shows that Kac symmetry alone does not control cocommutative elements in the universal completion.

We study quantum shadow inversion under Markovian locality constraints for four-partite systems arranged along the chain $A$–$B$–$C$–$D$. The goal is to reproduce the expectation value of a fixed endpoint observable $O_{AD}$ after an unknown global unitary, without requiring full unitary inversion. We formulate the task using Markov-admissible supermaps and introduce the Markov-implementable centralizer to describe the remaining endpoint gauge freedom. We show that unrestricted endpoint post-processing is too broad, and impose an endpoint-local refinement. Under this condition, every implementable endpoint unitary must factorize across $A|D$, so the Markov constraint strictly reduces the centralizer-induced shadow ambiguity whenever the full centralizer contains non-product unitaries. This provides a structural mechanism by which Markov locality enhances identifiability in quantum shadow inversion.

We define a $2\pi/3$-rotationally invariant 6-valent vertex in the $\mathfrak{gl}_N$ web category. When $N = 4, 5$, we provide a categorification of the 6-valent vertex using $\mathfrak{gl}_N$ foams and decompose the hexagon web into a direct sum of indecomposables. A similar decomposition is conjectured for $N \geq 6$.

The Levin–Wen model provides a lattice realization of topological orders associated with a given unitary fusion category. A longstanding open problem is to characterize Levin–Wen ground-state wave functions intrinsically, without assuming a priori categorical symmetry data or a Hamiltonian. We address this by proposing six axioms on a family of wave functions defined on lattices at multiple scales. These axioms allow us to reconstruct the underlying unitary fusion category and prove that the resulting wave functions map to nonzero Levin–Wen ground-state vectors of the emergent category.

We introduce 3-group-graded extensions of fusion 2-categories, where 3-groups are modeled by 2-crossed modules. From this data, we derive a state-sum invariant of 4-manifolds equipped with a homotopy class of maps to a homotopy 3-type, or equivalently of flat 3-bundles over 4-manifolds. This invariant is nontrivial and generalizes the Douglas-Reutter invariant of 4-manifolds. To construct our invariant, we encode homotopy classes of maps to the classifying space of a 3-group $\sigma$ via $\sigma$-colorings of triangulations, and we generalize Pachner's theorem in this context.

Let $H$ be a Hopf algebra with a bijective antipode over a field $\mathbf k$ and $B/A$ be a flat right $H$-Galois extension. Ştefan constructed a spectral sequence converging to the Hochschild cohomology $\mathrm{HH}^{p+q}(B, N)$ with $\mathrm{E}_2^{p,q} = \mathrm{H}^p(H, \mathrm{HH}^q(A, N))$. We show that when $B/A$ is faithfully flat, the Ştefan spectral sequence is multiplicative. More precisely, a new formalism of functorial multiplicative spectral sequences via lax monoidal functors over monoidal categories is introduced. A functorial multiplicative spectral sequence is constructed for any faithfully flat Hopf Galois extension. Identification with Ştefan's spectral sequence from the $E_2$-page follows from Künzer's comparison theorems. Applications include strongly graded algebras, smash products, crossed products, group and Lie algebra extensions, with new multiplicativity results in several cases.

We report a Lean 4 formal verification accompanying the paper "Subregular Affine Cells and the Level -1 Vertex Algebra of Type D" (arXiv:2608.11997). The formalization kernel-checks substantial internal parts of the proof architecture, including the Section 4 membership/descent chain, the type-D norm-gap argument, zero-orbit energy and signed-permutation rigidity, node-weight and numerical rigidity calculations, the exhaustion logic, the passage to the candidate quotient classification, the simple-object count, the final additive Grothendieck-group comparison, and the coefficient-substitution layer of the uniform character formula. Higher representation-theoretic results whose foundational infrastructure is not presently constructed in the file are isolated as explicit semantic interfaces rather than introduced as Lean axioms. Thus the precise claim is a kernel-checked internal deduction from explicit representation-theoretic boundary inputs, not a from-scratch formalization of vertex algebras, BRST reduction, finite W-algebras, or affine Hecke theory inside Mathlib.

We study lattice vertex algebras of type ADE over an algebraically closed field $\mathbb{F}$ of prime characteristic $p>2$ and their representations. Let $L$ be a root lattice of type ADE, and $G_{L}$ the Gram matrix of $L$. When $\det G_{L}\not \equiv 0\pmod{p}$, we establish an isomorphism between the lattice vertex algebra $V_{L,\mathbb{F}}$ and the level-one simple affine vertex algebra of the same type. Via this isomorphism, we classify the irreducible $\mathbb{N}$-graded modules of $V_{L,\mathbb{F}}$ viewed as an $\mathbb{N}$-graded vertex algebra. We also consider the case where $L$ is of type $A_n$ with $\det G_L=n+1\equiv 0\pmod{p}$. We show that $V_{L,\mathbb{F}}$ is not simple and determine the simple $\mathbb{N}$-graded quotient of $V_{L,\mathbb{F}}$. Furthermore, when $n+1=ap$ for some $a\in \mathbb{Z}_+$ with $\gcd(a,p)=1$, we give the classification of the irreducible $\mathbb{N}$-graded modules for the simple quotient of $V_{L,\mathbb{F}}$.

Jin recently claims that the distinguished level $-1$ vacuum block of $L_{-1}(D_\ell)$ realizes the specialized dual affine left-cell module attached to the subregular cell containing $s_0$. We show that the proof of this Grothendieck-group statement contains a gap. We give an explicit counterexample over the discrete valuation ring $\Bbbk[[t]]$, showing that the categorical implication used in the proof is false in general. Consequently, Jin's Theorem~6.3 does not follow from the argument given in his Section~6.2. We then supply a complete replacement proof for the grading-restricted level-$-1$ vacuum block. Combining the classification of simple objects, we prove finite length, construct an injective signed normalized-character map into a completed singular orbit module, and identify its image with the specialized dual subregular left-cell module. Thus the cell-module realization claimed is established for the grading-restricted vacuum block.

We introduce the notion of a total inner product on an $A_\infty$-algebra, and a cyclic Chern-Simons form associated to a topologically nilpotent element. The total inner product applied to the cyclic Chern-Simons form gives a superpotential function that is gauge invariant on solutions of the Maurer-Cartan equation known as bounding cochains. The derivative of the superpotential is computed. The definition of total inner product replaces strict symmetries that appear in previous notions of homotopy inner products with symmetries up to an infinite family of coherent homotopies. We show how to recover previous notions of homotopy inner products as special cases of total inner products. Total inner products are designed to facilitate the definition of descendent open Gromov-Witten invariants. The definitions of the total inner product and the cyclic Chern-Simons form use the total complex of cyclic codifferential forms, which gives a chain model for cyclic homology. We give explicit formulas for homotopy equivalences with other known models. We discuss also the notion of an $\infty$-trace, which arises from Connes' cyclic complex, and the associated $\infty$-modulus, which gives another gauge-invariant function on bounding cochains. In open Gromov-Witten theory, the $\infty$-modulus is used to normalize the bounding cochains to which the superpotential is applied. We develop our definitions for general curved Banach $A_\infty$-algebras, and give both unital and non-unital versions of the main results. In the unital setting, we work with weak bounding cochains, solutions of an inhomogeneous Maurer-Cartan equation involving the unit. Weak bounding cochains arise in the open Gromov-Witten theory of Lagrangian submanifolds with non-vanishing Maslov class. The derivative of the superpotential at a weak bounding cochain is related to the derivative of the $\infty$-modulus.

We provide some information about the Drinfeld center $\Z(\Ver_p)$ of the Verlinde category $\Ver_p$. We compute explicitly the graded Cartan matrix, bound quiver and cohomology of $\Z(\Ver_p^+)$, and prove that the category $\mathscr{Z}(\Ver_p^+)$ is wild for every $p\ge 7$. In the special case $p=5$, we show that $\Z(\Ver_5^+)$ has exactly $10$ non-isomorphic indecomposable objects, classify them and describe the Green ring of $\Z(\Ver_5^+)$, and compute the semisimplification of $\Z(\Ver_5^+)$.