Papers

We show that quasi-Poisson and quasi-BV structures on (super)manifolds $M$ admitting an action of a Lie (super)group $G$ can be identified with a special class of homotopy Poisson and respectively $BV_\infty$ structures on the associated differential (super)manifolds of the form ${\mathcal M}:=M\times \Pi g$, $g$ being the Lie (super)algebra of $G$. The construction is based on the proven existence of an essentially unique family of higher Jacobiators for any Lie (super)algebra $g$ with an invariant (odd or even) scalar product. We apply these arguments to the moduli spaces $M$ of flat $G$-connections on a surface with boundaries using Fock-Rosly construction.

In this work, we revisit the analytical structure of the classical affine Lie bialgebra underlying the integrable structures of AdS/CFT worldsheet scattering and the one-dimensional Hubbard model. Earlier work showed that the algebraic relations implied two singularity locations in the evaluation parameter plane besides the common expansion points infinity and zero for Laurent polynomials forming the basis for loop and affine algebras. Our aim is to formulate these singularities rigorously using meromorphic functions rather than by expansion into geometric series. We thus carefully analyse the analytical structure of the affine bialgebra relations in the complex plane paying attention to potential restrictions and extensions due to the additional singularities. As a first attempt with some shortcomings, we reformulate the affine bialgebra relations with the extra singularities serving as the expansion points for Laurent polynomials. We finally overcome the remaining issues by proposing three-point and four-point affine algebras for the rational and trigonometric cases. With a suitable choice of basis for meromorphic functions on the Riemann sphere with several punctures based on Krichever-Novikov algebras, we show that the multi-point affine algebras form consistent quasi-triangular Lie bialgebras.

In a (1+1)-dimensional QFT with non-invertible symmetry, the fusion rules of topological lines are often the most accessible data, yet they do not fully determine whether the symmetry is anomaly-free, i.e. whether it admits trivially gapped phases. Such phases are characterized by fiber functors of the corresponding fusion category. Building on Tannaka duality, we extract fiber functors from the fusion ring alone by solving simultaneously for monoidal structure maps and tensor automorphisms. This allows us to find categorifications admitting a fiber functor without solving the pentagon equations directly. We test this procedure on the known Tambara–Yamagami ring of $\mathbb{Z}_2\times\mathbb{Z}_2$. Our main result concerns the rank-11 fusion ring shared by the representation categories of the two extraspecial groups of order 27, which has a multiplicity 3 channel and is relevant to self-triality. In both cases we find more than one fiber functor, i.e. several non-invertible SPT phases. We use the explicit data of these fiber functors to derive the $F$-symbols of the two representation categories in a compact form depending on a non-degenerate alternating bicharacter and an invertible object $\epsilon$. Using the reconstructed data, we compute the anomalous algebra on the interface between two different SPT phases.

We establish that the skew-symmetric bicharacter on the group of $\ell$-weights of a quantum affine algebra is invariant under Chari's braid group action. Furthermore, we define an extension of this bicharacter to the group generated by prefundamental $\ell$-weights and show that this invariance continues to hold under the extended braid group action of Frenkel-Hernandez. As immediate applications, we obtain braid group actions by automorphisms on the quantum tori associated respectively with the category of finite-dimensional representations of the quantum affine algebra and with the category $\mathcal{O}^{\mathfrak{b}}$ of its Borel subalgebra. In addition, we derive explicit formulas for quantum $QQ$-systems in all simply-laced types. Finally, we show that our extended bicharacter yields a quantization matrix compatible with the Hernandez-Leclerc cluster algebra categorified by $\mathcal{O}^{\mathfrak{b}}$ for all finite Dynkin types, extending Bittmann's construction in the simply-laced setting.

Let $\mathfrak g$ be a simple Lie algebra of classical type. We construct a $\mathbb Z[1/2]$-integral form of the Yangian $Y(\mathfrak g)$ generated by divided powers of the positive and negative Drinfeld generators, and establish its triangular decomposition and PBW basis. Under the loop filtration, its associated graded algebra is the current algebra part of Garland's integral form. As applications, over an algebraically closed field of characteristic $p>3$, we identify the small Yangian with the restricted Yangian and classify their finite dimensional simple modules in terms of restricted Drinfeld polynomials.

Work by David Reutter and Christopher Schommer-Pries has shown that ordinary smooth TQFTs can distinguish (stable) diffeomorphism classes of closed, connected, even-dimensional manifolds subject to certain finiteness conditions. In particular, simply connected, closed, smooth 6-manifolds with finite $\pi_2$ are diffeomorphic if and only if they cannot be distinguished by ordinary TQFTs. The question of whether or not this result holds for all simply connected closed 6-manifolds was open. We consider TQFTs out of the topological and smooth bordism categories, as well as out of a new bordism category constructed in this paper called the formally smooth (FS) bordism category. We present a pair of simply connected closed 6-manifolds with infinite $\pi_2$ that are not homotopy equivalent, yet are indistinguishable by ordinary TQFTs in all three categories. We extend this counter-example to show that ordinary TQFTs cannot distinguish the homotopy type of simply connected closed manifolds in all dimensions greater than or equal to 6 and non-simply connected closed 5-manifolds. We show that our class of counter-example pairs is spinnable and present a non-spinnable class of counter-examples by introducing a $\mathbb{C}P^2$ connected summand. Lastly, we use more highly connected analogues of our counter-examples to show that, for all $n\geq2$ and $m\geq 2n$, there is a pair of $(2n-1)$-connected closed $(4n+m)$-manifolds that are not homotopy equivalent, yet cannot be distinguished by ordinary TQFTs.

We revisit an old non-diagonal modular invariant of $F_4$ at level $6$, discovered in 1989. At the time of its discovery, it was noticed that it could not be obtained from a conformal embedding of affine Lie algebras. Later, it was shown that it can be obtained from an extension of a vertex operator algebra which is not of Lie type. To this modular invariant, one can attach a full CFT defined (or formulated) as a module category $\mathcal M$ for the fusion category $\mathcal A = \mathcal C(F_4,6)$. More recently, it was recognized that $\mathcal M$ is associated with an étale algebra of $\mathcal A$. In other words, the resulting quantum module is a quantum subgroup (it is also said to have self-fusion, to be of type I, or to be flat). We study the dual $\mathcal O$ of $\mathcal A$ with respect to $\mathcal M$. In particular, we describe the algebra of quantum symmetries (we obtain the chiral generators and display the Ocneanu graphs for the fundamental irreps of $\mathcal A$). The chiral graphs are not connected, each one containing two pairs of isomorphic components. The vertex sets of these components are identified with the simple objects, up to equivalence, of two module categories $\mathcal M$ and $\mathcal M^\prime$ sharing the same modular invariant. $\mathcal M$ enjoys self-fusion, whereas $\mathcal M^\prime$ does not. The restriction of the action of the Verlinde algebra $A$ to the associated abelian groups $M$ and $M^\prime$ defines two families of matrices with non-negative integer entries (nimreps) and their adjacency graphs. As a by-product, we obtain the matrices describing the defect lines of the theory, the integer and rational fusion polynomials attached to $A$ and its modules, and the relevant numerical invariants, quantum dimensions, quantum masses, and induction-restriction rules.

We construct a presentation of the Khovanov arc algebras of type $D$ by a quiver with quadratic relations. As an application, we use this presentation and the Koszul dual to compute some Hochschild cohomology groups of $\mathbb{D}_4$, and show that $\mathbb{D}_4$ is intrinsically formal, that is, it admits no non-trivial $A_\infty$-deformations.

For a quantum affine superalgebra with a fixed Drinfeld current presentation, we study shifted quantum affine superalgebras through a two-sided shifted Drinfeld current presentation, in which the leading degrees of the Cartan–Drinfeld currents are modified while the remaining current and Serre relations are retained. We then focus on the standard-parity type A algebras $\mathcal U_q^{\mu_+,\mu_-}(\widehat{\mathfrak{sl}}_{m|n})$. For arbitrary shifts and $m,n\ge1$, $m\ne n$, we prove a triangular decomposition. When $m,n>1$, we obtain PBW–Drinfeld bases for arbitrary shifts and a Levendorskii-type finite presentation of the one-sided algebra $\mathcal U_q^{0,\mu}$ for every antidominant $\mu$. For antidominant two-sided shifts, we construct shifted RTT realizations and prove the corresponding Ding–Frenkel isomorphisms. We construct a shifted formal Drinfeld coproduct for arbitrary two-sided shifts, coassociative in a completed tensor product. Shifted RTT realizations yield non-completed one-sided coproducts in the antidominant region, made explicit on finitely many generators by the finite presentation; injective shift homomorphisms extend them to arbitrary one-sided shifts. Their ternary coassociativity holds whenever the middle shift is antidominant, including boundary cases. Finally, for two-sided antidominant shifts $\mu_+,\mu_-\in\Lambda^-$, a shifted full general-linear Ding–Frenkel isomorphism identifies the special-linear simple-current algebra as the fixed algebra under common scalar rescalings and yields coassociative two-sided non-completed coproducts.

We prove that every weakly group-theoretical modular fusion category, equipped with its canonical positive spherical structure, has generalized property F: the associated Reshetikhin-Turaev mapping class group representations have finite projective image for all compact oriented surfaces with labeled framed points. This generalizes earlier finite-image results for braid group representations associated with weakly group-theoretical braided fusion categories and for mapping class group representations associated with group-theoretical modular fusion categories and Ising categories. In the weakly group-theoretical setting, it also extends the known finiteness result for symmetric mapping class groups of closed surfaces to the full mapping class groups.

We establish a generalized quantum cluster algebra structure, defined by Bai–Chen–Ding–Xu, on the reduced Lê–Roger–Yang skein algebra of every quasi-triangulable punctured surface. We define Frobenius maps for generalized quantum cluster algebras at roots of unity and use these maps, together with the cluster algebra structure, to construct Frobenius maps for the (reduced) Lê–Roger–Yang skein algebras of all punctured surfaces. As applications of the Frobenius map, we determine the centers of both the Muller–Roger–Yang skein algebras and the reduced Lê–Roger–Yang skein algebras at roots of unity, and compute their ranks as modules over their centers.

We study supersymmetric semi-infinite cohomology and its $N = 2$ superconformal structure via the BRST reduction of an $N = {\frac{3}{2}}$ superconformal structure on the super-affine vertex algebra associated to a super Manin pair arising from the Kazama–Suzuki $N = 2$ condition. Moreover, we realize super coset models in terms of relative supersymmetric semi-infinite cohomology.

Let $Q$ be a bipartite Dynkin quiver of finite simply-laced type and $\mathscr C_Q$ the corresponding Hernandez–Leclerc category. We study a two-parameter specialization of the Fedele–Hernandez toroidal Grothendieck ring and identify its generic positive part with a Cartan-type multiparameter quantum group. The structural input is the identity \[ 2D^{(2)}-D_0=\omega_Q\circ(\deg_Q,\deg_Q), \] where $D_0,D^{(2)}$ are the two effective toroidal commutation forms and $\omega_Q$ is the antisymmetric Euler form of \(Q\). It realizes the specialized character algebra, after an auxiliary scalar extension, as a graded bicharacter twist of the one-parameter algebra at $v=t_0t^{1/2}$. This yields the required root grading and PBW graded dimensions. For every source–sink edge $r\to s$, these dimensions combine with the local \(A\)–\(Y\) commutation formula to give a quadratic annihilator for \(\operatorname{Ad}_{Z_r}\). Consequently \[ \operatorname{Spec}_{Z_r}(Z_s)=\{t_0t,t_0^{-1}\}, \] without thinness or type-specific character formulas. The two spectral values yield the oriented multiparameter Serre relations. With \[ q_{ii}=t_0^2t,\qquad q_{ij}=t_0^{-1},\qquad q_{ji}=t_0^{-1}t^{-1}\quad(i\to j), \] and \(q_{ij}=1\) on nonedges, we obtain \[ U_{\mathbf q}^{+}(\mathfrak g) \simeq \mathscr K_\infty(\mathscr C_Q)\otimes_RK \] for $\mathfrak g$ of type $A_n,D_n,E_6,E_7,E_8$.

In this paper we demonstrate that Hopf algebras and Frobenius algebras provide an appropriate setting for an analogue of the classical Lie correspondence. We introduce a compatibility structure between Hopf and Frobenius algebras called Hopf-Frobenius modules, and give constructions that can be applied to any such module, which produce corresponding Lie groups and Lie algebras. Further, we show that classical Lie algebras and Lie groups give rise to Hopf-Frobenius modules for which the corresponding Lie groups and Lie algebras produced by our construction are those of the ordinary Lie correspondence.

A recent trapped-ion experiment realised a lattice gauge theory in which the internal states of the ions carry the gauge field and their vibrations carry bosonic matter. In that experiment, a charge moving around a loop was frozen in place by Aharonov–Bohm interference. The noise of such a device is usually simulated after the measurements have been made. Here we show that it can be predicted before the machine is run. For each quantity that the experiment measures, we derive, from the reported noise rates alone, a formula for how it decays together with a bound it can never exceed. The results reveal a simple pattern. The local conservation law of the theory is a parity of the vibrational mode, so heating violates it at a rate that grows as the mode warms, and the violation accelerates rather than accumulating steadily. At zero electric field, each of the three noise processes of the device leaves its own fingerprint. Heating affects only the conservation law, motional dephasing only the interference, and qubit errors only the stored flux and the frozen state. Once the electric field is switched on, the frozen state begins to leak, and we write an exact formula and a guaranteed bound for that leak.

We study exact recovery for non-Abelian topological surface codes based on the quantum double $D(G)$ of any finite group $G$. We determine an orthogonal error basis, that is comprised of group multiplication and irreducible representation operators. Whenever the errors are supported on closed loops on the lattice or dual lattice, there is a redundancy in the basis, due to stabilizers. To remedy this, we introduce a gauge-fixing that results in a complete and orthogonal error basis on the code space. Assuming as input predetermined, neutral correctable error clusters, we construct charge and flux transfer circuits that move the errors onto ancillas that then get projected out. This is an exact and deterministic recovery protocol, applicable to surface codes for any finite, in particular non-Abelian, group $G$.

We show that $\mathbb{Z}_2$ is a quantum normal subgroup of $SU_q(2)\times SU_q(2)$ and that the quotient group is isomorphic to $SO_q(4)$, thereby establishing that $SU_q(2)\times SU_q(2)$ is a double cover of $SO_q(4)$.

We classify constant numerical solutions of the $\Z_2$-graded twisted reflection equation for the fundamental $GL$-type R-matrix and an arbitrary parity sequence, without assuming evenness or invertibility. The classification is based on a four-index locality principle established for this type of reflection equation. We determine the singular and invertible solutions and recover previously conjectured matrices for the minimal symmetric grading. Even invertible solutions comply with the recent classification of graded Satake diagrams. The classical stabilizer of a full-support lower-triangular non-even family is a Lie superalgebra whose even part is an electrical Lie algebra.

We prove a generalization of Grauert's higher coherence theorem for a class of curved differential graded (non-commutative) Fréchet algebras. This allows us to extend the Fourier-Mukai calculus to derived categories arising in many new contexts. We then apply it and prove equivalence of derived categories of dual generalized complex tori using a non-commutative version of the Poincaré line bundle. This lays the foundation for categories of generalized complex branes on generalized complex tori. Examples include complex tori, symplectic tori as well as their non-commutative and B-field deformations.

We establish that, for a binary operad Var, the Hadamard product of Var and the operad Nov of Novikov algebras is a binary operad (i.e., coincides with their Manin white product Nov $\circ$ Var) if and only if the quadratic part of Var governs a 2-variety. The condition Nov $\circ$ Var $=$ Nov $\otimes$ Var is known to be equivalent to the existence of an easily verifiable characterization of those differential Var-polynomials which lie in the free derived Var-algebra. For a binary quadratic operad, these conditions are also equivalent to the Dong property of formal distributions. We distinguish the white product defined as a generated sub-operad from its quadratic approximation and give an example in which the canonical epimorphism between them is not injective.

Let $\mathfrak{g}$ be a basic classical Lie superalgebra and let $\widehat{\mathfrak{g}}$ be the corresponding affine Lie superalgebra. In this paper, we first prove that a Cartan subalgebra acts semisimply on ordinary modules for the simple affine vertex operator superalgebra $L_{\widehat{\mathfrak{g}}}(k,0)$ at boundary admissible level $k$. Then we prove that the category $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ of ordinary $L_{\widehat{\mathfrak{g}}}(k,0)$-modules is finite, semisimple and $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ is exactly the category $KL_k(\mathfrak{g})$ of finite-length generalized modules for the affine vertex operator superalgebra $L_{\widehat{\mathfrak{g}}}(k,0)$.Thus $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ is a braided tensor supercategory. Furthermore, we obtain the rigidity of the supercategory $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ and thus it is a ribbon supercategory. Finally, we conclude that $\mathcal{O}_{k}^{ord}(\mathfrak{g})$ is a ribbon fusion supercategory.

We introduce the crystal groupoid of a compact semisimple Lie group $K$; it is the Deaconu-Renault groupoid of a higher-rank analogue of a subshift of finite type on an inverse limit of Kashiwara crystals. We show that its Steinberg algebra and its $C^*$-algebra realise the crystal limits of the polynomial function algebra of $K$ and its $C^*$-envelope, respectively, in the sense of Matassa and the second author. We show that the orbit space of the crystal groupoid is in bijection with the Weyl group of $K$, with topology determined by the Bruhat order on the Weyl group. We also define a crystal groupoid for flag manifolds and related Poisson homogeneous spaces, and prove the analogous structural results.

We equip Khovanov's colored $\mathfrak{gl}(2)$-homology groups with a degree-2 derivation. Over a field of characteristic $p>2$, the derivation specializes to a $p$-nilpotent differential, which leads to a categorification of colored $\mathfrak{gl}(2)$-homology at a $2p$th root of unity.

We discuss (2+1)d Chern-Simons Higgsing transitions based on conformal embeddings and their relation to the proliferation of anyons. These transitions preserve or enlarge the intrinsic topological order despite reducing the gauge group. When the topological order is enlarged, the two phases differ by anyon condensation, and the transition takes an Abelian topological order to a non-Abelian one. In the $SO(N)_{2} \hookrightarrow SU(N)_{1}$ family with $N \geq 3$, the two semiclassical regimes have equivalent intrinsic topological orders but distinct realizations of a global $U(1)$ symmetry. For odd $N$, these are non-Abelian realizations of transitions from fermionic $1/N$ Laughlin states to charge-$2e$ anyon superconductivity coexisting with the same chiral $\mathbb{Z}_{N}$ topological order. The scalar representations are always non-integrable with respect to the UV Chern-Simons level and therefore do not directly label anyons of the un-Higgsed phase, despite defining Wilson lines in the UV. Using heuristic adjoint screening, we propose candidate anyons in the un-Higgsed phase, while branching under the Higgs subgroup reveals channels corresponding to condensable anyons of the Higgs phase. We further construct a $\mathrm{Spin}(16)_{1} \to \mathrm{Spin}(9)_{2}$ transition admitting a condensable algebra that contains a non-Abelian anyon but not all of the allowed gaugable channels in its self-fusion.

We study exact sequences of representation categories of weak Hopf algebras over an arbitrary field. Given a sequence $A\overset{k}{\to} B\overset{\pi}{\to} H$, where $A$ and $B$ are weak Hopf algebras and $H$ is a Hopf algebra, we develop verifiable algebraic conditions on $k$ and $\pi$ under which there is an exact sequence $\mathrm{Rep}(H)\to\mathrm{Rep}(B)\to\mathrm{Rep}(A)$ of tensor categories in the sense of Bruguières and Natale (2011). Along the way, we develop a generalization of the restriction of scalars functor for maps $\pi:B\to H$ between associative algebras satisfying a weakened multiplicativity constraint depending on a relatively separable subalgebra $B_r\subseteq B$, as well as a theory of kernels and cokernels for weak Hopf algebras. We, in particular, find that the cokernel of a weak Hopf algebra homomorphism $k:A\to B$ always exists, is a Hopf algebra, and the cokernel map is surjective when $A$ is connected. We conclude by studying examples of such exact sequences built from groupoids, formal ribbon extensions of quasitriangular weak Hopf algebras, and cocycled crossed products of a Hopf algebra acting weakly on a weak Hopf algebra.

Let $B$ and $D$ be both algebras and coalgebras in a braided monoidal category $\mathcal{C}$. In the literature, there are equivalent conditions for $(B,D)$ to form a generalized smash biproduct (or cross product) denoted by $B\times D$, which were mainly given by Bespalov and Drabant in 1999 as well as by Bulacu, Caenepeel and Torrecillas in 2013. This paper is devoted to constructing another biproduct $D\times B^\ast$ when $B$ has a left dual object $B^\ast$ in $\mathcal{C}$. As results, we show that $B\times D$ is left smash over $D\times B^\ast$, and establish a specific isomorphism ${}_{B\times D}\mathfrak{YD}(\mathcal{C})^{B\times D}\cong{}_{D\times B^\ast}\mathfrak{YD}(\mathcal{C})^{D\times B^\ast}$ between the categories of the (left-right) Yetter-Drinfeld modules in $\mathcal{C}$. The constructions and results are provided in three levels: (1) $B\times D$ is an algebra and a coalgebra; (2) $B\times D$ is a bialgebra; (3) $B\times D$ is a Hopf algebra.

Given any tracial von Neumann algebra $\mathcal{A}$, the first author has defined a unitary tensor functor $\Phi:{\rm Rep}(\mathcal{Q}_{\rm aut}(\mathcal{A}))\to {\rm Bimod}(\mathcal{A})$ in his paper [Gos24]. In this paper, we consider $\mathcal{A}$ to be the underlying von Neumann algebra of $C^*(S_3)$ and show that given any DQG $\mathcal{Q}$, which contains $C^*(S_3)$ as a quantum subgroup and has an outer action $\tilde{\Phi}$ (in the sense of Definition 3.4}) on ${\rm Bimod}(\mathcal{A})$, which agrees with $\Phi$ when restricted on ${\rm Rep}(C^*(S_3))$, is monoidally equivalent to the DQG $C^*(S_3)$. Furthermore, given any DQG $\mathcal{Q}$ with an outer action $\Gamma:{\rm Rep}(\mathcal{Q})\to {\rm Bimod}(\mathcal{A})$, such that $ {\rm Out}(\mathcal{A})\subseteq {\rm Im}(\Gamma)$, is either monoidally equivalent to Out$(\mathcal{A})$ or monoidally equivalent to $C^*(S_3)$ itself.

Radchenko and Wheeler (RW) recently proved a finite pentagon relation for real quadratic special values of the modular quantum dilogarithm and used it to establish the rank-$1$ twisted convolution identity conjectured by the current authors. RW gave an explicit argument in the principal case and remarked that their proof holds for all rank-$1$ admissible tuples. We extend their proof to all rank-$r$ admissible tuples and provide an explicit dictionary between the modular quantum dilogarithm and the Shintani-Faddeev modular cocycle conventions in the respective papers. Thus, we establish that, if $d,r$ are positive integers such that $r<\frac{d-1}{2}$ and $\frac{d^2-1}{r(d-r)} \in \mathbb{Z}$, then there exist ghost $r$-SICs: i.e., configurations of $d^2$ rank-$r$ subspaces in $\mathbb{C}^d$ that satisfy a non-Hermitian equichordal condition. Under the Stark conjecture, these configurations are Galois conjugate to Hermitian equichordal configurations called $r$-SICs (or rank-$r$ SIC-POVMs).

We introduce the notion of $K_{\mathfrak{q}}$-algebra, a conjectural axiomatization of the fusion algebra of rotation-equivariant BPS line defects in a 4d ${\cal N}=2$ Supersymmetric Quantum Field Theory. We also introduce the notion of RG flow of $K_{\mathfrak{q}}$-algebras, a conjectural axiomatization of the action of Seiberg-Witten type RG flows on line defects. These definitions enrich the theory of cluster algebras, $K$-theoretic Coulomb branch algebras, skein algebras and quantum groups.

Surface braids are a generalization of braids to surfaces in the 4-disk. They are described by planar graphs called charts, and sequences of braid words called braid systems, that represent monodromies of branch points in projections. Yang-Baxter operators (YBOs) have been used extensively in knot theory for constructions of Jones and other polynomials, defined by braid group representations. We propose to use YBOs for constructing surface braid invariants through braid systems. The intersections of eigenspaces are used for this construction. We further use 3-cocycles of the Yang-Baxter cohomology theories via charts, in an analogue of quandle cocycle invariants but more generally utilizing tensor products of modules. An approach towards defining knotted surface invariants through closures of surface braids is discussed.