Papers

A scalar field alone does not determine a quantum theory: states, effects, processes, symmetries, composition, and discard are equally structural. Realification illustrates this point: an orthogonal complex structure J^2 = -I selects the physical real operators and the balanced composite. Quaternionic quantum mechanics has an analogous exact representation on a doubled complex space selected by an antiunitary symplectic structure Theta^2 = -I. Within that sector, a Choi fixed-point condition characterizes when a complex channel admits quaternionic Kraus operators, while exact correction of a finite-dimensional right-quaternionic code is characterized by compression coefficients in the real center and admits an explicit recovery. For octonions, nonassociativity precludes a unique continuation; para-linear, categorical, sectorial, Jordan, Clifford-envelope, and Moufang models are compared by the operational structures they retain and the additional data required to define composites, channels, or recovery. Across these cases, an exact ambient representation does not erase the complex or symplectic structure that selects the physical theory.

Let $A$ be a commutative simple algebra in a braided finite tensor category $\mathcal{B}$. We identify the largest transparent subalgebra of $A$ as the algebra induced by a central lift of the free-module functor. This identification gives formulas for the Frobenius-Perron dimension and the Müger center of the category of local $A$-modules. These formulas give criteria for nondegeneracy, symmetry, and modularity, together with sharp bounds on $\mathrm{FPdim}_{\mathcal{B}}(A)$. We also realize the Müger center of $\mathcal{B}$ as a category of local modules over an adjoint algebra. Finally, we prove a relative-center factorization and deduce that taking the category of local modules preserves the relative Witt class.

There is a well-known circle of conjectures relating unitary solutions of the Yang–Baxter equation, unitary braided fusion categories, topological quantum computation, and link invariants. Progress is limited by the lack of a classification of unitary Yang–Baxter operators. We propose a conjectural classification with three generating sources: monomial solutions, group-type solutions arising from Yetter–Drinfeld modules, and solutions from twisted group-algebra towers. We conjecture that, up to unit scalars and local unitary basis changes, all solutions are generated by these sources. We provide evidence for our classification conjecture by means of computational searches for unitary Yang–Baxter operators among Clifford groups and a polynomial-in-Pauli ansatz. This search yields only a few solutions that do not appear to be equivalent to monomial or group type solutions. Direct calculations identify their link invariants with specializations of the HOMFLYPT and BMW/Kauffman invariants; the latter is a squared Jones specialization at a primitive sixth root, and the corresponding qutrit operator strictly localizes the semisimple quotient of the BMW algebra specializations $\mathcal{C}_n(q^3,e^{\pi\ii/6})$ associated with the braided fusion category $SO(4)_2$. The quaternionic Family III specialization realizes a two-eigenvalue class and gives tensor-power realization of the semisimple Hecke algebra tower associated with \(SU(3)_3\). Significantly, this answers a recent existence question of Lechner, and verifies an open case of the Rowell-Wang localization conjecture from 2010.

We propose a refinement of the quantum Hikita conjecture of Kamnitzer, McBreen, and Proudfoot that bridges the representation theory of Coulomb branches with the enumerative geometry of Higgs branches. We also introduce a general framework for proving it, which we carry out for ADE quiver gauge theories with minuscule framings and for the gauge theory corresponding to the Jordan quiver. As an application, we use the resulting quantum Hikita isomorphisms to give a geometric description of graded traces on quantized Coulomb branches.

We define a family of Turaev-Viro type invariants for hyperbolic $3$-manifolds with cusp ends, extending the invariants introduced in \cite{LMSWY} for hyperbolic $3$-manifolds with totally geodesic boundary. These invariants are constructed from what we call the ideal $\mathrm{U}_{q\tilde q}\mathfrak{sl}(2;\mathbb R)$-$6j$ symbols, which are variants of the $6j$-symbols associated with the positive representations of the modular double of $\mathrm{U}_q\mathfrak{sl}(2;\mathbb R)$. We also prove that these invariants decay exponentially, with the exponential decay rate determined by the hyperbolic volume of the manifold.

We propose a definition of the quantized Coulomb branches of separated cotangent type, and prove that the corresponding classical construction recovers the non-cotangent Coulomb branch. We also obtain a formula for quasi-minuscule monopole operators in arbitrary cotangent type. Applying these results, we compute the monopole operators for orthosymplectic quivers and construct a homomorphism from the shifted twisted Yangian of split ADE type to the corresponding quantized Coulomb branch algebra.

Sections 2–3 of this note were prepared primarily for the author's private use, as a means of organizing the materials in (generally non-holomorphic) conformal field theory (CFT) on 2-dimensions in the language of the vertex operator algebra, and also of collecting subtle details. The choice of topics is intended to supplement introductory physics textbooks on the subject. Section 4 deals with (generally non-holomorphic) Type 0 CFTs in the language of vertex operator superalgebra (SVOA). Formulations of crossing symmetry and intertwining superoperators [resp. CFT vertex superoperators] of states in the Ramond (SVOA-twisted) sector [resp. the Ramond–Ramond sector] are presented by using two auxiliary parameters. The formulation allows systematic generalization to the class of simple current extensions characterized by four conditions: (1) the vanishing third cohomology, (2) skew-symmetric bimultiplicative form, (3) fixed-point-free action of simple currents, and (4) one additional condition described in the main text.

In arXiv:1405.4105 it was asked whether an affine Hopf algebra with finite Gelfand-Kirillov dimension is necessarily Noetherian. It is well-known that the converse is not true –take the group algebra of a polycyclic group which is not nilpotent-by-finite. However, it was conjectured in arXiv:2301.04428 that a pointed affine Noetherian Hopf algebra whose group of group-likes is nilpotent-by-finite necessarily has finite Gelfand-Kirillov dimension. In the present paper we conjecture that a post-Nichols algebra over a polycyclic-by-finite group is Noetherian if and only if it is affine and has finite Gelfand-Kirillov dimension. Partial results supporting this conjecture are presented; and their consequences for the preceding questions and conjectures is analyzed.

We propose a categorification of the Lusztig integral form of the quantum Cartan subalgebra for $sl(2)$ via a colimit of categories of equivariant coherent sheaves on finite-dimensional projective spaces. The nonequivariant version of our construction yields a nonsemisimple abelian categorification of the ring of integer-valued polynomials.

In this paper, when the quiver Hecke algebra R is symmetric, we present a new construction of quantum monoidal seeds $ \mathscr{S}_{w,v}$ for $\mathcal{C}_{wv}$ using the reflection functors $\mathcal{F}_i$ and the newly introduced operators $\mathcal{K}_i$. The monoidal seed $ \mathscr{S}_{w,v}$ is obtained as a subseed of the monoidal seed of $\mathcal{C}_w$ constructed by applying $\mathcal{K}_i$ and $\mathcal{F}_i$ along the special KF sequence determined by a reduced expression of $w$ and $v$. We further prove that the monoidal seed $\mathscr{S}_{w,v}$ coincides with the set of all prime factors of the determinantial modules $M(w_{\le k } \Lambda_{i_k}, v_{\le k} \Lambda_{i_k} )$. We prove that the Grothendieck ring $K(\mathcal{C}_{wv}) $ lies between the cluster algebra and the upper cluster algebra.

Let $G=F\Gamma$ be a factorization of a finite group, with neither factor assumed normal and with $F\cap\Gamma$ allowed to be nontrivial. We prove that $\exp(G)$ divides $\operatorname{lcm}(|F|,|\Gamma|)$, or equivalently that $\gcd([G:F],[G:\Gamma])\exp(G)$ divides $|G|$. This answers a cohomological divisibility question posed by Natale. Combining the group-theoretic divisibility with Natale's exponent bound and a lifting argument, we prove Kashina's exponent conjecture, in the arbitrary-field formulation of Etingof and Gelaki, for every finite-dimensional semisimple and cosemisimple Hopf algebra $H$ over a field $k$ for which $\operatorname{Rep}(H\otimes_k\overline{k})$ is group-theoretical. The same argument proves the corresponding degree-three cohomological divisibility for coefficients in an arbitrary $G$-module. For complex group-theoretical categories, we also establish Frobenius-Schur exponent divisibility under a cohomological factorization hypothesis, without assuming a fiber functor. We derive applications to low-dimensional Hopf algebras and abelian extensions.

We consider quantum group generalizations of the action of monomial matrices on complex Hadamard matrices. This gives rise to various notions of quantum symmetries and quantum equivalences of Hadamard matrices. We show that if one acts by a certain largest monomial quantum group, then all Hadamard matrices of a given size become quantum equivalent. Taking a more restrictive quantization leads to a notion of $s$-quantum equivalence. We exhibit examples of Butson matrices of the same size and order that are not $s$-quantum equivalent for any choice of $s$. We also show that $s$-quantum equivalence is operationally modeled by a synchronous non-local “Hadamard equivalence” game. Our methods are largely graphical calculus based, using categories generated by complementary spiders. We use these same tools to study quantum affine equivalence of quantum groups, and prove that all finite quantum groups of the same size are quantum affinely equivalent. We also provide a short graphical proof of a result of Kasprzak–Sołtan–Woronowicz asserting that quantum symmetries of finite quantum groups must be classical.

Let $L_2=\mathbb{Z}\alpha$ be the rank-one root lattice with $(\alpha,\alpha)=2$, and let $A_5$ act on the lattice vertex operator algebra $V_{L_2}$ through an icosahedral subgroup of $\operatorname{Aut}(V_{L_2})\cong PSL_2(\mathbb{C})$. We prove that the fixed-point vertex operator algebra $V_{L_2}^{A_5}$ is strongly rational.

Recently we started building a bridge between two cases of Langlands duality. The latter is one of the most influential topics in mathematical research. It has many different appearances and influential subtopics. Yet there is a topic that until now seems unrelated to the Langlands program. That is the topic of invariant differential operators. That is strange since both items are deeply rooted in Harish-Chandra's representation theory of semisimple Lie groups. We started with the case of the group ~$SL(2n)$. In the present paper we deal with the group $SL(2n+1)$. The two mentioned groups are similar, but their representation theories is rather different.

We construct three derived endofunctors on the derived category of graded integrable representations of the current algebra of a flat degeneration of $D(2|1,\alpha)$, and prove that their classes in the Grothendieck group are the genus two Macdonald operators. Computing the graded $\operatorname{Ext}$ pairing explicitly, we deduce the self-adjointness of the genus two Macdonald operators and the orthogonality of genus $2$ Macdonald polynomials with respect to the certain skew-bilinear form.

We construct phase transitions that proliferate condensable anyons in general 2+1d topological orders, including non-abelian ones. The central tool that provides a systematic approach to this question is the Symmetry Topological Field Theory (SymTFT). For a given topological order, we identify the relevant symmetry from the transparent lines generated by the condensable anyons, and thereby realize the topological order in terms of a 3+1d SymTFT sandwich. The proliferation phase transition is realized by coupling scalar fields to the anyons purely on the symmetry boundary of the SymTFT. We illustrate the construction for abelian theories, as well as non-abelian ones, $D(S_3)$ and $SU(2)_k$ Chern-Simons theories, and extend it to anomalous anyons.

In this paper, we continue our research on the double Yangian and reflection algebras of the Lie superalgebra $\mathfrak{gl}_{M|N}$. Extending the Etingof-Kazhdan construction, we introduce the structure of $h$-adic quantum vertex superalgebra on the vacuum module $V^c(\mathfrak{gl}_{M|N})$ over the double Yangian of level $c\in\mathbb{C}$. Next, we construct families of central elements in the completed double Yangian for $\mathfrak{gl}_{M|N}$ at the critical level. Finally, we show that the level $c$ restricted modules over the reflection algebra of $\mathfrak{gl}_{M|N}$ are naturally equipped with the structure of quasi $V^{2c}(\mathfrak{gl}_{M|N})$-module. By using this structure, we find explicit formulas for families of central elements in the completed reflection algebra of $\mathfrak{gl}_{M|N}$ at the critical level.

We prove the simple-object prediction of Shan–Yan–Zhao and a basis-preserving dual-cell realization for the distinguished vacuum block of the simple affine vertex algebras $L_{-1}(D_\ell)$, $\ell\ge5$. The block has exactly $\ell+1$ simple objects, indexed by the subregular affine left cell containing $s_0$. The proof combines a primitive-ideal inclusion, an independent exhaustion argument, and a finite-length step. A noncritical Sugawara lift supplies finite-dimensional weight-space detectors in the original Shan–Yan–Zhao category-$\mathcal O$ block, so dévissage applies to its ordinary Grothendieck group. We then identify this group, basis by basis, with the $q=1$ specialization of the corresponding dual affine left-cell module. An injective signed normalized-character realization identifies the resulting image with the canonical dual-cell image in the completed singular-orbit module and hence supplies the corresponding abstract $\widehat W$-module structure. We do not identify this action with a functorial action arising from affine twisting functors or Kashiwara–Tanisaki localization. The subregular inverse Kazhdan–Lusztig calculation of Bezrukavnikov–Kac–Krylov also yields uniform character formulas.

In our previous article [arxiv:2410.05120], we introduced the notion of a finite dimensional 3-Hilbert space, categorifying Baez's 2-Hilbert spaces. In this article, by further categorifying Baez's higher linear algebra, we provide useful tools for working with 3-Hilbert spaces, including, generalized scalar multiplication, orthonormal bases, and unitary adjoints for operators. We use these tools to endow the $\mathrm{C}^*$-3-category of 3-Hilbert spaces with a self-enrichment. We prove a Unitary Yoneda Lemma/Riesz Representation Theorem for 3-Hilbert spaces: the Yoneda embedding is an isometric equivalence. Finally, we define a unitary version of the Deligne product on 3-Hilbert spaces and prove that it satisfies an isometric version of the folding trick.

The odd Vassiliev conjecture asserts that all Jacobi diagrams with an odd number of legs vanish modulo the AS and IHX relations. Equivalently, the conjecture asserts that Vassiliev invariants cannot distinguish a knot from its inverse. The conjecture is elementary for one 1 or 2 loops, and was proven by Moskovich and Ohtsuki for 3 loops. We establish the conjecture for 4 loops. Our proof uses Jacobi diagrams and weight systems with polynomial coefficients, commuting colored legs, and several vanishing criteria.

Given a Lie group $G$, a level $k$, and a Lie subgroup $H$ one can construct 2d conformal field theories by either 1.) gauging a nonanomalous $H$ symmetry of the WZW model constructed from $(G,k)$ or 2.) using an algebraic procedure known as the GKO coset construction. The two models are closely related, but not precisely the same: The gauged WZW model is identified with the corresponding GKO model coupled to a 2d topological field theory. The topological theory is characterized by a commutative Frobenius algebra derived from the endomorphisms of an algebra object in a modular tensor category constructed from $(G,H,k)$. The partition function on the torus of the two models differ by a factor of the dimension of this algebra of endomorphisms. Concrete examples are constructed and some applications to string theory and 2d Yang-Mills coupled to nonanomalous matter are briefly discussed. This paper is a summary of a longer companion paper.

In the first part of this work, we have, for a not necessarily semisimple modular category, generalized the action of the mapping class groups of surfaces on the spaces of conformal blocks to the so-called derived block spaces. In the second part presented here, we compute this action explicitly in the case of Drinfel'd doubles of finite groups over fields of positive characteristic. To do that, we connect Lyubashenko's approach to mapping class group representations with the theory of representation varieties. In this way, we are able to show that the mapping class group representations on the derived block spaces are in general different from those on the ordinary block spaces.

This is an introduction to probability, written with a quantum idea in mind, namely that the semicircle law comes first. We first discuss discrete probability, notably with the binomial and hypergeometric laws, positive and negative, and the Poisson and compound Poisson laws. Then we get into the continuous case, with the basics of the theory explained, and with as starting examples the exponential, semicircle and beta distributions. Afterwards, we investigate the central limits and normal variables, both real and complex, and with a look at Rayleigh laws, and hyperspherical laws too. Finally, we discuss a number of more specialized distributions, and more specialized techniques too, and we end with an introduction to random matrices and freeness.

For any finite modular tensor category A, we show that the associated derived $\infty$-category D(A) supports a topological quantum field theory in dimension 3. This TQFT takes the form of a symmetric monoidal functor from an $\infty$-category of surfaces with markings by objects in D(A), and appropriately decorated bordisms, to the $\infty$-category of dg vector spaces. We show that the state spaces in this theory are naturally identified with linearized mapping spaces for D(A). Though our TQFT requires markings from the derived $\infty$-category, we show that all markings can be removed after taking a homotopy truncation. The resulting unmarked TQFT produces projective mapping class group actions on cohomology in dimension 2, and in particular a projective SL_2(Z)-action on Hochschild cohomology. We expect these mapping class group actions to recover those of Lentner et al. arxiv:2003.06527 and Schweigert-Woike arxiv:2004.14343. In dimension 3 we obtain power-series valued knot invariants, and power-series valued invariants for modular tensor categories. We also obtain power-series invariants for closed 3-manifold, though these can already be calculated at the abelian level. Our derived field theories are proposed as mathematical formalizations for topological A-twists of certain N=4 supersymmetric QFTs, in dimension 3. Following physical principles, we discuss the possibility of deforming our TQFTs along local systems via an analogous (conjectural) deformation of quantum group representations along the Langlands dual group.

We categorify a lifting of the integral form of the Cartan subalgebra for quantum sl(2), via categories of equivariant coherent sheaves on projective spaces.

In Weinstein's symplectic "category", objects are symplectic manifolds and morphisms are canonical relations, which may not be composable; it is therefore only morally a category. We construct a true differential graded category which is morally dual to Weinstein's "category". The objects in this category are prequantum systems, associated to integral symplectic manifolds. The morphisms are a complex of differential forms twisted by a prequantum line bundle. We also consider the cohomology category, which turns out to be an (ordinary) linear category. In each case, Weinstein's morphisms are associated with currents dual to the forms. For the differential graded category, these are isotropic, or if preferred Lagrangian, currents, twisted by a section of the prequantum line bundle; in the case of cohomology, these currents are supported on integral Lagrangian submanifolds, equipped with a global covariant constant section of the prequantum line bundle. We then apply our methods to quantization, and show that for Kahler manifolds, a quantization in this category is related to holomorphic quantization. Along the way we show that in the case of prequantum systems, the Lefschetz action of ${\mathfrak {sl}}(2,\R)$ on differential forms, studied by Brylinski, Mathieu, Guillemin, and Tseng-Yau in the symplectic case, is promoted to an action of the superalgebra ${\mathfrak {osp}}(1|2)$ on twisted differential forms, with ${\mathfrak {sl}}(2,\R)$ as the even subalgebra. A first application of this superalgebra action is the vanishing theorem which shows the cohomology category is supported in middle dimension.

This paper introduces a two-variable polynomial invariant for oriented twisted knots, denoted by $Q_{K}^{z}(s,t)$, refining the $Q$-polynomial of N. Kamada and S. Kamada \cite{NaoSei}. We exhibit an infinite family of twisted knots indistinguishable by the $Q$-polynomial but separated by the $Q^z$-polynomial. To prove invariance, we first determine a generating set of oriented Reidemeister moves for twisted knot diagrams, extending the result of Ali \cite{Dan} for oriented virtual knots; this result is new and of independent interest, as it provides the minimal framework needed to verify invariance of any oriented twisted knot invariant. As further applications, we derive an explicit crossing change formula, obtain lower bounds on the Gordian distance between homotopic twisted knots, examine the existence of cosmetic crossings in a twisted knot diagram, and finally prove that $Q^{z}_{K}(s,t)$ is a Vassiliev invariant of order one.

Starting from the $(E_8,T_1)$ Nahm sum for the vacuum character of $T_{10}M_{\rm eff}(11,2)$, we construct a three-dimensional (3d) $\mathcal N=2$ $U(1)^8$ Chern–Simons matter theory $\mathcal T$ with level matrix $C_{E_8}$. Its $A$-twisted vacuum half-index reproduces the character exactly, while four distinguished Wilson loop half-indices agree with the remaining characters to the orders checked. Applying particle–vortex duality and gauging yields a second 3d $\mathcal N=2$ Chern–Simons matter theory $\mathcal T^\vee$ with level matrix $C_{E_8}^{-1}$. Exact superconformal-index matching supports the claim that the two theories flow to 3d $\mathcal N=4$ rank-zero mirror SCFTs. For the $(\mathcal D,D_c)$ boundary condition, the mirror map between the $A$-twisted Wilson loop half-indices of $\mathcal T$ and the $B$-twisted Wilson loop half-indices of $\mathcal T^\vee$ coincides componentwise with the Zagier transformation, realizing a Zagier duality between the complete $(E_8,T_1)$ and $(T_1,E_8)$ Nahm systems. Using the Bethe-equation formalism, we show that the corresponding $A/B$-twisted TQFT data agree up to orientation reversal and the $E_8$ framing phase $e^{-2\pi \mathrm{i}/3}$. The duality interface gives the torus bilinear $Z_{\mathcal I}(\tau)=\sum_i\chi_i(\tau)\chi_i^\vee(\tau)=\chi_{(E_8)_1}(\tau)$, where $\{\chi_i\}$ and $\{\chi_i^\vee\}$ are the corresponding five-component character bases. Thus Zagier duality is realized as part of a concrete 3d mirror and interface structure.

We give an explicit construction of monoidal structures on derived categories of right $A_\infty$-modules over an $A_\infty$-algebra $A$ equipped with a $B_\infty$-structure. Given such a $B_\infty$-algebra $A$, we construct an induction functor \[\iota\colon \mathcal{D}^{\rm{r}}_\infty(A) \longrightarrow \mathcal{D}^{\rm{bi}}_\infty(A)\] from right $A_\infty$-modules to $A_\infty$-bimodules and define \[M\boxtimes_A N=M\overset{\infty}{\otimes}_A\iota(N).\] We prove that $(\mathcal{D}^{\rm{r}}_\infty(A),\boxtimes_A,A)$ is a monoidal triangulated category: the unit and associativity constraints are induced by explicit quasi-isomorphisms of $A_\infty$-bimodules, including \[\iota(A)\simeq A \qquad\text{and}\qquad \iota(M)\overset{\infty}{\otimes}_A\iota(N)\simeq \iota(M\boxtimes_A N).\] We apply this construction to finite-dimensional Hopf algebras $H$, the Yoneda dg algebra $\mathcal{Y}(\Bbbk,\Bbbk)$ of the trivial $H$-module carries a natural brace $B_\infty$-structure, and hence its derived category carries the monoidal structure constructed above. We show that the Koszul duality functor \[\rm{Hom}_H(\mathcal{Y}(H,\Bbbk),-)\colon \mathcal{K}(\rm{Inj}\text{-}H)\longrightarrow \mathcal{D}(\mathcal{Y}(\Bbbk,\Bbbk))\] is triangulated lax monoidal, and that its restriction to the localizing subcategory generated by the injective resolution $\mathcal{Y}(H,\Bbbk)$ is a monoidal triangulated equivalence. If $H$ is local, this localizing subcategory is all of $\mathcal{K}(\rm{Inj}\text{-}H)$. In particular, this gives an alternative, purely algebraic proof of the monoidal equivalence conjectured by Krause and established by Benson–Krause through the classifying space $BG$. Finally, our examples recover the usual tensor product for graded commutative algebras and show that the resulting brace $B_\infty$ and monoidal structures can depend essentially on the chosen Hopf structure.

Let $\Sigma$ be a compact connected oriented surface of genus at least $3$ with at most one boundary component. Santharoubane associated to certain presentations of the mapping class group modulo its center a finitely presented algebra equipped with a canonical surjection onto the generic Kauffman bracket skein algebra of $\Sigma$, and conjectured that a suitable choice yields an algebra isomorphic to the skein algebra. We show that every algebra arising from this construction admits an augmentation character, whereas the generic skein algebra of $\Sigma$ admits no unital character over $\mathbb Q(A)$. The latter obstruction follows from the intersection-one Dehn-twist identity together with a $4$-holed-sphere skein relation. Consequently, the conjectured isomorphism does not hold as stated.