arXiv research

Quantum theory constructs a group and skein module for knot complements.

Quantum trace map connects Teichmüller theory and quantum groups.

Constructs quantum representations of mapping class groups using skein theory.

Paper connects Stokes phenomena to quantum groups and Poisson-Lie groups.

In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra $\mathfrak{sl}(2|1)$. This construction based on nilpotent irreducible finite dimensional representations of quantum group $\mathcal{U}_ξ\mathfrak{sl}(2|1)$ where $ξ$ is a root of unity of odd …

Modified Hennings invariant defined using quantum groups and integrals.

New quantum invariants derived from unrolled quantum groups match existing Hennings invariants.

Quantum Frobenius map for $SL_3$ skein modules constructed and described.

Witt algebra acts on categorified quantum groups in type A.

We show how networks of Wilson lines realize quantum groups U_q(sl(m)), for arbitrary m, in 3d SU(N) Chern-Simons theory. Lifting this construction to foams of surface operators in 4d theory we find that rich structure of junctions is encoded in combinatorics of planar diagrams. For a particular choice of surface opera…

We explore homotopies in quantum field theory formalism.

New framework for quantum invariants of 3-manifolds using homology.

Spin networks boost quantum algorithms solving SU(2) symmetric problems.

We construct knot invariants categorifying the quantum knot variants for all representations of quantum groups. We show that these invariants coincide with previous invariants defined by Khovanov for sl(2) and sl(3) and by Mazorchuk-Stroppel and Sussan for sl(n). Our technique uses categorifications of the tensor produ…

We adapt some of the methods of quantum Teichmüller theory to construct a family of representations of the pure braid group of the sphere.

Quantum affine bundles are quantum principal bundles with affine quantum structure groups. A general theory of quantum affine bundles is presented. In particular, a detailed analysis of differential calculi over these bundles is performed, including the description of a natural differential calculus over the structure …

We construct knot invariants from the radical part of projective modules of restricted quantum groups. We also show a relation between these invariants and the colored Alexander invariants.

In this paper we construct the quantum group, at roots of unity, of abelian Chern-Simons theory. We then use it to model classical theta functions and the actions of the Heisenberg and modular groups on them.

Quantum link invariants derived from skein algebras.

A Hermitian TQFT from non-semisimple quantum sl(2) modules.

Constructs 3D topological field theories from a specific quantum group, linking to physics invariants.

Constructs manifolds from quantum codes with novel geometric properties.

Refines quantum annular homology using stable homotopy methods.

Homological model for quantum representations of mapping class groups.

Study of quantum decorated character stacks and their quantizations.

Extends quantum annular homology to infinite sets.

Quantum gravity yields mapping class group representations.

The abstract semiclassicalises quantum group principal bundles to Poisson geometry.

It has been conjectured that every $(2+1)$-TQFT is a Chern-Simons-Witten (CSW) theory labelled by a pair $(G,λ)$, where $G$ is a compact Lie group, and $λ\in H^4(BG;Z)$ a cohomology class. We study two TQFTs constructed from Jones' subfactor theory which are believed to be counterexamples to this conjecture: one is the…

The relationship between expectation and price is commonly established with two principles: no-arbitrage, which asserts that both maps are positive; and equivalence, which asserts that the maps share the same null events. Constructed from the Arrow-Debreu securities, classical and quantum models of economics are then d…

For a group G, the notion of a ribbon G-category was introduced by the second author in a previous work with a view towards constructing 3-dimensional homotopy quantum field theories (HQFT's) with target K(G,1). We discuss here how to derive ribbon G-categories from a simple complex Lie algebra g where G is the center …

In this paper we construct the Differential calculus on the Hopf Group Coalgebra introduced by Turaev [10]. We proved that the concepts introduced by S.L.Woronowicz in constructing Differential calculus on Hopf Compact Matrix Pseudogroups (Quantum Groups)[7] can be adapted to serve again in our construction.

We consider two different quantizations of the character variety consisting of all representations of surface groups in SL_2. One is the skein algebra considered by Przytycki-Sikora and Turaev. The other is the quantum Teichmuller space introduced by Chekhov-Fock and Kashaev. We construct a homomorphism from the skein …

Curved Rickard complexes extend link homologies to arbitrary representations.

New smooth 2-group extensions from bundle gerbes on manifolds.

We construct a certain cross product of two copies of the braided dual $\tilde H$ of a quasitriangular Hopf algebra $H$, which we call the elliptic double $E_H$, and which we use to construct representations of the punctured elliptic braid group extending the well-known representations of the planar braid group attache…

Researchers develop a new quantum invariant using a matrix dilogarithm for 3-manifolds.

In this paper, we give a precise and workable definition of a quantum knot system, the states of which are called quantum knots. This definition can be viewed as a blueprint for the construction of an actual physical quantum system. Moreover, this definition of a quantum knot system is intended to represent the "quantu…

Arithmetic Dijkgraaf-Witten theory constructs analogues in Chern-Simons TQFT.

The usual construction of link invariants from quantum groups applied to the superalgebra D_{2 1,alpha} is shown to be trivial. One can modify this construction to get a two variable invariant. Unusually, this invariant is additive with respect to connected sum or disjoint union. This invariant contains an infinity of …

Researchers create projective representations of Hecke groups using TQFT.

We study quantum invariant Z(M) for cusped hyperbolic 3-manifold M. We construct this invariant based on oriented ideal triangulation of M by assigning to each tetrahedron the quantum dilogarithm function, which is introduced by Faddeev in studies of the modular double of the quantum group. Following Thurston and Neuma…

Refined invariants for 4D 2-handlebodies, linking quantum groups and cohomology.

Introduces noncommutative geometry for modeling quantum spacetime.

Given a discrete group G and a spherical G-fusion category whose neutral component has invertible dimension, we use the state-sum method to construct a 3-dimensional Homotopy Quantum Field Theory (HQFT) with target the Eilenberg-MacLane space K(G,1).

Develops a framework for designing quantum neural networks that respect symmetries.

Quantum invariants derived from $\mathcal{U}_q(\mathfrak{sl}_2)$ link holonomy.

These lectures given in Montreal in Summer 1997 are mainly based on, and form a condensed survey of, the book by N. Chriss and V. Ginzburg: `Representation Theory and Complex Geometry', Birkhauser 1997. Various algebras arising naturally in Representation Theory such as the group algebra of a Weyl group, the universal …