arXiv research

Proves rigidity of SU(2) and SO(3) quantum representations at prime levels.

Quantum representations of mapping class groups are locally rigid at prime levels.

New properties established for SO(3) quantum representations, showing density and surjectivity.

The paper connects knot homology, quantum 6j-symbols, and complements of knots.

We give an irreducible decomposition of the so-called local representations (see arXiv:0707.2151) of the quantum Teichmüller space $\mathcal{T}_q(Σ)$ where $Σ$ is a punctured surface of genus $g>0$ and $q$ is a primitive $N$-th root of unity with $N$ odd. As an application, we construct a family of representations of t…

Study on kernels of SO(3) WRT representations for surfaces of genus g≥3.

The paper proves properties of quantum representations and their Toledo invariants.

A cluster variety of Fock and Goncharov is a scheme constructed by gluing split algebraic tori, called seed tori, via birational gluing maps called mutations. In quantum theory, the ring of functions on seed tori are deformed to non-commutative rings, represented as operators on Hilbert spaces. Mutations are quantized …

For $N \geq 2$, we study a certain sequence $(ρ_p^{(c_p)})$ of N-dimensional representations of the mapping class group of the one-holed torus arising from SO(3)-TQFT, and show that the conjecture of Andersen, Masbaum, and Ueno \cite{1} holds for these representations. This is done by proving that, in a certain basis a…

We prove that for each sufficiently complicated orientable surface $S$, there exists an infinite image linear representation $ρ$ of $π_1(S)$ such that if $γ\inπ_1(S)$ is freely homotopic to a simple closed curve on $S$, then $ρ(γ)$ has finite order. Furthermore, we prove that given a sufficiently complicated orientable…

Defines a new link invariant for type D webs.

Fast algorithm for braid group Hecke representation, applied to knot invariants.

Quantum mechanics applied to credit loans for better repayment schedules.

We quantize the interaction of gravity with Yang-Mills and spinor fields, hence offering a quantum theory incorporating all four fundamental forces of nature. Using canonical quantization we obtain solutions of the Wheeler-DeWitt equation in a vector bundle and the method of second quantization leads to a symplectic ve…

Let $L$ be a link and $Φ^{A}_{L}(q)$ its link invariant associated with the vector representation of the quantum (super)algebra $U_{q}(A)$. Let $F_{L}(r,s)$ be the Kauffman link invariant for $L$ associated with the Birman--Wenzl--Murakami algebra $BWM_{f}(r,s)$ for complex parameters $r$ and $s$ and a sufficiently lar…

Andersen, Masbaum and Ueno conjectured that certain quantum representations of surface mapping class groups should send pseudo-Anosov mapping classes to elements of infinite order (for large enough level $r$). In this paper, we relate the AMU conjecture to a question about the growth of the Turaev-Viro invariants $TV_r…

This work introduces a new quantum kernel, quantum tangent kernel, for improved performance.

Proves a conjecture about matrix orders for pseudo-Anosov maps.

Holonomy invariants from $\mathrm{SL}_2(\mathbb{C})$ link complements detect link geometry.

Restricts quantum representations of mapping class groups to integral coefficients.

The generalized Legendre transform method of Lindstrom and Rocek yields hyperkaehler metrics from holomorphic functions. Its main ingredients are sections of ${\cal O}(2j)$ bundles over the twistor space satisfying a reality condition with respect to antipodal conjugation on the hyperkaehler sphere of complex structure…

The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular H…

The Jones-Witten theory gives rise to representations of the (extended) mapping class group of any closed surface Y indexed by a semi-simple Lie group G and a level k. In the case G=SU(2) these representations (denoted V_A(Y)) have a particularly simple description in terms of the Kauffman skein modules with parameter …

We derive the quantum Teichmüller space, previously constructed by Kashaev and by Fock and Chekhov, from tensor products of a single canonical representation of the modular double of the quantum plane. We show that the quantum dilogarithm function appears naturally in the decomposition of the tensor square, the quantum…

The paper explores mapping class groups and their quantum field theory representations.

This work constructs a finite-dimensional projective representation for a quantum Teichmüller model.

We show that when $r \geq 5$ is prime, the SO(3) Witten-Reshetikhin-Turaev quantum invariants for three-manifolds at the level $r$ form a dense set in the complex plane. This confirms a conjecture of Larsen and Wang.

We generalize the asymptotic faithfulness of the skein quantum $SU(2)$ representations of mapping class groups of orientable closed surfaces to skein $SU(3)$. Skein quantum representations of mapping class groups are different from the Reshetikin-Turaev ones from quantum groups or geometric quantization because they ar…

Let $q$ be a $2N$th root of unity where $N$ is odd. Let $U_q(sl_2)$ denote the quantum group with large center corresponding to the lie algebra $sl_2$ with generators $E,F,K$, and $K^{-1}$. A semicyclic representation of $U_q(sl_2)$ is an $N$-dimensional irreducible representation $ρ:U_q(sl_2)\rightarrow M_N(\mathbb{C}…

We develop the general theory for the construction of Extended Topological Quantum Field Theories (ETQFTs) associated with the Costantino-Geer-Patureau quantum invariants of closed 3-manifolds. In order to do so, we introduce relative modular categories, a class of ribbon categories which are modeled on representations…

GQML uses symmetries from representation theory to improve quantum machine learning.

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…

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

Direct formula found for ADO invariants from homological representations.

New structure for quantum algebra representations.

The paper establishes isomorphisms and constructs colored versions of Lawrence representations.

QCML uses quantum geometry to represent data.

Post-quantum cryptography needed for blockchain security.

This note is a write-up of a talk given by the author at the Meeting of the Sociedade Portuguesa de Matematica in July 2012. We describe Jaeger's HOMFLY-PT expansion of the Kauffman polynomial and how to generalize it to other quantum invariants using the so-called "branching rules" for Lie algebra representations. We …

New spin on Khovanov-Rozansky homology categorifies spin link polynomial.

By now it is well established that the quantum dimensions of descendants of the adjoint representation can be described in a universal form, independent of a particular family of simple Lie algebras. The Rosso-Jones formula then implies a universal description of the adjoint knot polynomials for torus knots, which in p…

Propose a model-independent axiomatic framework for derived skein theory.

In this thesis, we give a unification of the quantum WRT invariants. Given a rational homology 3-sphere M and a link L inside, we define the unified invariants, such that the evaluation of these invariants at a root of unity equals the corresponding quantum WRT invariant. In the SU(2) case, we assume the order of the f…

Study of webs in quantum type C, proving equivalence to quantum representations.

Homological model for quantum representations of mapping class groups.

Quantum theory uses modular group representations to assign invariants to 3-manifolds.

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

Researchers define new quantum representations for a Lorentz algebra and study their Clebsch-Gordan decomposition.