Study topological quantum mechanics on orbifolds with geometric interpretation.
problem Quantum mechanical models on symplectic orbifolds.
method Explicit orbifold version of quantum HKR map and exact semi-classical approximation.
result Geometric and quantum field theoretic interpretation of orbifold algebraic index.
The HKR (Hennings-Kauffman-Radford) framework is used to construct invariants of 4-thickenings of 2-dimensional CW complexes under 2-deformations (1- and 2- handle slides and creations and cancellations of 1-2 handle pairs). The input of the invariant is a finite dimensional unimodular ribbon Hopf algebra A and an elem…
Paper connects two invariants of 3D manifolds using Hopf algebras.
problem Establishing a relation between two invariants of 3D manifolds.
method Using spherical Hopf algebras and their Drinfeld doubles, the paper connects the chromatic spherical invariant and the Hennings-Kauffman-Radford invariant.
result The chromatic spherical invariant is equal to the Hennings-Kauffman-Radford invariant for a specific type of Hopf algebra.
Invariants for 3-manifolds with embedded links using Hopf algebras.
problem Constructing invariants for 3-manifolds with embedded framed links.
method Using involutory Hopf algebras and representations of Drinfeld double.
result The invariant recovers known invariants for specific cases.
The paper shows compatibility between two quantum maps for surfaces and 3-manifolds.
problem Connecting quantum trace and UV-IR maps for surfaces and 3-manifolds.
method Analyzing compatibility under triangulation changes and using skein modules.
result Compatibility of quantum trace and UV-IR maps for surfaces and 3-manifolds.
Quantum machine learning models can approximate any continuous function.
problem Theoretical understanding of quantum feature maps in machine learning.
method Proving universal approximation property of quantum machine learning models in quantum-enhanced feature spaces.
result Quantum machine learning models are universal approximators of continuous functions.
Quantum trace map defined for 3-manifolds with torus boundaries.
problem Quantifying topological structures of 3-manifolds with torus boundaries.
method Defining a quantum trace map from skein module to a quantum torus module.
result Established a 3D quantum trace map for 3-manifolds with torus boundaries.
Restricts quantum representations of mapping class groups to integral coefficients.
problem Integrality of non-semisimple quantum representations of mapping class groups.
method Exhibits explicit bases of states spaces that span Z[ζ]-lattices invariant under mapping class groups. result Restricts quantum representations to integral coefficients from Q(ζ) to Z[ζ]. 3D quantum trace map connects 3-manifold quantizations.
problem Quantization of 3-manifold character varieties.
method Study of stated skein modules and face suspensions.
result Existence of 3D quantum trace map proved.
Quantum trace maps for surfaces are shown to be compatible under triangulations.
problem Constructing and understanding quantum trace maps for surfaces.
method Developed quantum mutation maps between subalgebras of quantum torus algebras for different triangulations.
result Quantum trace maps are natural and independent of triangulation choices.
Quantum duality map extended to general marked surfaces and its compatibility with skein algebras proven.
problem Generalizing quantum duality map to general marked surfaces and proving its compatibility with skein algebras.
method Generalized quantum duality map, reduced stated skein algebras, quantum trace maps, skein lifting.
result Compatibility of quantum duality map with skein algebras proven.
Quantum Frobenius map for SL3 skein modules constructed and described.
problem Constructing a quantum Frobenius map for SL3 skein modules. method Using threading polynomials and the Frobenius map of Parshall-Wang for quantum group Oq(SL3). result Described the quantum Frobenius map for SL3 skein modules. We study quantum moment maps of G-invariant star products, which are a quantum analogue of the moment map for classical Hamiltonian systems. Introducing an integral representation, we show that any quantum moment map for a G-invariant star product is differentiable. This property gives us a new method for the class…
Defines a map connecting 3d-index and skein module.
problem Connecting mathematical physics predictions with topological quantum field theory.
method Defines a map from skein module to Laurent series ring.
result The map fulfills a supersymmetry prediction and is part of a conjectural topological quantum field theory.
Quantum trace map defines invariants for knots and links, confirming a length conjecture.
problem Defining invariants for knots and links in hyperbolic 3-manifolds.
method Introducing a quantum trace map for ideally triangulated knot complements, combining with state-integral models.
result Perturbative invariants determine an asymptotic expansion of the Jones polynomial, confirming the length conjecture.
A new hybrid framework reduces quantum runtime and noise effects.
problem Challenges in deploying deep QFMs on real quantum hardware.
method Iterative Quantum Feature Maps (IQFMs) combining shallow QFMs and classical augmentation weights.
result Numerical experiments show IQFMs outperforming quantum convolutional neural networks.
The paper explores mapping class groups and their quantum field theory representations.
problem Understanding finite dimensional representations of mapping class groups.
method Survey of topological quantum field theory aspects.
result Discussion of finite dimensional representations in quantum field theory.
Propose a new 3d quantum trace map that agrees with Garoufalidis and Yu's construction and extends to certain manifolds with ideal triangulated boundaries.
problem Relationship between two constructions of 3d quantum trace maps.
method Propose a new 3d quantum trace map.
result Proposed 3d quantum trace map agrees with Garoufalidis and Yu's construction and extends to certain manifolds with ideal triangulated boundaries.
Quantum trace map connects Teichmüller theory and quantum groups.
problem Connecting quantum groups to Teichmüller theory for knots.
method Quantum snakes technology to relate Fock-Goncharov monodromy matrices to quantum SL_n.
result Quantized Fock-Goncharov matrices satisfy quantum SL_n relations.
Quantum traces embed into quantum tori for surface skein algebras.
problem Embedding stated skein algebras into quantum tori.
method Two different embeddings using quantum trace maps and lambda length coordinates.
result Quantum cluster algebra of Muller equals reduced stated skein algebra.
Quantum gravity yields mapping class group representations.
problem Quantization of 3-manifold metrics and mapping class group invariance.
method Quantum dilogarithm functions and mapping class group action.
result Families of unitary representations of mapping class groups.
Proves rigidity of SU(2) and SO(3) quantum representations at prime levels.
problem Quantum representations of mapping class groups at prime levels.
method Ocneanu rigidity of modular categories and harmonic representatives in Hodge theory.
result Rigidity of SU(2) and SO(3) quantum representations at all prime levels for closed surfaces of genus at least 7.
Diffusion maps help learn complex quantum phase transitions from data.
problem Learning quantum phase transitions from experimental data is challenging.
method Diffusion maps for nonlinear dimensionality reduction and spectral clustering.
result Diffusion maps can learn complex phase transitions unsupervised.
Quantum representations of mapping class groups are locally rigid at prime levels.
problem Locally rigid properties of quantum representations of mapping class groups.
method Proving local rigidity for Fibonacci representations of mapping class groups at prime levels.
result Local rigidity of Fibonacci representations of mapping class groups at prime levels.
We show how the quantum trace map of Bonahon and Wong can be constructed in a natural way using the skein algebra of Muller, which is an extension of the Kauffman bracket skein algebra of surfaces. We also show that the quantum Teichmüller space of a marked surface, defined by Chekhov-Fock (and Kashaev) in an abstract …
New connections found for quantum flag manifolds modules.
problem Unique connections for relative line modules over quantum flag manifolds.
method Applied general results on quantum principal bundles to Heckenberger-Kolb calculi.
result Found bimodule connections with invertible maps.
Authors prove quantum invariant conjecture for figure-eight knot complement.
problem Connecting quantum invariants of surface diffeomorphisms to hyperbolic volumes.
method Analyzes the simplest case of a one-puncture torus and figure-eight knot complement.
result Proves conjecture linking quantum invariant to hyperbolic volume.
New quantum kernels avoid overfitting by combining local and global components.
problem Exponential concentration in quantum kernels leads to overfitting.
method Local-global quantum kernels combining small subsystem and full-system measurements.
result Demonstrated benign overfitting in local-global quantum kernels.
A Hermitian TQFT from non-semisimple quantum sl(2) modules.
problem Constructing a Hermitian TQFT from a non-semisimple category.
method Endowed a non-semisimple category of quantum sl(2) modules with a Hermitian structure and proved the resulting TQFT is Hermitian.
result Projective representations of the mapping class group in indefinite unitary matrices.
We define a canonical map from a certain space of laminations on a punctured surface into the quantized algebra of functions on a cluster variety. We show that this map satisfies a number of special properties conjectured by Fock and Goncharov. Our construction is based on the "quantum trace" map introduced by Bonahon …
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…
We prove that either the images of the mapping class groups by quantum representations are not isomorphic to higher rank lattices or else the kernels have a large number of normal generators. Further we show that the images of the mapping class groups have nontrivial 2-cohomology, at least for small levels. For this pu…
Quantum autoencoders allow for reducing the amount of resources in a quantum computation by mapping the original Hilbert space onto a reduced space with the relevant information. Recently, it was proposed to employ approximate quantum adders to implement quantum autoencoders in quantum technologies. Here, we carry out …
We prove asymptotic faithfulness for the quantum Sp(4) mapping class group representation. This provides the first example of asymptotic faithfulness lying outside of the An family. The methods used are generalized from the proof of asymptotic faithfulness for skein SU(2)k mapping class group represent…
Homological model for quantum representations of mapping class groups.
problem Investigate linearity of mapping class groups using quantum representations.
method Homological action on configuration space with twisted coefficients.
result Identify subrepresentation equivalent to quantum sl2 representation. In this thesis we study the classical and quantum momentum maps and the theory of reduction. We focus on the notion of momentum map in Poisson geometry and we discuss the classification of the momentum map in this framework. Furthermore, we describe the so-called Poisson Reduction, a technique that allows us to reduce …
Defines map from skein module to Habiro ring for quantum modularity.
problem Quantum modularity conjecture and its arithmetic aspects.
method Defining a map from Kauffman bracket skein module to Habiro ring using Witten's conjecture.
result Image is an effectively computable module of finite rank.
The volume conjecture is extended for surface diffeomorphisms with quantum invariants.
problem Extending the volume conjecture for quantum invariants of surface diffeomorphisms.
method Relating asymptotics of quantum invariants to hyperbolic cone structures on mapping tori.
result The conjecture is proven for a specific case of the once-punctured torus bundle.
Quantum map counts BPS states in special theories.
problem Computing protected spin characters in class S theories.
method Geometric approach from 5D supersymmetric Yang-Mills theory.
result Explicit computation of protected spin characters in various examples.
Quantum kernel improves probabilistic time series forecasting.
problem Quantifying uncertainty in probabilistic time series predictions.
method Integrates quantum kernel with Gaussian process regression.
result Quantum kernel enhances forecasting performance.
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…
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 …
New framework for quantum invariants of 3-manifolds using homology.
problem Quantum invariants of 3-manifolds.
method Homological construction of mapping class group representations extended to cobordisms.
result Construction of a (2+1)-dimensional TQFT assigning twisted homology to surfaces. Quantum states associated with subsets of product manifolds are separable.
problem Characterizing quantum states associated with subsets of product manifolds.
method Using holomorphic sections of quantum line bundles and restriction maps.
result Quantum states associated with finite unions of products are separable.
The central extension of the mapping class groups of punctured surfaces of finite type that arises in quantum Teichmüller theory is 12 times the Meyer class plus the Euler classes of the punctures. This is analogous to the result obtained in \cite{FS} for the Thompson groups.
The paper studies properties of stated SL(n)-skein algebras and their centers.
problem Properties of stated SL(n)-skein algebras and their centers.
method Quantum trace maps and embeddings into quantum tori.
result Finitely generation and PI-degrees of centers of stated SL(n)-skein algebras.
We show that the reduced quantum hyperbolic invariants of pseudo-Anosov diffeomorphisms of punctured surfaces are intertwiners of local representations of the quantum Teichmüller spaces. We characterize them as the only intertwiners that satisfy certain natural cut-and-paste operations of topological quantum field theo…
Quantum traces map skein algebras to Fock-Goncharov spaces.
problem Establishing quantum traces between skein algebras and Fock-Goncharov spaces.
method Defining and proving properties of quantum traces for SLn-skein algebras. result Existence and properties of quantum traces for SLn-skein algebras.