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.
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 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.
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.
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 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.
Unified framework combines trace-induced quantum kernels for improved machine learning models.
problem Improving performance of quantum machine learning models using trace-induced kernels.
method Developed a unified framework combining various trace-induced quantum kernels, including global fidelity and local projected kernels, as Lego kernels.
result Local projected kernels can achieve comparable performance to global fidelity kernels with fewer quantum resources.
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.
Center identified in stated skein algebra for quantum traces.
problem Understanding the center of the stated skein algebra.
method Analyzing the algebra as a generalization of Kauffman bracket skein algebra, focusing on the case when the quantum parameter is a root of unity.
result Simple description and dimension calculation of the center over the center module.
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.
The abstract discusses new 3-manifold invariants and ETQFTs from Lie superalgebra representations.
problem Developing new 3-manifold invariants and ETQFTs from Lie superalgebra representations.
method Examining two m-traces in the category of representations over quantum sl(m∣n), considering quotients, and conjecturing generalizations. result Quotients of perturbative modules over quantum sl(m∣n) lead to 3-manifold invariants and ETQFTs. Direct formula found for ADO invariants from homological representations.
problem Computing ADO invariants from quantum group representations.
method Direct homological formula for ADO invariants using partial traces of homological representations.
result Direct formula for ADO invariants without further truncations.
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.
We study new invariants of elliptic partial differential operators acting on sections of a vector bundle over a closed Riemannian manifold that we call the relativistic heat trace and the quantum heat traces. We obtain some reduction formulas expressing these new invariants in terms of some integral transforms of the u…
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. 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 …
Develops trace class operators and inverse Laplacian theory for infinite dimensions.
problem Understanding trace class operators and inverse Laplacian on infinite dimensional spaces.
method Presentation of trace class operators and construction of inverse Laplacian on closed manifolds.
result Original trace computations involving the inverse Laplacian on the torus.
Extends quantum trace map to SL3(C) for 3D surfaces.
problem Generalizing quantum trace map to higher dimensions.
method Definition of SL3(C) quantum trace invariant.
result Construction of SL3(C) quantum trace map.
New methods derive a generalized Frenkel trace formula for Lie groups.
problem Deriving a generalized Frenkel trace formula for Lie groups.
method Applying supersymmetric localization to quantum mechanical and gauged sigma models.
result Presented two complementary approaches for the derivation of the trace formula.
Quantization of the Teichmüller space of a punctured Riemann surface S is an approach to 3-dimensional quantum gravity, and is a prototypical example of quantization of cluster varieties. Any simple loop γ in S gives rise to a natural trace-of-monodromy function I(γ) on the Teichmüller space. For any…
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 …
Quantizes geodesic lengths in Teichmüller spaces using algebraic methods.
problem Constructing quantized geodesic lengths for Teichmüller spaces.
method Developed quantum trace maps and investigated algebraic structures.
result Showed a recursion relation and commutation properties for quantized trace-of-monodromy.
Introduces a new model for mapping matrices to matrices, subsuming linear regression.
problem Learning matrix-to-matrix mappings from data.
method Partial trace regression model, leveraging quantum information theory.
result Relevance demonstrated in matrix-to-matrix regression and positive semidefinite matrix completion.
Witt algebra acts on categorified quantum groups in type A.
problem Action of Witt algebra on categorified quantum groups.
method Construction of action on categorified quantum group and foams.
result Action of Witt algebra on foams recovers previous results.
Proves super-version of index theorem from algebraic cobordism invariants.
problem Cobordism invariants in supersymmetric quantum mechanics.
method Trace methods for deformation quantization.
result Recovery of cobordism invariant using trace methods.
Spectral sequence connects knot homologies via algebraic geometry.
problem Connecting algebraic and geometric knot homologies.
method Bigraded spectral sequence from gl(0)-homology to knot Floer homology.
result Constructs a Bockstein-type spectral sequence.
Motivated by topology, we develop a general theory of traces and shadows for an endobicategory, which is a~pair: bicategory C and endobifunctor Σ:C→C. For a graded linear bicategory and a fixed invertible parameter q, we quantize this theory by using the endofunctor Σq such th…
Quantum ML predicts data with improved speed and accuracy.
problem Predicting data using maximum likelihood in a quantum setting.
method Quantum states embedding and minimization of quantum relative entropy.
result Unified framework for classical and quantum LLMs with performance guarantees.
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…
In earlier work, Helen Wong and the author discovered certain "miraculous cancellations" for the quantum trace map connecting the Kauffman bracket skein algebra of a surface to its quantum Teichmueller space, occurring when the quantum parameter q is a root of unity. The current paper is devoted to giving a more repr…
Quantum stochastic flow computes heat kernel traces for Ricci flat manifolds.
problem Computing heat kernel traces for Ricci flat manifolds.
method Quantum stochastic differential equation (qsde) on Fock space over L2 differential 1-forms, adapted flow construction. result Trace of the connection Laplacian heat kernel can be computed over any compact Ricci-flat Riemannian manifold.
Introduces LRY skein algebras generalizing Kauffman bracket and Roger-Yang skein algebras.
problem Generalizing skein algebras for surfaces with arbitrary ground rings.
method Constructs LRY skein algebras, quantum traces, and Dehn-Thurston coordinates.
result LRY skein algebras are domains, have degenerations to monomial subalgebras of quantum tori, and are orderly finitely generated.
Quantum channels' contraction under privacy constraints studied.
problem Understanding the privacy constraints on quantum channel contractions.
method Established upper bounds on contraction coefficients for specific divergences under QLDP constraints.
result Upper bounds and full characterization of contraction coefficients for specific quantum distances.
We establish a relation between the trace evaluation in SO(3) topological quantum field theory and evaluations of a topological Tutte polynomial. As an application, a generalization of the Tutte golden identity is proved for graphs on the torus.
Let F be a finite type surface and ζ a complex root of unity. The Kauffman bracket skein algebra Kζ(F) is an important object in both classical and quantum topology as it has relations to the character variety, the Teichmüller space, the Jones polynomial, and the Witten-Reshetikhin-Turaev Topological Quantum Fie…
We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the quantum Hamiltonian dynamics associated with classical Hamiltonian flows over closed prequantized symplectic manifolds in the context of geometric quantization of Kostant and Souriau. We express the associated evolution operators via paral…
By introducing a finer version of the Kauffman bracket skein algebra, we show how to decompose the Kauffman bracket skein algebra of a surface into elementary blocks corresponding to the triangles in an ideal triangulation of the surface. The new skein algebra of an ideal triangle has a simple presentation. This gives …
We study the geometry of determinant line bundles associated to Dirac operators on compact odd dimensional manifolds. Physically, these arise as (local) vacuum line bundles in quantum gauge theory. We give a simplified derivation of the commutator anomaly formula using a construction based on noncyclic trace extensions…
Quantum invariants for surfaces in 4D 2-handlebodies.
problem Quantum invariants of ribbon surfaces in 4D 2-handlebodies.
method Unimodular ribbon categories, labeled Kirby graphs, and modified traces.
result Recovery and generalization of existing invariants.
The density matrices are positively semi-definite Hermitian matrices of unit trace that describe the state of a quantum system. The goal of the paper is to develop minimax lower bounds on error rates of estimation of low rank density matrices in trace regression models used in quantum state tomography (in particular, i…
We construct a Hennings type logarithmic invariant for restricted quantum sl(2) at a 2p-th root of unity. This quantum group U is not braided, but factorizable. The invariant is defined for a pair: a 3-manifold M and a colored link L inside M. The link L is split into two parts colored…
Quantum probability theory reveals hidden structure in joint probability distributions.
problem Understanding hidden structure in joint probability distributions.
method Modeling joint probability distributions as density operators and applying partial trace.
result Decoding extra information in reduced density operators that captures subsystem interactions.
Study centers of quantum tori and skein algebras for even roots of unity.
problem Understanding the center of quantum tori and skein algebras for even roots of unity.
method Analyzing quantum tori and skein algebras, computing PI-degree, and decomposing matrices.
result PI-degrees of quantum tori and skein algebras are the same.
In this article we construct link invariants and 3-manifold invariants from the quantum group associated with Lie superalgebra sl(2∣1). This construction based on nilpotent irreducible finite dimensional representations of quantum group Uξsl(2∣1) where ξ is a root of unity of odd …
New proof of SL(n) skein algebra for twice punctured sphere, showing it's a polynomial algebra.
problem Proving the structure of SL(n) skein algebra for a specific surface.
method Constructing a linear basis of explicit SL(n) webs, proving spanning and linear independence.
result SL(n) skein algebra of twice punctured sphere is a commutative polynomial algebra in n-1 generators.
We prove that the balanced Chekhov-Fock algebra of a punctured triangulated surface is isomorphic to a skein algebra which is a deformation of the algebra of regular functions of some abelian character variety. We first deduce from this observation a classification of the irreducible representations of the balanced Che…
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 …