Basing on evaluation of the Racah coefficients for SU_q(3) (which supported the earlier conjecture of their universal form) we derive explicit formulas for all the 5-, 6- and 7-strand Wilson averages in the fundamental representation of arbitrary SU(N) group (the HOMFLY polynomials). As an application, we list the answ…
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48 results for “quantum Racah coefficients”
New method for calculating colored HOMFLY-PT polynomials for links with different symmetric representations.
problem Calculating colored HOMFLY-PT polynomials for links with arbitrary symmetric representations.
method Using quantum Racah coefficients (6j-symbols) of to simplify the evaluation.
result Multi-colored link polynomials for a specific link L7a3 are successfully evaluated.
The paper calculates Racah matrices for up to 3 strands of knots and links.
problem Systematic description of colored knot and link invariants.
method Highest weight method and use of Racah matrices.
result Explicit answers for Racah matrices and colored polynomials for 3-strand knots and links.
We describe the inclusive Racah matrices for the first non-(anti)symmetric rectangular representation R=[2,2] for quantum groups U_q(sl_N). Most of them have sizes 2, 3, and 4 and are fully described by the eigenvalue hypothesis. Of two 6x6 matrices, one is also described in this way, but the other one corresponds to t…
Study reveals hidden structure behind Racah matrices for twisted knots.
problem Understanding non-associativity in representation products of twisted knots.
method Analysis of quantum R-matrices and their eigenvalues to decompose Racah matrices.
result Discovery of pentad structure associated with universal R-matrix.
New findings on knot polynomials for specific representations.
problem Understanding HOMFLY polynomials for twist knots and their representations.
method Differential expansion of HOMFLY polynomials for twist knots and analysis of Racah matrices.
result Deviation of a specific coefficient from skew dimension in R=[333] representation.
New formula simplifies evolution of twist knots and calculates Racah matrices for rectangular representations.
problem Simplifying evolution of twist knots and calculating Racah matrices for rectangular representations.
method Developed a universal formula for triangular evolution matrix applicable to rectangular representations . Used skew characters and Macdonald polynomials.
result Explicit knowledge of twist-family evolution leads to a nearly explicit answer for Racah matrix in arbitrary rectangular representation .
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…
This paper is a next step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the Racah matrices, i.e. the whole set of mixing matrices in channels with all possible , for …
The paper calculates R and Racah matrices for SO(5) and finds Kauffman polynomials.
problem Generalizing Reshetikhin-Turaev approach to SO(2n+1) case.
method Provided R and Racah matrices for SO(5) symmetric representation.
result Found Kauffman polynomials for SO(5) symmetric representation.
Differential Expansion for antiparallel triple pretzels: the way the factorization is deformedhep-th
Factorization of DE coefficients is violated in antiparallel triple pretzels, but described elegantly.
problem Understanding the origins of factorization in double braids and its extension to antiparallel triple pretzels.
method Defect-preserving deformation from trefoil to antiparallel triple pretzels, analysis of DE coefficients.
result Factorization of DE coefficients is violated but described by an elegant formula for symmetric representations.
Quantum cluster algebras for surfaces with coefficients defined using skein theory.
problem Defining quantum cluster algebras for surfaces with coefficients.
method Introducing a skein algebra and proving it has a quantum cluster structure.
result The skein algebra of a walled surface naturally generalizes quantum cluster algebras of marked surfaces.
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.
Quantum modularity proven for specific theta series.
problem Proving quantum modularity for partial theta series with periodic coefficients.
method Explicit proof using Kontsevich-Zagier series and colored Jones polynomials.
result Kontsevich-Zagier series is a weight 3/2 quantum modular form.
Continuing the quest for exclusive Racah matrices, which are needed for evaluation of colored arborescent-knot polynomials in Chern-Simons theory, we suggest to extract them from a new kind of a double-evolution -- that of the antiparallel double-braids, which is a simple two-parametric family of two-bridge knots, gene…
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 -lattices invariant under mapping class groups.
result Restricts quantum representations to integral coefficients from to .
New quantum states capture more information, enabling advanced processing tasks.
problem Quantum information processing challenges with limited statistical information.
method Introducing Random-Coefficient Pure States (RCPS) and exploiting their higher-order statistics.
result RCPS provide richer information than density operators, enabling new quantum tasks.
We construct a general procedure to extract the exclusive Racah matrices S and \bar S from the inclusive 3-strand mixing matrices by the evolution method and apply it to the first simple representations R =[1], [2], [3] and [2,2]. The matrices S and \bar S relate respectively the maps (R\otimes R)\otimes \bar R\longrig…
Factorization of the differential expansion coefficients for HOMFLY-PT polynomials of double braids, discovered in arXiv:1606.06015 in the case of rectangular representations , is extended to the first non-rectangular representations and . This increases chances that such factorization will take p…
This work constructs a finite-dimensional projective representation for a quantum Teichmüller model.
problem Quantum Teichmüller theory and its finite-dimensional representation.
method Explicit construction using cyclic quantum dilogarithm and mutations of coefficients.
result Reconstruction of quantum Teichmüller space with explicit intertwiners.
Simpler equations derived for knot polynomials coefficients, forming a ring.
problem Complexity of knot polynomials colored with symmetric representations.
method Deriving two difference equations for quantum C-polynomials coefficients.
result Quantum C-polynomials form a ring and are much simpler than colored polynomials.
Racah matrices and higher -symbols are used in description of braiding properties of conformal blocks and in construction of knot polynomials. However, in complicated cases the logic is actually inverted: they are much better deduced from these applications than from the basic representation theory. Following the re…
Character expansion expresses extended HOMFLY polynomials through traces of products of finite dimensional R- and Racah mixing matrices. We conjecture that the mixing matrices are expressed entirely in terms of the eigenvalues of the corresponding R-matrices. Even a weaker (and, perhaps, more reliable) version of this …
Study on quantum invariant for positive links.
problem Characterizing and understanding the quantum invariant of positive links.
method Skein theory of -webs, explicit formulae, diagrammatic quantities, obstructions.
result Positive links are fibered if and only if the second coefficient of the polynomial is 1.
Quantum basis coefficients are positive integers for framed local systems on surfaces.
problem Positivity of quantum basis coefficients for framed local systems on surfaces.
method Introduced a graph to solve a combinatorial ordering problem about ideal triangulations and closed curves on surfaces.
result Laurent coefficients of quantum basis elements are positive integers.
Quantum K-theory of quintic 3-fold conjectured with non-polynomial coefficients.
problem Reconstructing quantum K-theory for quintic 3-fold.
method Formulated explicit conjecture for small J-function and its q-difference equation.
result Coefficients of q-difference equations are non-polynomial functions of Gopakumar-Vafa invariants.
This paper is a new step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R^3->Q with all possible Q, for R=[3,1]. The calculation is made possible …
Efficient algorithms for WRT invariants of torus bundles using algebraic structures.
problem Computing topological invariants of 3-manifolds is generally intractable.
method Embedding skein algebra into symmetric subalgebra at roots of unity for polynomial-time classical computation and using quantum algorithms for exponential space advantage.
result Polynomial-time classical computation and quantum algorithms for WRT invariants of torus bundles.
Quantum invariants for fibered links determined by genus and Hopf invariant.
problem Quantum invariants of fibered links in .
method Genus bounds and Giroux-Goodman theorem on fiber surfaces.
result Top coefficient of ADO invariant is determined by Hopf invariant.
Quantum heat tracesmath-ph
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…
New method uses resurgent analysis to determine growth rate of quantum field theory coefficients.
problem Determining the growth rate of quantum field theory coefficients.
method Resurgence analysis on the Stokes line, leading to transseries decomposition and continued across natural boundary.
result Essential exponent of growth has Cardy-like interpretation as effective central charge.
In this paper we present some conditions for the (strong) stabilizability of an n-D Quantum MIMO system P(X). It contains two parts. The first part is to introduce the n-D Quantum MIMO systems where the coefficients vary in the algebra of Q-meromorphic functions. Then we introduce some conditions for the stabilizabilit…
Lossy compression of statistical data using quantum annealing.
problem Efficiently compressing statistical floating-point data.
method Representation learning with binary variables, classical optimization of basis vectors, quantum annealing for coefficients, bias correction.
result Quantum annealing shows promising results with 3.5x better compression than neural-network autoencoders.
Quantum cluster algebra constructed from web skein relations on surfaces.
problem Quantization of cluster structures on moduli spaces of SL3 local systems.
method Constructing a quantum cluster algebra inside the skew-field of a skein algebra of unpunctured surfaces.
result Laurent expressions of webs in clusters have positive coefficients.
We propose a version of the non-relativistic quantum mechanics in which the pure states of a quantum system are described as sections of a Hilbert (generally infinitely-dimensional) fibre bundle over the space-time. There evolution is governed via (a kind of) a parallel transport in this bundle. Some problems concernin…
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.
A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich -parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformatio…
Quantum algorithm estimates mean with sub-Gaussian error.
problem Estimating mean of quantum-computed random variables.
method Quantum mean estimation algorithm with sub-Gaussian error rate.
result Achieves nearly-optimal quadratic speedup over classical methods.
The Quantum Modularity Conjecture of Zagier predicts the existence of a formal power series with arithmetically interesting coefficients that appears in the asymptotics of the Kashaev invariant at each root of unity. Our goal is to construct a power series from a Neumann-Zagier datum (i.e., an ideal triangulation of th…
We present a cocycle model for elliptic cohomology with complex coefficients in which methods from 2-dimensional quantum field theory can be used to rigorously construct cocycles. For example, quantizing a theory of vector bundle-valued fermions yields a cocycle representative of the elliptic Thom class. This construct…
Quantum methods improve option pricing accuracy.
problem Pricing financial derivatives using Monte Carlo integration.
method Hybrid classical-quantum methods using Fourier series and QML.
result Quantum methods achieve remarkable accuracy in option pricing.
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.
External edge condition and group cohomologies associated with the quantum Clebsch-Gordan conditionmath.GT
In this article we determine the structure of a twisted first cohomology group of the first homology of a trivalent graph with a coefficient associated with the quantum Clebsch-Gordan condition. As an application we give a characterization of a combinatorial property, the external edge condition, which is defined by th…
Short note observes quantum Hochschild homology as a composition of known operations.
problem Quantum Hochschild homology as a new invariant of annular links.
method Observes quantum Hochschild homology as a composition of two known operations.
result Quantum Hochschild homology is a valid invariant of annular links.
Applications of Quantum Tunneling effect have long gone beyond the traditional physical meaning. Initially created by Gamow to explain α-decay of nuclear particles, along the time, quantum tunneling found fertile domain of research in chemistry and recently in biology, where the new discipline of Quantum Biology emerge…
Study calculates instanton homology for simple braids, linking to Fano variety quantum cohomology.
problem Calculating instanton homology for specific braids.
method Using local coefficients and algebraic curves, calculates homology groups and their module structures.
result Equivalent to computing quantum cohomology of a moduli space of parabolic bundles.
Quantum-inspired method optimizes portfolio selection.
problem Optimizing asset allocation in finance.
method Combining quantum-inspired and conventional optimization methods.
result Faster and more accurate portfolio optimization solutions.
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 representation.