Quantum Group SU Research Articles

Tensor Network Renormalization with Fusion Charges—Applications to 3D Lattice Gauge Theory

Tensor network methods are powerful and efficient tools for studying the properties and dynamics of statistical and quantum systems, in particular in one and two dimensions. In recent years, these methods have been applied to lattice gauge theories, yet these theories remain a challenge in ( 2 + 1 ) dimensions. In this article, we present a new (decorated) tensor network algorithm, in which the tensors encode the lattice gauge amplitude expressed in the fusion basis. This has several advantages—firstly, the fusion basis does diagonalize operators measuring the magnetic fluxes and electric charges associated to a hierarchical set of regions. The algorithm allows therefore a direct access to these observables. Secondly the fusion basis is, as opposed to the previously employed spin network basis, stable under coarse-graining. Thirdly, due to the hierarchical structure of the fusion basis, the algorithm does implement predefined disentanglers. We apply this new algorithm to lattice gauge theories defined for the quantum group SU ( 2 ) k and identify a weak and a strong coupling phase for various levels k . As we increase the level k , the critical coupling g c decreases linearly, suggesting the absence of a deconfining phase for the continuous group SU ( 2 ) . Moreover, we illustrate the scaling behaviour of the Wilson loops in the two phases.

Recurrence relation for the 6j-symbol of suq(2) as a symmetric eigenvalue problem

A well-known recurrence relation for the 6j-symbol of the quantum group su q(2) is realized as a tridiagonal, symmetric eigenvalue problem. This formulation can be used to implement an efficient numerical evaluation algorithm, taking advantage of existing specialized numerical packages. For convenience, all formulas relevant for such an implementation are collected in Appendix A. This realization is a byproduct of an alternative proof of the recurrence relation, which generalizes a classical (q = 1) result of Schulten and Gordon and uses the diagrammatic spin network formalism of Temperley–Lieb recoupling theory to simplify intermediate calculations.

POISSON BOUNDARIES OVER LOCALLY COMPACT QUANTUM GROUPS

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a noncommutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SU q(2) arising from measures on its spectrum.

Character expansion for HOMFLY polynomials. II. Fundamental representation. Up to five strands in braid

Character expansion is introduced and explicitly constructed for the (non-colored) HOMFLY polynomials of the simplest knots. Expansion coefficients are not the knot invariants and can depend on the choice of the braid realization. However, the method provides the simplest systematic way to construct HOMFLY polynomials directly in terms of the variable A=q^N: a much better way than the standard approach making use of the skein relations. Moreover, representation theory of the simplest quantum group SU_q(2) is sufficient to get the answers for all braids with m<5 strands. Most important we reveal a hidden hierarchical structure of expansion coefficients, what allows one to express all of them through extremely simple elementary constituents. Generalizations to arbitrary knots and arbitrary representations is straightforward.

Dimensional Reduction Over the Quantum Sphere and Non-Abelian q-Vortices

We extend equivariant dimensional reduction techniques to the case of quantum spaces which are the product of a Kaehler manifold M with the quantum two-sphere. We work out the reduction of bundles which are equivariant under the natural action of the quantum group SU_q(2), and also of invariant gauge connections on these bundles. The reduction of Yang-Mills gauge theory on the product space leads to a q-deformation of the usual quiver gauge theories on M. We formulate generalized instanton equations on the quantum space and show that they correspond to q-deformations of the usual holomorphic quiver chain vortex equations on M. We study some topological stability conditions for the existence of solutions to these equations, and demonstrate that the corresponding vacuum moduli spaces are generally better behaved than their undeformed counterparts, but much more constrained by the q-deformation. We work out several explicit examples, including new examples of non-abelian vortices on Riemann surfaces, and q-deformations of instantons whose moduli spaces admit the standard hyper-Kaehler quotient construction.

The Baum–Connes conjecture for free orthogonal quantum groups

We prove an analogue of the Baum–Connes conjecture for free orthogonal quantum groups. More precisely, we show that these quantum groups have a γ-element and that γ = 1 . It follows that free orthogonal quantum groups are K-amenable. We compute explicitly their K-theory and deduce in the unimodular case that the corresponding reduced C ⁎ -algebras do not contain nontrivial idempotents. Our approach is based on the reformulation of the Baum–Connes conjecture by Meyer and Nest using the language of triangulated categories. An important ingredient is the theory of monoidal equivalence of compact quantum groups developed by Bichon, De Rijdt and Vaes. This allows us to study the problem in terms of the quantum group SU q ( 2 ) . The crucial part of the argument is a detailed analysis of the equivariant Kasparov theory of the standard Podleś sphere.

THE 3D SPIN GEOMETRY OF THE QUANTUM TWO-SPHERE

We study a three-dimensional differential calculus [Formula: see text] on the standard Podleś quantum two-sphere [Formula: see text], coming from the Woronowicz 4D+ differential calculus on the quantum group SU q(2). We use a frame bundle approach to give an explicit description of [Formula: see text] and its associated spin geometry in terms of a natural spectral triple over [Formula: see text]. We equip this spectral triple with a real structure for which the commutant property and the first order condition are satisfied up to infinitesimals of arbitrary order.

Characterization of SU q (ℓ + 1)-equivariant spectral triples for the odd dimensional quantum spheres

The quantum group SU q (ℓ + 1) has a canonical action on the odd dimensional sphere S q 2ℓ + 1 . All odd spectral triples acting on the L 2 space of S q 2ℓ + 1 and equivariant under this action have been characterized. This characterization then leads to the construction of an optimum family of equivariant spectral triples having nontrivial K -homology class. These generalize the results of Chakraborty and Pal for SU q (2).

Families of Noncommutative Instantons

These notes aim at a pedagogical introduction to recent work on deformation of spaces and deformation of vector bundles over them, which are relevant both in mathematics and in physics, notably monopole and instanton bundles. We first decribe toric noncommutative manifolds (also known as isospectral deformations) and give a detailed introduction to gauge theories on a toric four-sphere. This includes a Yang-Mills action functional with associated equations of motion and self-duality equations. We construct a particular class of instanton solutions on a SU(2) bundle with a suitable use of twisted conformal symmetries. In the second part, we describe a different deformation of an instanton bundle over the classical four-sphere by constructing a quantum group SU_q(2) bundle on a sphere which is different from the toric one.

The Dirac Operator on SUq(2)

We construct a 3+-summable spectral triple over the quantum group SU q (2) which is equivariant with respect to a left and a right action of The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.

A TWO-PARAMETER DEFORMED SUSY ALGEBRA FOR SUp/q(n)-COVARIANT (p,q)-DEFORMED FERMIONIC OSCILLATORS

We construct a two-parameter deformed SUSY algebra by constructing SUSY generators which are bilinears of n (p,q)-deformed fermions covariant under the quantum group SU p/q(n) and n undeformed bosons. The Fock space representation of the algebra constructed is discussed and the total deformed Hamiltonian for such a system is obtained. Some physical applications of the quantum group covariant two-parameter deformed fermionic oscillator algebra are also considered.

The WZW model on random Regge triangulations

By exploiting a correspondence between random Regge triangulations (i.e., Regge triangulations with variable connectivity) and punctured Riemann surfaces, we propose a possible characterization of the SU(2) Wess–Zumino–Witten model on a triangulated surface of genus g. Techniques of boundary CFT are used for the analysis of the quantum amplitudes of the model at level κ=1. These techniques provide a non-trivial algebra of boundary insertion operators governing a brane-like interaction between simplicial curvature and WZW fields. Through such a mechanism, we explicitly characterize the partition function of the model in terms of the metric geometry of the triangulation, and of the 6 j symbols of the quantum group SU(2) Q , at Q= e −1 π/3 . We briefly comment on the connection with bulk Chern–Simons theory.

Exact analytical expression for magnetoresistance using quantum groups

We obtain an exact analytical expression for magnetoresistance using noncommutative geometry and quantum groups. Then we will show that there is a deep relationship between magnetoresistance and the quantum group su q (2), from which we understand the quantum interpretation of the quantum corrections to the conductivity.

The quantum q-Krawtchouk and q-Meixner polynomials and their related D-functions for the quantum groups SUq(2) and SUq(1,1)

The complete comparative analysis of the quantum q-Krawtchouk and q-Meixner polynomials of a discrete variable on a nonuniform grid ( x( s)= q 2 s ) and the D-function for the quantum groups SU q (2) and SU q (1,1) is done. The complete set of characteristics of these polynomials (i.e., orthogonality relations, normalization factors, recurrent relations, the explicit analytical expressions, the Rodrigues formulas, the formulas of difference derivatives, various particular values and cases) are calculated. The correlations between the properties of the polynomials mentioned above and the D-functions for the quantum groups SU q (2) and SU q (1,1) are established. In the case of SU q (1,1) only D-functions for the positive discrete series of the unitary irreducible representations are considered. It is known that on the nonuniform grid x( s)= q 2 s there are two kinds of Krawtchouk and Meixner polynomials. Also the properties of the second kind of these polynomials which are not connected to the D-functions are discussed.

The Big q -Jacobi Function Transform

Abstract. We give a detailed description of the resolution of the identity of a second-order q -difference operator considered as an unbounded self-adjoint operator on two different Hilbert spaces. The Hilbert spaces are weighted L 2 -spaces for a discrete measure on [-1,∞) and [-1,z] , and we expand an element of the L 2 -spaces in terms of eigenfunctions of L . We establish the inversion formula by studying the spectral decomposition of L . In the first case we establish the big q -Jacobi function transform as an analogue of the Jacobi function transform. In the second case we obtain the known orthogonality relations for the big q -Jacobi polynomials. The dual orthogonality relations in the first case give a one-parameter family of nonextremal orthogonality measures for the continuous dual q -1 -Hahn polynomials with q -1 >1 , and explicit sets of functions which complement these polynomials to orthogonal bases of the corresponding weighted L 2 -spaces. The q -difference operator and the two choices of Hilbert spaces arise from the harmonic analysis on the quantum groups SU q (1,1) and SU q (2) .

Quantum Instantons with Classical Moduli Spaces

We introduce a quantum Minkowski space-time based on the quantum group SU(2)q extended by a degree operator and formulate a quantum version of the anti-self-dual Yang-Mills equation. We construct solutions of the quantum equations using the classical ADHM linear data, and conjecture that, up to gauge transformations, our construction yields all the solutions. We also find a deformation of Penrose's twistor diagram, giving a correspondence between the quantum Minkowski space-time and the classical projective twistor space.

Thermodynamics of two parameter quantum group gases

We investigate the thermodynamics of the two parameter generalized bosonic and fermionic quantum gases, whose particle algebras are invariant under the quantum group SU p/ q ( d) for all p, q and d. As well as considering the high temperature behavior, we also discuss the low temperature limit of the deformed fermionic gas.

An appropriate candidate for exact distribution of closed random walks using quantum groups

We show that the structure of the quantum group su q (2) is intimately related to the random walks on a two-dimensional lattice. Using this connection we obtain an appropriate candidate for the exact area distribution of closed random walks of length N on a two-dimensional square lattice. We compare our results with exact enumeration.

Non-compact quantum groups and quantum Harish-Chandra modules

An important problem of the quantum group theory is to construct and classify the Harish-Chandra modules; it is discussed in this work. The way of producing the principal non-degenerate series representations of the quantum group SU n,n is sketched. A q-analogue for the Penrose transform is described.

A novel coproduct of suq(2) and its relation to XXZ spin chain model

While it is normal to restrict the deformation parameter q in the coproduct of the quantum group su q (2) to the positive reals, the naive replacement of q by − q results in a coproduct that does not respect algebra homomorphism. However, it is possible to define a twisted coproduct in this case that respects the algebra homomorphism. Application to XXZ spin chain model is discussed.