Noncommutative Instantons on the 4-Sphere¶from Quantum Groups

We describe an approach to the noncommutative instantons on the 4-sphere based on quantum group theory. We quantize the Hopf bundle ?⁷→?⁴ making use of the concept of quantum coisotropic subgroups. The analysis of the semiclassical Poisson–Lie structure of U(4) shows that the diagonal SU(2) must be conjugated to be properly quantized. The quantum coisotropic subgroup we obtain is the standard SU _q (2); it determines a new deformation of the 4-sphere ∑⁴ _q as the algebra of coinvariants in ? _q ⁷. We show that the quantum vector bundle associated to the fundamental corepresentation of SU _q (2) is finitely generated and projective and we compute the explicit projector. We give the unitary representations of ∑⁴ _q , we define two 0-summable Fredholm modules and we compute the Chern–Connes pairing between the projector and their characters. It comes out that even the zero class in cyclic homology is non-trivial. Received: 3 January 2001 / Accepted: 14 November 2001     

Noncommutative Manifolds, the Instanton Algebra¶and Isospectral Deformations

We give new examples of noncommutative manifolds that are less standard than the NC-torus or Moyal deformations of ℝ ⁿ . They arise naturally from basic considerations of noncommutative differential topology and have non-trivial global features. The new examples include the instanton algebra and the NC-4-spheres S ⁴ _θ. We construct the noncommutative algebras ?=C ^∞ (S ⁴ _θ) of functions on NC-spheres as solutions to the vanishing, ch _j (e) = 0, j < 2, of the Chern character in the cyclic homology of ? of an idempotent e∈M ₄ (?), e ²=e, e=e ^*. We describe the universal noncommutative space obtained from this equation as a noncommutative Grassmannian as well as the corresponding notion of admissible morphisms. This space Gr contains the suspension of a NC-3-sphere S ³ _θ distinct from quantum group deformations SU _q (2) of SU (2). We then construct the noncommutative geometry of S _θ ⁴ as given by a spectral triple ?, ℋ, D) and check all axioms of noncommutative manifolds. In a previous paper it was shown that for any Riemannian metric g _μν on S ⁴ whose volume form is the same as the one for the round metric, the corresponding Dirac operator gives a solution to the following quartic equation,

The Variational Principle for a Class of Asymptotically Abelian C*-Algebras

Let (A,α) be a C^*-dynamical system. We introduce the notion of pressure P _α(H) of the automorphism α at a self-adjoint operator H∈A. Then we consider the class of AF-systems satisfying the following condition: there exists a dense α-invariant *-subalgebra ? of A such that for all pairs a,b∈? the C^*-algebra they generate is finite dimensional, and there is p=p(a,b)∈ℕ such that [α ^j (a),b]= 0 for |j|≥p. For systems in this class we prove the variational principle, i.e. show that P _α(H) is the supremum of the quantities h _φ(α) −φ(H), where h _φ(α) is the Connes–Narnhofer–Thirring dynamical entropy of α with respect to the α-invariant state φ. If H∈A, and P _α(H) is finite, we show that any state on which the supremum is attained is a KMS-state with respect to a one-parameter automorphism group naturally associated with H. In particular, Voiculescu's topological entropy is equal to the supremum of h _φ(α), and any state of finite maximal entropy is a trace. Received: 19 April 2000 / Accepted: 14 June 2000     

Examples of gauged Laplacians on noncommutative spaces

We present two (classes of) examples of gauged Laplacian operators. The first one is a model of spin-Hall effect on a noncommutative four-sphere S _ϑ ⁴ with isospin degrees of freedom, coming from a noncommutative instanton, and invariant under the quantum group SO _ϑ (5). The second one, a Hall effect on a quantum 2-dimensional sphere S _q ², describes ‘excitations moving on the quantum sphere’ in the field of a magnetic monopole with symmetry coming from the quantum group SU _q (2). For both models, ample symmetries provide a complete diagonalization.     

Improvement of perturbation theory in QCD for e + e − →hadrons and the problem of α s freezing

A method of improving perturbation theory in QCD is developed which can be applied to any polarization operator. The case of the polarization operator Π(q ²), corresponding to the process e ⁺ e ⁻→ hadrons, is considered in detail. By the use of the analytical properties of Π(q ²) and a perturbation expansion of Π(q ²) for q ²<0, the function ImΠ(q ²) at q ²>0 is defined in such a way that the infrared pole is eliminated. The convergence of the perturbation series for R(q ²)=σ(e ⁺ e ⁻ →hadrons)/(e ⁺ e ⁻→μ ⁺ μ ⁻) is improved. After substitution of R(q ²) into the dispersion relation an improved Adler function D(q ²) is obtained, having no infrared pole and a frozen α _s (q ²). Good agreement with experiment is achieved. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 3, 167–170 (10 August 1999) Published in English in the original Russian journal. Edited by Steve Torstveit.     

Determination of the neutron electric form factor from the reaction 3 He(e,e'n) at medium momentum transfer

The electric form factor of the neutron G_En has been determined in double polarized exclusive ³ He(e,e'n) scattering in quasi–elastic kinematics by measuring asymmetries A _⊥, A _∥ of the cross section with respect to helicity reversal of the electron, with the nuclear spin being oriented perpendicular to the momentum transfer q in case of A_⊥ and parallel in case of A_∥. The experiment was performed at the 855 MeV c. w. microtron MAMI at Mainz. The degree of polarization of the electron beam and of the gaseous ³ He target were each about 50%. Scattered electrons and neutrons were detected in coincidence by detector arrays covering large solid angles. Quasi–elastic scattering events were reconstructed from the measured electron scattering angles ϑ_e, φ_e and the neutron momentum vector p _n ^′ in the plane wave impulse approximation. We obtain the result <G _En>_{(0.27 < Q2c2/GeV2 < 0.5)}= 0.0334 ± 0.0033_stat± 0.0028_syst which is averaged over the indicated range of Q ², the squared momentum transfer. This G _En value is significantly smaller than measured from the D(e,e'n) reaction under similar kinematical conditions. To what extent final state interactions in ³He quench the G _En result is subject of calculations currently in progress elsewhere. Received: 29 April 1999     

Construction of Fredholm representations and a modification of the Higson-Roe corona

The Fredholm representation theory is well adapted to the construction of homotopy invariants of non-simply-connected manifolds by means of the generalized Hirzebruch formula [σ(M)] = 〈L(M)ch _A f*ξ, [M]〉 ∈ K _A ⁰(pt) ⊗ Q, where A = C*[π] is the C*-algebra of the group π, π = π ₁(M). The bundle ξ ∈ K _A ⁰(Bπ) is the canonical A-bundle generated by the natural representation π → A. Recently, the first author constructed a natural family of Fredholm representations that lead to a symmetric vector bundle on the completion of the fundamental group with a modification of the Higson-Roe corona, provided that the completion is a closed manifold.     

Strong Connections and Chern–Connes Pairing¶in the Hopf–Galois Theory

We reformulate the concept of connection on a Hopf–Galois extension B⊆P in order to apply it in computing the Chern–Connes pairing between the cyclic cohomology HC ^{2 n} (B) and K ₀ (B). This reformulation allows us to show that a Hopf–Galois extension admitting a strong connection is projective and left faithfully flat. It also enables us to conclude that a strong connection is a Cuntz–Quillen-type bimodule connection. To exemplify the theory, we construct a strong connection (super Dirac monopole) to find out the Chern–Connes pairing for the super line bundles associated to a super Hopf fibration. Received: 8 March 2000 / Accepted: 5 January 2001     

Asymptotic behavior of orbits of C 0-semigroups and solutions of linear and semilinear abstract differential equations

In this paper, we study the asymptotic behavior of solutions of semilinear abstract differential equations (*) u′(t) = Au(t) + t ⁿ f(t, u(t)), where A is the generator of a C ₀-semigroup (or group) T(·), f(·, x) ∈ A for each x ∈ X, A is the class of almost periodic, almost automorphic or Levitan almost periodic Banach space valued functions ϕ: ℝ → X and n ∈ {0, 1, 2, ...}. We investigate the linear case when T(·)x is almost periodic for each x ∈ X; and the semilinear case when T(·) is an asymptotically stable C ₀-semigroup, n = 0 and f(·, x) satisfies a Lipschitz condition. Also, in the linear case, we investigate (*) when ϕ belongs to a Stepanov class S ^p-A defined similarly to the case of S ^p-almost periodic functions. Under certain conditions, we show that the solutions of (*) belong to A _u:= A ∩ BUC(ℝ, X) if n = 0 and to t ⁿ A _u ⊕ w _n C ₀ (ℝ, X) if n ∈ ℕ, where w _n(t) = (1 + |t|)ⁿ. The results are new for the case n ∈ ℕ and extend many recent ones in the case n = 0. Dedicated to the memory of B. M. Levitan     

Electromagnetic form factors of nucleons in a relativistic square-root potential model of independent quarks

The nucleon electromagnetic form factorsG _E ^P (q²),G _M ^P (q²) and the axial-vector form factor G_A(q²) are studied in a relativistic model of independent quarks confined by an equally mixed scalar-vector square root potentialV _q(r)=1/2(1+γ ⁰)(ar ^1/2+ν ₀) taking into account the appropriate centre-of-mass corrections. The respective root-mean-square radii associated withG _E ^P (q²) and G _A (q²) come out as [〈r ²〉 _E ^P ]^1/2=0.86 fm and 〈r _A ² 〉^1/2=0.88 fm. Restoration of chiral symmetry in this model is discussed to derive the pion-nucleon form factorG _πNN(q²) and consequently the pion-nucleon coupling constant is obtained asg _πNN(q²)=12.81 as compared tog _πNN(q²)_exp⋍13.     

Multiple Instantons Representing Higher-Order Chern–Pontryagin Classes

It has been shown in the work of Chakrabarti, Sherry and Tchrakian that the chiral SO _±(4 p) Yang–Mills theory in the Euclidean 4 p (p≥ 2) dimensions allows an axially symmetric self-dual system of equations similar to Witten's instanton equations in the classical 4-dimensional SU(2)∼SO _±(4) theory and the solutions represent a new class of instantons. However the rigorous existence of these higher-dimensional instanton solutions has remained open except for the solution of unit charge representing a single instanton. In this paper we establish an existence and uniqueness theorem for multi-instantons of arbitrary charges in the case p≥ 2. These solutions are the first known instantons, with the Chern–Pontryagin index greater than one, of the Yang–Mills model in higher dimensions. Our approach is a study of a nonlinear variational equation defined on the Poincaré half plane. Received: 20 May 1996 / Accepted: 30 April 1997     

Quantum Brownian Motion on Non-Commutative Manifolds: Construction,Deformation and Exit Times

We begin with a review and analytical construction of quantum Gaussian process (and quantum Brownian motions) in the sense of Franz (The Theory of Quantum Levy Processes, [math.PR], 2009), Schürmann (White noise on bioalgebras. Volume 1544 of Lecture Notes in Mathematics. Berlin: Springer-Verlag, 1993) and others, and then formulate and study in details (with a number of interesting examples) a definition of quantum Brownian motions on those non-commutative manifolds (a la Connes) which are quantum homogeneous spaces of their quantum isometry groups in the sense of Goswami (Commun Math Phys 285(1):141–160, 2009). We prove that bi-invariant quantum Brownian motion can be ‘deformed’ in a suitable sense. Moreover, we propose a non-commutative analogue of the well-known asymptotics of the exit time of classical Brownian motion. We explicitly analyze such asymptotics for a specific example on non-commutative two-torus A_q{\mathcal{A}_\theta} , which seems to behave like a one-dimensional manifold, perhaps reminiscent of the fact that A_q{\mathcal{A}_\theta} is a non-commutative model of the (locally one-dimensional) ‘leaf-space’ of the Kronecker foliation.     

Critical gravity in the Chern–Simons modified gravity

We perform the perturbation analysis of the Chern–Simons modified gravity around the AdS₄ spacetime (its curvature radius ℓ) to obtain the critical gravity. In general, we could not obtain an explicit form of perturbed Einstein equation which shows a massive graviton propagation clearly, but for the Kerr–Schild perturbation and Chern–Simons coupling θ=kx/y, we find the AdS wave as a single massive solution to the perturbed Einstein equation. Its mass squared is given by M ²=[−9+(2ℓ ²/k−1)²]/4ℓ ². At the critical point of M ²=0 (k=ℓ ²/2), the solution takes the log-form and the linearized excitation energies vanish.     

Geometrical Tools for Quantum Euclidean Spaces

We apply one of the formalisms of noncommutative geometry to ℝ ^N _q , the quantum space covariant under the quantum group SO _q (N). Over ℝ ^N _q there are two SO _q (N)-covariant differential calculi. For each we find a frame, a metric and two torsion-free covariant derivatives which are metric compatible up to a conformal factor and which have a vanishing linear curvature. This generalizes results found in a previous article for the case of ℝ³ _q . As in the case N=3, one has to slightly enlarge the algebra ℝ ^N _q ; for N odd one needs only one new generator whereas for N even one needs two. As in the particular case N=3 there is a conformal ambiguity in the natural metrics on the differential calculi over ℝ ^N _q . While in our previous article the frame was found “by hand”, here we disclose the crucial role of the quantum group covariance and exploit it in the construction. As an intermediate step, we find a homomorphism from the cross product of ℝ ^N _q with U _q so(N) into ℝ ^N _q , an interesting result in itself. Received: 4 March 2000 / Accepted: 11 October 2000     

Bistability of quantum magnetotransport in a multilayer Ge/p-Ge1−x Six heterostructure with wide potential wells

Two metastable states of a multilayer Ge/p-Ge_1−x Si_x heterosystem with wide (∼ 35 nm) potential wells (Ge) are observed in strong magnetic fields B at low temperatures. In the first state, the Hall resistivity exhibits an inflection near the value ρ_xy=h/e ² scaled to one Ge layer. The longitudinal magnetoresistivity ρ_xx(B) possesses a minimum in the range of fields where this inflection occurs. The temperature evolution of the inflection in ρ_xy(B), the minimum of ρ _xx(B), and the value of ρ_xy at the inflection indicates a weakly expressed state of the quantum Hall effect with a uniform current distribution over the layers. In the second metastable state, an unusually wide plateau near h/2e ² with a very weak field dependence is observed in ρ_xy(B). Estimates show that in these samples the Fermi level lies below but close to the top of the inflection in the bottom of the well. For this reason, the second state can be explained by separation of a hole gas in the Ge layers into two sublayers, and the saturation of ρ_xy(B) near h/2e ² can be explained by the formation of a quantum Hall insulator state. Pis’ma Zh. éksp. Teor. Fiz. 70, No. 4, 290–297 (25 August 1999)     

Projective Module Description of the q-Monopole

The Dirac q-monopole connection is used to compute projector matrices of quantum Hopf line bundles for arbitrary winding number. The Chern–Connes pairing of cyclic cohomology and K-theory is computed for the winding number −1. The non-triviality of this pairing is used to conclude that the quantum principal Hopf fibration is non-cleft. Among general results, we provide a left-right symmetric characterization of the canonical strong connections on quantum principal homogeneous spaces with an injective antipode. We also provide for arbitrary strong connections on algebraic quantum principal bundles (Hopf–Galois extensions) their associated covariant derivatives on projective modules. Received: Received: 4 September 1998 / Accepted: 16 October 1998     

A Hopf Bundle Over a Quantum Four-Sphere from the Symplectic Group

We construct a quantum version of the SU(2) Hopf bundle S⁷→S⁴. The quantum sphere S⁷_q arises from the symplectic group Sp_q(2) and a quantum 4-sphere S⁴_q is obtained via a suitable self-adjoint idempotent p whose entries generate the algebra A(S⁴_q) of polynomial functions over it. This projection determines a deformation of an (anti-)instanton bundle over the classical sphere S⁴. We compute the fundamental K-homology class of S⁴_q and pair it with the class of p in the K-theory getting the value −1 for the topological charge. There is a right coaction of SU_q(2) on S⁷_q such that the algebra A(S⁷_q) is a non-trivial quantum principal bundle over A(S⁴_q) with structure quantum group A(SU_q(2)).     

Nonadiabatic effects in the phonon spectra of superconductors

The nonadiabatic corrections to the self-energy part Σ_s(q, ω) of the phonon Green’s function are studied for various values of the phonon vectors q resulting from electron-phonon interactions. It is shown that the long-range electron-electron Coulomb interaction has no direct influence on these effects, aside from a possible renormalization of the corresponding constants. The electronic response functions and Σ_s(q, ω) are calculated for arbitrary vectors qand energy ω in the BCS approximation. The results obtained for q=0 agree with previously obtained results. It is shown that for large wave numbers q, vertex corrections are negligible and Σ_s(q, ω) possesses a logarithmic singularity at ω=2Δ, where Δ is the superconducting gap. It is also shown that in systems with nesting, Σ_s(Q, ω) (where Q is the nesting vector) possesses a square-root singularity at ω=2Δ, i.e., exactly of the same type as at q=0. The results are used to explain the recently published experimental data on phonon anomalies, observed in nickel borocarbides in the superconducting state, at large q. It is shown, specifically, that in these systems nesting must be taken into account in order to account for the emergence of a narrow additional line in the phonon spectral function S(q, ω)≈−π ⁻¹ Im D _s (q, ω), where D _s (q, ω) is the phonon Green’s function, at temperatures T<T _c . Zh. éksp. Teor. Fiz. 115, 1799–1817 (May 1999)     

On braided poisson and quantum inhomogeneous groups

The well known incompatibility between inhomogeneous quantum groups and the standardq-deformation is shown to disappear (at least in certain cases) when admitting the quantum group to be braided. Braided quantumISO(p, N - p) containingSO _q (p, N - p) with |q|=1 are constructed forN=2p, 2p + 1, 2p + 2. Their Poisson analogues (obtained first) are presented as an introduction to the quantum case. Presented at the 6th International Colloquium on Quantum Groups: “Quantum Groups and Integrable Systems”, Prague, 19–21 June 1997.