Gauge theory on nonassociative spaces

We show how to do gauge theory on the octonions and other nonassociative algebras such as `fuzzy $R^4$' models proposed in string theory. We use the theory of quasialgebras obtained by cochain twist introduced previously. The gauge theory in this case is twisting-equivalent to usual gauge theory on the underlying classical space. We give a general U(1)-Yang-Mills example for any quasi-algebra and a full description of the moduli space of flat connections in this theory for the cube $Z_2^3$ and hence for the octonions. We also obtain further results about the octonions themselves; an explicit Moyal-product description of them as a nonassociative quantisation of functions on the cube, and a characterisation of their cochain twist as invariant under Fourier transform.Comment: 24 pages latex, two .eps figure

Braided Cyclic Cocycles and Non-Associative Geometry

We use monoidal category methods to study the noncommutative geometry of nonassociative algebras obtained by a Drinfeld-type cochain twist. These are the so-called quasialgebras and include the octonions as braided-commutative but nonassociative coordinate rings, as well as quasialgebra versions \CC_{q}(G) of the standard q-deformation quantum groups. We introduce the notion of ribbon algebras in the category, which are algebras equipped with a suitable generalised automorphism $\sigma$, and obtain the required generalisation of cyclic cohomology. We show that this \emph{braided cyclic cocohomology} is invariant under a cochain twist. We also extend to our generalisation the relation between cyclic cohomology and differential calculus on the ribbon quasialgebra. The paper includes differential calculus and cyclic cocycles on the octonions as a finite nonassociative geometry, as well as the algebraic noncommutative torus as an associative example.Comment: 36 pages latex, 9 figure

Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group

Braided groups and braided matrices are novel algebraic structures living in braided or quasitensor categories. As such they are a generalization of super-groups and super-matrices to the case of braid statistics. Here we construct braided group versions of the standard quantum groups $U_q(g)$. They have the same FRT generators $l^\pm$ but a matrix braided-coproduct \und\Delta L=L\und\tens L where $L=l^+Sl^-$, and are self-dual. As an application, the degenerate Sklyanin algebra is shown to be isomorphic to the braided matrices $BM_q(2)$; it is a braided-commutative bialgebra in a braided category. As a second application, we show that the quantum double D(\usl) (also known as the `quantum Lorentz group') is the semidirect product as an algebra of two copies of \usl, and also a semidirect product as a coalgebra if we use braid statistics. We find various results of this type for the doubles of general quantum groups and their semi-classical limits as doubles of the Lie algebras of Poisson Lie groups.Comment: 45 pages. Revised (= much expanded introduction

Continuous-wave, multimilliwatt, mid-infrared source tunable across 6.4–7.5 μm based on orientation-patterned GaAs

We report a continuous-wave (cw) source of tunable mid-infrared radiation providing tens of milliwatt of output power in the 6460–7517 nm spectral range. The source is based on difference-frequency generation (DFG) in orientation-patterned (OP)-GaAs pumped by a Tm-fiber laser at 2010 nm and a 1064 nm-Yb-fiber-pumped cw optical parametric oscillator. Using a 25.7-mm-long OP-GaAs crystal, we have generated up to 51.1 mW of output power at 6790 nm, with >40 mW and >20 mW across 32% and 80% of the mid-infrared tuning range, respectively, which is to the best of our knowledge the highest tunable cw power generated in OP-GaAs in this spectral range. The DFG output at maximum power exhibits passive power stability better than 2.3% rms over more than 1 h and a frequency stability of 1.8 GHz over more than 1 min, in high spatial beam quality. The system and crystal performance at high pump powers have been studiedPostprint (published version

Optical emission near a high-impedance mirror

Solid state light emitters rely on metallic contacts with high sheet-conductivity for effective charge injection. Unfortunately, such contacts also support surface plasmon polariton (SPP) excitations that dissipate optical energy into the metal and limit the external quantum efficiency. Here, inspired by the concept of radio-frequency (RF) high-impedance surfaces and their use in conformal antennas we illustrate how electrodes can be nanopatterned to simultaneously provide a high DC electrical conductivity and high-impedance at optical frequencies. Such electrodes do not support SPPs across the visible spectrum and greatly suppress dissipative losses while facilitating a desirable Lambertian emission profile. We verify this concept by studying the emission enhancement and photoluminescence lifetime for a dye emitter layer deposited on the electrodes

Differential and Twistor Geometry of the Quantum Hopf Fibration

We study a quantum version of the SU(2) Hopf fibration $S^7 \to S^4$ and its associated twistor geometry. Our quantum sphere $S^7_q$ arises as the unit sphere inside a q-deformed quaternion space $\mathbb{H}^2_q$. The resulting four-sphere $S^4_q$ is a quantum analogue of the quaternionic projective space $\mathbb{HP}^1$. The quantum fibration is endowed with compatible non-universal differential calculi. By investigating the quantum symmetries of the fibration, we obtain the geometry of the corresponding twistor space $\mathbb{CP}^3_q$ and use it to study a system of anti-self-duality equations on $S^4_q$, for which we find an `instanton' solution coming from the natural projection defining the tautological bundle over $S^4_q$.Comment: v2: 38 pages; completely rewritten. The crucial difference with respect to the first version is that in the present one the quantum four-sphere, the base space of the fibration, is NOT a quantum homogeneous space. This has important consequences and led to very drastic changes to the paper. To appear in CM

Quantum statistics and noncommutative black holes

We study the behaviour of a scalar field coupled to a noncommutative black hole which is described by a $\kappa$-cylinder Hopf algebra. We introduce a new class of realizations of this algebra which has a smooth limit as the deformation parameter vanishes. The twisted flip operator is independent of the choice of realization within this class. We demonstrate that the $R$-matrix is quasi-triangular up to the first order in the deformation parameter. Our results indicate how a scalar field might behave in the vicinity of a black hole at the Planck scale.Comment: 8 pages, no figures, revtex4; in v2 some points are explained in more detail, few typos corrected and one reference adde