Equivariant Kaehler Geometry and Localization in the G/G Model

We analyze in detail the equivariant supersymmetry of the $G/G$ model. In spite of the fact that this supersymmetry does not model the infinitesimal action of the group of gauge transformations, localization can be established by standard arguments. The theory localizes onto reducible connections and a careful evaluation of the fixed point contributions leads to an alternative derivation of the Verlinde formula for the $G_{k}$ WZW model. We show that the supersymmetry of the $G/G$ model can be regarded as an infinite dimensional realization of Bismut's theory of equivariant Bott-Chern currents on K\"ahler manifolds, thus providing a convenient cohomological setting for understanding the Verlinde formula. We also show that the supersymmetry is related to a non-linear generalization (q-deformation) of the ordinary moment map of symplectic geometry in which a representation of the Lie algebra of a group $G$ is replaced by a representation of its group algebra with commutator $[g,h] = gh-hg$. In the large $k$ limit it reduces to the ordinary moment map of two-dimensional gauge theories.Comment: LaTex file, 40 A4 pages, IC/94/108 and ENSLAPP-L-469/9

Derivation of the Verlinde Formula from Chern-Simons Theory and the G/G model

We give a derivation of the Verlinde formula for the $G_{k}$ WZW model from Chern-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function $Z_{\Sigma\times S^{1}}$ of $\Sigma\times S^{1}$ with an arbitrary number of labelled punctures. By a suitable gauge choice, $Z_{\Sigma\times S^{1}}$ is reduced to the partition function of an Abelian topological field theory on $\Sigma$ (a deformation of non-Abelian BF and Yang-Mills theory) whose evaluation is straightforward. This relates the Verlinde formula to the Ray-Singer torsion of $\Sigma\times S^{1}$. We derive the $G_{k}/G_{k}$ model from Chern-Simons theory, proving their equivalence, and give an alternative derivation of the Verlinde formula by calculating the $G_{k}/G_{k}$ path integral via a functional version of the Weyl integral formula. From this point of view the Verlinde formula arises from the corresponding Jacobian, the Weyl determinant. Also, a novel derivation of the shift k\ra k+h is given, based on the index of the twisted Dolbeault complex.Comment: 47 pages (in A4 format), LaTex file, (original was truncated by the mailer - apologies, m.b.), IC/93/8

Boundary states for WZW models

The boundary states for a certain class of WZW models are determined. The models include all modular invariants that are associated to a symmetry of the unextended Dynkin diagram. Explicit formulae for the boundary state coefficients are given in each case, and a number of properties of the corresponding NIM-reps are derived.Comment: 34 pages, harvmac (b), 4 eps-figures. One reference added; some minor typos, as well as the $A_2$ embedding into $D_4$, are correcte

Stringing Spins and Spinning Strings

We apply recently developed integrable spin chain and dilatation operator techniques in order to compute the planar one-loop anomalous dimensions for certain operators containing a large number of scalar fields in N =4 Super Yang-Mills. The first set of operators, belonging to the SO(6) representations [J,L-2J,J], interpolate smoothly between the BMN case of two impurities (J=2) and the extreme case where the number of impurities equals half the total number of fields (J=L/2). The result for this particular [J,0,J] operator is smaller than the anomalous dimension derived by Frolov and Tseytlin [hep-th/0304255] for a semiclassical string configuration which is the dual of a gauge invariant operator in the same representation. We then identify a second set of operators which also belong to [J,L-2J,J] representations, but which do not have a BMN limit. In this case the anomalous dimension of the [J,0,J] operator does match the Frolov-Tseytlin prediction. We also show that the fluctuation spectra for this [J,0,J] operator is consistent with the string prediction.Comment: 27 pages, 4 figures, LaTex; v2 reference added, typos fixe

CERN summer school 2015-report

As summer student of the 2015 CERN summer school programme, I was enlisted to work at the ISOLDE-CERN nuclear research facility as a member of the experimental group CRIS (collinear resonance ionisation spectroscopy). The experimental set up used at CRIS is a high resolution, efficiency and selectivity laser spectroscopy experiment designed to scan the hyperfine structure of low yield very exotic nuclei with very low background, and therefore acquiring nuclear electromagnetic moment data in the region far from stability. This data is of vital importance to nuclear theories trying to capture physics in these exotic nuclei. My work consisted of updating certain parts of the beam-lin

Efficient, high-resolution resonance laser ionization spectroscopy using weak transitions to long-lived excited states

Laser spectroscopic studies on minute samples of exotic radioactive nuclei require very efficient experimental techniques. In addition, high resolving powers are required to allow extraction of nuclear structure information. Here we demonstrate that by using weak atomic transitions, resonance laser ionization spectroscopy is achieved with the required high efficiency (1%–10%) and precision (linewidths of tens of MHz). We illustrate experimentally and through the use of simulations how the narrow experimental linewidths are achieved and how distorted resonance ionization spectroscopy line shapes can be avoided. The role of the delay of the ionization laser pulse with respect to the excitation laser pulse is crucial: the use of a delayed ionization step permits the best resolving powers and line shapes. A high efficiency is maintained if the intermediate level has a lifetime that is at least of the order of the excitation laser pulse width. A model that describes this process reproduces well the observed features and will help to optimize the conditions for future experiments. The simulation code is available upon request to the authors.11 pages, 11 figuresstatus: publishe

Observation of the radiative decay of the 229Th^{229}Th nuclear clock isomer

The radionuclide thorium-229 features an isomer with an exceptionally low excitation energy that enables direct laser manipulation of nuclear states. It constitutes one of the leading candidates for use in next-generation optical clocks$^{1–3}$. This nuclear clock will be a unique tool for precise tests of fundamental physics$^{4–9}$. Whereas indirect experimental evidence for the existence of such an extraordinary nuclear state is substantially older$^{10}$, the proof of existence has been delivered only recently by observing the isomer’s electron conversion decay$^{11}$. The isomer’s excitation energy, nuclear spin and electromagnetic moments, the electron conversion lifetime and a refined energy of the isomer have been measured$^{12–16}$. In spite of recent progress, the isomer’s radiative decay, a key ingredient for the development of a nuclear clock, remained unobserved. Here, we report the detection of the radiative decay of this low-energy isomer in thorium-229 ($^{229m}$Th). By performing vacuum-ultraviolet spectroscopy of $^{229m}$Th incorporated into large-bandgap CaF$_{2}$ and MgF$_{2}$ crystals at the ISOLDE facility at CERN, photons of 8.338(24) eV are measured, in agreement with recent measurements$^{14–16}$ and the uncertainty is decreased by a factor of seven. The half-life of $^{229m}$Th embedded in MgF$_{2}$ is determined to be 670(102) s. The observation of the radiative decay in a large-bandgap crystal has important consequences for the design of a future nuclear clock and the improved uncertainty of the energy eases the search for direct laser excitation of the atomic nucleus.The nucleus of the radioisotope thorium-229 (${}^{229}$Th) features an isomer with an exceptionally low excitation energy that enables direct laser manipulation of nuclear states. For this reason, it is a leading candidate for use in next-generation optical clocks. This nuclear clock will be a unique tool, amongst others, for tests of fundamental physics. While first indirect experimental evidence for the existence of such an extraordinary nuclear state is significantly older, the proof of existence has been delivered only recently by observing the isomer's electron conversion decay and its hyperfine structure in a laser spectroscopy study, revealing information on the isomer's excitation energy, nuclear spin and electromagnetic moments. Further studies reported the electron conversion lifetime and refined the isomer's energy. In spite of recent progress, the isomer's radiative decay, a key ingredient for the development of a nuclear clock, remained unobserved. In this Letter, we report the detection of the radiative decay of this low-energy isomer in thorium-229 (${}^{229\mathrm{m}}$Th). By performing vacuum-ultraviolet spectroscopy of ${}^{229\mathrm{m}}$Th incorporated into large-bandgap CaF${}_2$ and MgF${}_2$ crystals at the ISOLDE facility at CERN, the photon vacuum wavelength of the isomer's decay is measured as 148.71(42) nm, corresponding to an excitation energy of 8.338(24) eV. This value is in agreement with recent measurements, and decreases the uncertainty by a factor of seven. The half-life of ${}^{229\mathrm{m}}$Th embedded in MgF${}_2$ is determined to be 670(102) s. The observation of the radiative decay in a large-bandgap crystal has important consequences for the design of a future nuclear clock and the improved uncertainty of the energy eases the search for direct laser excitation of the atomic nucleus

First Offline Results from the S3 Low-Energy Branch

We present the first results obtained from the S3 Low-Energy Branch, the gas cell setup at SPIRAL2-GANIL, which will be installed behind the S3 spectrometer for atomic and nuclear spectroscopy studies of exotic nuclei. The installation is currently being commissioned offline, with the aim to establish optimum conditions for the operation of the radio frequency quadrupole ion guides, mass separation and ion bunching, providing high-efficiency and low-energy spatial spread for the isotopes of interest. Transmission and mass-resolving power measurements are presented for the different components of the S3-LEB setup. In addition, a single-longitudinal-mode, injection-locked, pumped pulsed-titanium–sapphire laser system has been recently implemented and is used for the first proof-of-principle measurements in an offline laser laboratory. Laser spectroscopy measurements of erbium, which is the commissioning case of the S3 spectrometer, are presented using the 4f126s23H6→4f12(3H)6s6p optical transition.peerReviewe