ANNUAL REPORT 2005 - Universiteit Utrecht

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ANNUAL REPORT 20053 PhD programmeand the “so-called” boost parameters h of the corresponding transition functions ofthe geometric structure. Such an object is an element g of the Lorentz group SO(2,1).Particles are easily added to this model by cutting out wedges from the polygons correspondingto a new pair of glued edges meeting at the ”point particle”. The conditionfor the angle at the tip of the wedge is Tr g=cos(m) in suitable units, where m is interpretedas the mass of the particle, which determines the angle. The deficit from2p in the rest frame of the particle is equal to its mass. These vertices carry threedimensional curvature singularities as opposed to the trivalent ones.This thesis contains two main results. One concerns the case of a higher genus spacelikeslice without particles. It is shown that the h variables parametrize Teichmllerspace, (which is the reduced configuration space of physically distinct universes) ifthey are interpreted as geodesic lengths in hyperbolic space. Then the dual graph ofthe skeleton yields an ideal triangulation of a uniformizing hyperbolic with the sametopology as the physical equal time surface and with constant curvature -1. Usingthis, one can show that the constraints corresponding to the closure of the Euclideanpolygon (in case there is only one) can always be solved by a clever choice of the chartcorresponding to basepoint of the geodesic loops on the uniformizing surface. It isvery likely that each physically distinct universe admits a slicing in terms of only onepolygon. The question of generalization for particles is more difficult, and explicitconfigurations are found where the construction does not work.The second result is the derivation of the postulated symplectic structure from thefirst order action of general relativity by means of symplectic reduction. It is donein two steps: the first is a reduction from the (infinite dimensional) dynamical fieldsof the action to a finite set of variables describing a non-planar polygonal contour ina background Minkowski space. This covariant model is then gauge-fixed to obtainthe ’t Hooft polygon model. In the last subsection aspects of quantization proceduresfrom these and related starting points are discussed.Nogradi, D. (UL)thesis title: Multi-calorons and their moduliadvisor: prof. dr. P.J. van Baaldate: 29 June 2005present position: postdoctoral fellow, Wuppertal University, GermanyPure Yang-Mills instantons are considered on S 1 × R 3 – so-called calorons. Theholonomy – or Polyakov loop around the thermal S 1 at spatial infinity – is assumedto be a non-centre element of the gauge group SU(n) as most appropriate for QCDapplications in the confined phase. It is shown that a charge k caloron can be seen asa collection of nk massive magnetic monopoles – coming in n types – each carryingfractional topological charge. This interpretation offers a physically appealing way ofintroducing monopole degrees of freedom into pure gluodynamics: as constituents offinite temperature instantons.Using the Nahm transform an elaborate treatment is given for arbitrary topologicalcharge and new exact and explicit solutions are found for SU(2) and charge 2. Thek zero-modes of the Dirac operator in the background of a charge k caloron are computed,and are shown to ‘hop’ between the n types of monopoles as a function of
ANNUAL REPORT 20053 PhD programmethe temporal boundary condition. The abelian limit – where it is assumed that themassive field components can be dropped – is analysed in great detail and is shownthat the abelian charge distribution of each monopole type coincides with the correspondingfermion zero-mode density.The 4nk dimensional hyperkahler moduli space is identified as an algebraic variety,defined by four matrices obeying a constraint, modulo a natural adjoint action. Thismoduli space is proved to be the same as the moduli space of stable holomorphicbundles over the complex projective plane which are trivial on two complex lines. Adescription is given for its twistor space which allows for the computation of the exacthyperkahler metric – at least in principle.Finally, lattice gauge theoretic applications are mentioned and is explicitly demonstratedhow to obtain calorons on the lattice using the method of cooling.Papadimitriou, I. (UvA)thesis title: Aspects of the gauge/gravity duality and holographyadvisors: prof. dr. J. de Boer; dr. K. Skenderisdate: 6 September 2005present position: postdoctoral fellow, DESY Theory Group, Hamburg, GermanyOne of the most important developments in string theory in recent years has been thediscovery of the correspondence or duality between certain string theories and certaingauge theories. Such ‘gauge/string dualities’ are ‘strong/weak coupling dualities’since they relate the strongly coupled regime of one theory to the weakly coupled limitof the other and vice versa. As a result, the gauge/string dualities have allowed fora deeper understanding of both string theory and gauge theory in regimes that wereinaccessible by other techniques.In this thesis we focus on a subclass of ‘gauge/string dualities’ where the relevantstring theory admits a supergravity description at low energies and hence the dualitybecomes a ‘gauge/gravity’ duality. The supergravity in question is usually Type IIBsupergravity in ten dimensions or eleven dimensional supergravity, which can be consistentlytruncated to some gauged supergravity in lower dimensions. Such gaugedsupergravities typically admit asymptotically locally Anti de Sitter (AldS) solutions,which, via the gauge/gravity duality are dual to the gauge theory living on the conformalboundary of the AlAdS space.In particular, correlation functions of BPS gauge-invariant operators of the (stronglycoupled) gauge theory, in a particular vacuum of the theory, can be computed from theon-shell supergravity action, which is a functional of the arbitrary Dirichlet boundaryconditions of the supergravity fluctuations around the solution corresponding to thegauge theory vacuum in question. Such computations, however, are obstructed by thedivergences of the on-shell supergravity action, which arise due to the infinite volumeof the AlAdS spaces.In Chapter 3 of this thesis we review the method of Holographic Renormalization,which allows for a systematic removal of the infrared divergences of the supergravityaction by constructing a set of covariant boundary counterterms, and we develop amore efficient Hamiltonian version of holographic renormalization. In Chapter 4 weapply this method to the computation of various correlation functions in N = 4 super
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ANNUAL REPORT 2005 - Universiteit Utrecht

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