Three-dimensional Gauge Research Articles

Numerical study of duality and universality in a frozen superconductor

The three-dimensional integer-valued lattice gauge theory, which is also known as a "frozen superconductor," can be obtained as a certain limit of the Ginzburg-Landau theory of superconductivity, and is believed to be in the same universality class. It is also exactly dual to the three-dimensional XY model. We use this duality to demonstrate the practicality of recently developed methods for studying topological defects, and investigate the critical behavior of the phase transition using numerical Monte Carlo simulations of both theories. On the gauge theory side, we concentrate on the vortex tension and the penetration depth, which map onto the correlation lengths of the order parameter and the Noether current in the XY model, respectively. We show how these quantities behave near the critical point, and that the penetration depth exhibits critical scaling only very close to the transition point. This may explain the failure of superconductor experiments to see the inverted XY model scaling.

Numerical studies of the two- and three-dimensional gauge glass at low temperature

We present results from Monte Carlo simulations of the two- and three-dimensional gauge glass at low temperature using the parallel tempering Monte Carlo method. Our results in two dimensions strongly support the transition being at T_c=0. A finite-size scaling analysis, which works well only for the larger sizes and lower temperatures, gives the stiffness exponent theta = -0.39 +/- 0.03. In three dimensions we find theta = 0.27 +/- 0.01, compatible with recent results from domain wall renormalization group studies.

Topological quantum memory

We analyze surface codes, the topological quantum error-correcting codes introduced by Kitaev. In these codes, qubits are arranged in a two-dimensional array on a surface of nontrivial topology, and encoded quantum operations are associated with nontrivial homology cycles of the surface. We formulate protocols for error recovery, and study the efficacy of these protocols. An order-disorder phase transition occurs in this system at a nonzero critical value of the error rate; if the error rate is below the critical value (the accuracy threshold), encoded information can be protected arbitrarily well in the limit of a large code block. This phase transition can be accurately modeled by a three-dimensional Z2 lattice gauge theory with quenched disorder. We estimate the accuracy threshold, assuming that all quantum gates are local, that qubits can be measured rapidly, and that polynomial-size classical computations can be executed instantaneously. We also devise a robust recovery procedure that does not require measurement or fast classical processing; however, for this procedure the quantum gates are local only if the qubits are arranged in four or more spatial dimensions. We discuss procedures for encoding, measurement, and performing fault-tolerant universal quantum computation with surface codes, and argue that these codes provide a promising framework for quantum computing architectures.

Monte Carlo simulations of the four-dimensionalXYspin glass at low temperatures

We report on the results for simulations of the four-dimensional $\mathrm{XY}$ spin glass using the parallel tempering Monte Carlo method at low temperatures for moderate sizes. Our results are qualitatively consistent with earlier work on the three-dimensional gauge glass as well as three- and four-dimensional Edwards-Anderson Ising spin glass. An extrapolation of our results would indicate that large-scale excitations cost only a finite amount of energy in the thermodynamic limit. The surface of these excitations may be fractal, although we cannot rule out a scenario compatible with replica symmetry breaking in which the surface of low-energy large-scale excitations is space filling.

D-Brane probes of Special Holonomy Manifolds, and Dynamics of Script N = 1 Three-Dimensional Gauge Theories

Using D2-brane probes, we study various properties of M-theory on singular, non-compact manifolds of G_2 and Spin(7) holonomy. We derive mirror pairs of N=1 supersymmetric three-dimensional gauge theories, and apply this technique to realize exceptional holonomy manifolds as both Coulomb and Higgs branches of the D2-brane world-volume theory. We derive a ``G_2 quotient construction'' of non-compact manifolds which admit a metric of G_2 holonomy. We further discuss the moduli space of such manifolds, including the structure of geometrical transitions in each case. For completeness, we also include familiar examples of manifolds with SU(3) and Sp(2) holonomy, where some of the new ideas are clarified and tested.

Monte Carlo simulations of vector spin glasses at low temperatures

In this paper I report results for simulations of the three-dimensional gauge glass and the four-dimensional XY spin glass using the parallel tempering Monte Carlo method at low temperatures for moderate sizes. The results are qualitatively consistent with earlier work on the three- and four-dimensional Edwards-Anderson Ising spin glass. I find evidence that large-scale excitations may cost only a finite amount of energy in the thermodynamic limit. The surface of these excitations is fractal, but I cannot rule out for the XY spin glass a scenario compatible with replica symmetry breaking where the surface of low-energy large-scale excitations is space filling.

Nature of the spin-glass state in the three-dimensional gauge glass

We present results from simulations of the gauge-glass model in three dimensions using the parallel tempering Monte Carlo technique. Critical fluctuations should not affect the data since we equilibrate down to low temperatures, for moderate sizes. Our results are qualitatively consistent with earlier work on the three- and four-dimensional Edwards-Anderson Ising spin glass. We find that large-scale excitations cost only a finite amount of energy in the thermodynamic limit and that those excitations have a surface whose fractal dimension is less than the space dimension, consistent with a scenario proposed by Krzakala and Martin, and Palassini and Young.

Lattice study of3Dcompact QED at finite temperature

We study the deconfinement phase transition and monopole properties in a finite temperature three-dimensional compact Abelian gauge model on the lattice. We predict the critical coupling as a function of the lattice size in a simplified model to describe monopole binding. We demonstrate numerically that monopoles are sensitive to the transition. In the deconfinement phase the monopoles appear in the form of a dilute gas of magnetic dipoles. In the confinement phase both monopole density and string tension differ from semiclassical estimates if monopole binding is neglected. However, an analysis of the monopole clusters shows that the relation between the string tension and the density of monopoles in charged clusters is in reasonable agreement with predictions. We study the cluster structure of the vacuum in both phases of the model.

Supersymmetric non-singular fractional D2-branes and NS-NS 2-branes

We obtain regular deformed D2-brane solutions with fractional D2-branes arising as wrapped D4-branes. The space transverse to the D2-brane is a complete Ricci-flat 7-manifold of G_2 holonomy, which is asymptotically conical with principal orbits that are topologically CP^3 or the flag manifold SU(3)/(U(1) x U(1)). We obtain the solution by first constructing an L^2 normalisable harmonic 3-form. We also review a previously-obtained regular deformed D2-brane whose transverse space is a different 7-manifold of G_2 holonomy, with principal orbits that are topologically S^3 x S^3. This describes D2-branes with fractional NS-NS 2-branes coming from the wrapping of 5-branes, which is supported by a non-normalisable harmonic 3-form on the 7-manifold. We prove that both types of solutions are supersymmetric, preserving 1/16 of the maximal supersymmetry and hence that they are dual to {\cal N}=1 three-dimensional gauge theories. In each case, the spectrum for minimally-coupled scalars is discrete, indicating confinement in the infrared region of the dual gauge theories. We examine resolutions of other branes, and obtain necessary conditions for their regularity. The resolution of many of these seems to lie beyond supergravity. In the process of studying these questions, we construct new explicit examples of complete Ricci-flat metrics.

Supergravity duals of gauge field theories from SU(2) × U(1) gauged supergravity in five dimensions

We study the SO(4)-symmetric solution of the five-dimensional SU(2) x U(1) gauged N=4 supergravity theory obtained in [hep-th/0101202]. This solution contains purely magnetic non-Abelian and electric Abelian fields. It can be interpreted as a reduction of seven-dimensional gauged supergravity on a torus, which comes from type IIB supergravity on S^3. We also show how to obtain that solution from six-dimensional Romans' theory on a circle. We then up-lift the solution to massless type IIA supergravity. The dual gauge field theory is twisted and is defined on the worldvolume of a NS-fivebrane wrapped on S^3. Two other spatial directions of the NS-fivebrane are on a torus. In the IR limit it corresponds to a three-dimensional gauge field theory with two supercharges.

Short range linear potential in 3D lattice compact QED

We study the static potential between electric charges in the finite temperature three-dimensional compact gauge theory on the lattice. We show that in the deconfinement phase at small separations between the charges the potential contains a linearly rising piece which goes over into the Coulomb potential as the distance between the charges is increased. The linear potential is due to the gas of magnetic dipoles which are realized as monopole–anti-monopole bound states.

Moduli space of twoU(1)instantons on noncommutativeR4andR3×S1

We employ the Afiyah-Drinfield-Hitchin-Manin method to derive the moduli space of two instantons in $U(1)$ gauge theory on a noncommutative space. We show by an explicit hyper-K\"ahler quotient construction that the relative metric of the moduli space of two instantons on ${R}^{4}$ is the Eguchi-Hanson metric and find a unique threshold bound state. For two instantons on ${R}^{3}\ifmmode\times\else\texttimes\fi{}{S}^{1},$ otherwise known as calorons, we give the asymptotic metric and conjecture a completion. We further discuss the relationship of caloron moduli spaces of A, D, and E groups to the Coulomb branches of three-dimensional gauge theory. In particular, we show that the Coulomb branch of the $\mathrm{SU}(2)$ gauge group with a single massive adjoint hypermultiplet coincides with the above two caloron moduli space.

Consistency relations for an implicitn-dimensional regularization scheme

We extend an implicit regularization scheme to be applicable in the n-dimensional space-time. Within this scheme divergences involving parity violating objects can be consistently treated without recoursing to dimensional continuation. Special attention is paid to differences between integrals of the same degree of divergence, typical of one loop calculations which are, in principle, undetermined. We show how to use symmetries in order to fix these quantities consistently. We illustrate with examples in which regularization plays a delicate role in order to both corroborate and elucidate the results in the literature for the case of $\mathrm{CPT}$ violation in extended ${\mathrm{QED}}_{4},$ topological mass generation in three-dimensional gauge theories, the Schwinger model, and its chiral version.

Conformal field theories on K3 and three-dimensional gauge theories

According to a recent conjecture, the moduli space of the heterotic conformal field theory on a $G\subset$ ADE singularity of an ALE space is equivalent to the moduli space of a pure $\cx N=4$ supersymmetric three-dimensional gauge theory with gauge group G. We establish this relation using geometric engineering of heterotic strings and generalize it to theories with non-trivial matter content. A similar equivalence is found between the moduli of heterotic CFT on isolated Calabi--Yau 3-fold singularities and two-dimensional Kazama-Suzuki coset theories.

Mirror symmetry and toric geometry in three-dimensional gauge theories

We study three dimensional gauge theories with N=2 supersymmetry. We show that the Coulomb branches of such theories may be rendered compact by the dynamical generation of Chern-Simons terms and present a new class of mirror symmetric theories in which both Coulomb and Higgs branches have a natural description in terms of toric geometry.

Finite temperature ordering in the three-dimensional gauge glass

We present results of Monte Carlo simulations of the gauge glass model in three dimensions using exchange Monte Carlo. We show for the first time clear evidence of the vortex glass ordered phase at finite temperature. Using finite size scaling we obtain estimates for the correlation length exponent, nu = 1.39 +/- 0.20, the correlation function exponent, eta = -0.47 +/- 0.07, and the dynamic exponent z = 4.2 +/- 0.6. Using our values for z and nu we calculate the resistivity exponent to be s = 4.5 +/- 1.1. Finally, we provide a plausible lower bound on the the zero-temperature stiffness exponent, theta >= 0.18.

Collision of SU(2) gauge fields in 3+1 dimensions

The Yang–Mills equations provide a classical mean field description of the non-Abelian soft glue field dynamics in heavy ion collisions at high energies. A C++ code for the numerical solution on a three-dimensional gauge lattice is presented. Non-perturbative time-dependent solutions in position space are calculated in the weak coupling regime of a gauge lattice within the Hamiltonian approach of Kogut and Susskind. Applications are demonstrated by simulations of the Yang–Mills field dynamics in collisions of initially uniformly color polarized Gaussian wave packets. The collisions are evolved in time. The time dependent energy densities, energy flows and energy currents in longitudinal and transverse directions with respect to the collision axis are presented. Simulations of this type have recently revealed an interesting solution which exhibits a time-delayed sudden increase of the transverse energy currents in the “non-Abelian case” at short times after the collision.

Branes and supersymmetry breaking in three dimensional gauge theories

It is shown that supersymmetry is spontaneously broken in certain three-dimensional supersymmetric gauge theories, by using the s-rule in their string theory realization as brane configurations. In particular, supersymmetry is broken in N=3 supersymmetric Yang-Mills-Chern-Simons theory with gauge group SU(n) and CS coefficient k, as well as in its N=2 and N=1 deformations, when n>|k|. In addition, supersymmetry is broken in the N=1 mass deformation of N=2 supersymmetric Yang-Mills theory with gauge group SU(n) and one matter multiplet when n>1. In the latter case the breaking is induced by an instanton-generated repulsive potential.

A focusing multilayer analyser for local diffraction studies

A novel X-ray diffraction technique for the local structural characterization of thick specimens is presented. Broad energy-band focusing elements are used both on the incoming and exit (diffracted) side of the sample. The geometry allows imaging, and magnification, of a line through the thickness of the sample. In comparison with conventional methods of defining three-dimensional gauge volumes the new technique provides superior depth resolution, higher flux, and a remedy for some systematic errors occurring in strain measurements due to, for example, grain size effects. The technique is validated by a synchrotron test experiment using a bent and meridionally graded multilayer as the focusing analyser element. The incoming beam is monochromated, at 30 keV, and focused to a 15 µm spot size by means of a bent Laue crystal. The resulting depth profile from the (222) reflection of a 21 µm-thick rolled Au foil has a width of 44 µm. The depth resolution, magnification and reflectivity as a function of the energy bandwidth are found to be well matched by theory. The prospect of the technique and the associated aberrations are discussed.

String multiplets from an invariant 2-brane

We study a three-dimensional gauge theory obtained from the dimensional reduction of a D4-brane world volume theory in the background of space-time moduli. An $\mathrm{SL}(3)$ symmetry in this theory, which acts on fields as well as coupling constants, is identified. By comparing the energies with the string tensions, we show that certain 1/2 supersymmetric classical solutions of this theory can be identified as $\mathrm{SL}(3,Z)$ multiplets of type II strings in eight dimensions. The results are then generalized to the non-linear Born-Infeld action. We also discuss the possibility of 1/8 BPS states in this theory and their representations in terms of string networks.