P.G. Lauwers

Lagrangians of N = 2 supergravity-matter systems

We present explicit expressions for general actions of vector and scalar multiplets coupled to N = 2 supergravity. We outline their construction which is based on the superconformal tensor calculus. The vector multiplets may be associated with a gauge group G which may also act on the scalar multiplets. The latter are naturally described in terms of quaternions; in the simplest case their kinetic terms define a nonlinear sigma model of a quaternionic projective space. We give an extension of the vector multiplet action which is not obtained from a chiral superspace density, and contains a Chern-Simons-type term. Transformation rules are given and the conditions for supersymmetry breaking are defined.

GAUGE AND MATTER FIELDS COUPLED TO N = 2 SUPERGRAVITY

We consider the potential of a general matter system of N = 2 vector and scalar multiplets coupled to supergravity. For lagrangians that are initially quadratic in the matter fields we prove that the potential is either positive or unbounded from below. Our results have been obtained in the framework of a superconformal multiplet calculus, and we have verified that they can be derived from each of the three off-shell representations. As an example we consider SO(6) Yang-Mills theory coupled to scalar multiplets in the 10 + 10 representation, which, for suitably chosen parameters, leads to the potential of gauged N = 8 supergravity. We discuss possibilities for residual nonabelian symmetry groups after breaking of N = 8 super-symmetry to N = 1 or 2.

Fermion simulations using parallel transported multigrid

Abstract Using a parallel transported multigrid (PTMG) technique, we compute the chiral symmetry order parameter and the pseudo-scalar meson mass in the quenched Schwinger model. In our PTMG method, the coarse level gauge and “fermionic” fields are derived from the fine level fields using the appropriate gauge fields as parallel transporters. For small quark masses and moderate correlation lengths we obtain a very rapid convergence (about one order of magnitude per cycle). In the same range of parameters, the one-level relaxation algorithms are slower by many orders of magnitude. We then show how to use this solver for dynamical fermions.

Noncompact N=2 supergravity

A massive spin-one multiplet with central charge is coupled to N=2 supergravity. Compared to conventional gauge fields the anomalous magnetic moment of the spin-one particles is of the opposite sign. The construction of this theory is based on an N=2 supersymmetric gauge theory associated with the noncompact group SO(2,1). As a byproduct we present a convenient expression for the N=2 Einstein-Yang-Mills lagrangian.

Inverting the Dirac matrix for SU(2) lattice gauge theory by means of parallel transported multigrid

We compare the convergence rates for inverting the Dirac matrix of a non-abelian gauge theory by the parallel transported multigrid algorithm (PTMG) and by the conjugate gradient (CG) approach. The test is carried out for SU(2) lattice gauge theory in two dimensions. The gauge field configurations are generated in the quenched approximation for various values of the gauge field correlation length ξ(β). We have performed runs for three different lattice sizes with the ratio ξ/L kept fixed; L denotes the linear dimension of the square lattices used. As L increases, the PTMG becomes more and more efficient. On a 256×256 lattice with the bare quark mass mbare = 0.0005 and ξ = 20, the PTMG is at least ten times faster than the CG. It seems reasonable to expect similar results also in four dimensions.

Parallel-transported multigrid and its application to the schwinger model

Abstract We present an efficient multigrid algorithm to invert the Dirac-Operator in simulations of lattice gauge theories. The coarse-level gauge and “fermionic” fields are derived from the fine-level fields using the appropriate gauge fields as parallel transporters; for this reason the method is called Parallel-Transported Multigrid (PTMG). We demonstrate the algorithm by computing the chiral symmetry order parameter and the pseudoscalar meson mass in the quenched Schwinger Model. In a considerable part of the physically relevant parameter space the method is superior over the one-level relaxation algorithms by several orders of magnitude. The method can be generalized in a straightforward way to SU (3) lattice gauge theory in 4 dimensions.

Variational investigations of euclidean SU(2) lattice gauge theory

Abstract We have extended our earlier investigations of lattice systems based on analytical variational techniques to the SU(2) gauge theory in 3 and 4 dimensions. Results are presented for the average plaquette and the specific heat.

An analytical variational study of the phase structure of compact QED on the lattice

Abstract An analytical variational approach is applied to lattice gauge theories with Jastrow type functionals as ansatze. The results are presented for compact QED in three and four dimensions.

Parallel-Transported Multi-Grid for inverting the Dirac-operator: variants of the method and their efficiency☆

Abstract Using U(1) lattice gauge theory in two dimensions as a test case, we discuss in detail several variants of the Parallel-Transported Multi-Grid (PTMG) with different restriction and interpolation procedures, resulting in radically different efficiencies. In particular, it is optimal to stop the coarsening once the coarse mesh size is of the order of the largest correlation length ξ. A dependence of the efficiency on the topological charge Q of the two-dimensional U(1) configurations was not detected, at least not for the most efficient PTMG variant in the physically relevant range of the parameters. In order to investigate this phenomenon in a reliable way, we developed and applied a Monte Carlo simulation algorithm using global update steps, which ensures ergodic covering of all topological sectors of the theory. For large enough systems and correlation lengths, an optimally tuned PTMG algorithm yields a very high convergence rate (one order of magnitude per cycle) and it beats by far the Conjugate-Gradient method in CPU time. Similar conclusions hold for other relevant lattice gauge theories.

A numerical study of the SU(2) Higgs model

Abstract Preliminary results of a simulation of the SU(2) theory with a Higgs field in the fundamental representation are presented. For an 8 3 × 16 and a 12 3 × 24 lattice we investigated the parameter range 2.4 ≤ β ≤ 3.0, λ = ∞. In the confinement region huge autocorrelations prevent us from obtaining reliable results. In the Higgs region the W mass m w turns out to have larger finite size effects than previously expected. Besides the W and Higgs masses, we estimate for β = 3.0 on the 12 3 × 24 lattice the renormalized gauge coupling g т and (unfortunately only up to an unknown waave function renormalization) the Higgs expectation value ν. The latter is obtained from the vacuum overlap order parameter. Technical problems encountered in its computation are discussed in some detail. Finally we discuss our results; in particular we show that the tree level relation m w 2 = g τ 2 ν 2 /2 is fulfilled remarkably well.