Paolo Valtancoli

STABILITY OF THE FUZZY SPHERE SOLUTION FROM MATRIX MODEL

We consider a matrix model depending on the parameter λ which permits the fuzzy sphere as a classical background. By expanding the bosonic matrices around this background one recovers a U(1) (U(n)) noncommutative gauge theory on the fuzzy sphere. To check the classical stability of this background, we look for new classical solutions of this model and find them for λ < 1, that make the fuzzy sphere solution unstable for and stable otherwise.

Classical scattering in 2+1 gravity with N point sources

Abstract The classical dynamics of N point sources in three-dimensional gravity is considered. An explicit solution is found in the first-order formalism, where the dreibein and the spin connection correspond to a Chern-Simons gauge field on the Poincare group. If we are not interested in a smooth metric, as in the gauge theory, such a solution exists for arbitrary trajectories and conventional scattering is not defined. Only particle exchanges are meaningful, and are described by the braid group acting on the holonomies of the gauge field. On the other hand, the metric in the Einstein theory must be smooth and invertible, outside the particle trajectories, and fulfil proper asymptotic conditions. We argue that these requirements constrain the asymptotic motion of the particles, so that the two-body scattering problem is well defined. We determine the scattering angle in some special limits and we argue its exact form in the massless case.

Isomonodromic properties of the Seiberg-Witten solution

Abstract The Seiberg-Witten solution of N = 2 supersymmetric SU (2) gauge theories with matter is analyzed as an isomonodromy problem. We show that the holomorphic section describing the effective action can be deformed by moving its singularities on the moduli space while keeping their monodromies invariant. Well-known examples of isomonodromic sections are given by thev correlators of two-dimensional rational conformal field theories - the conformal blocks. The Seiberg-Witten section similarly admits the operations of braiding and fusing of its singularities, which obey the Yang-Bacter and Pentagonal identities, respectively. Using them, we easily find the complete expressions of the monodromies with affine team, and the full quantum numbers of the BPS spectrum. While the braiding describes the quark-monopole transmutation, the fusing implies the superconformal points in the moduli space. In the simplest case of three singularities, the supersymmetric sections are directly related to the conformal blocks of the logarithmic minimal models.

Dirac's Observables for the Higgs Model I: The Abelian Case

We search a canonical basis of Diracs observables for the classical Abelian Higgs model with fermions in the case of a trivial U(1) principal bundle. The study of the Gauss law first class constraint shows that the model has two disjoint sectors of solutions associated with two physically different phases. In the electromagnetic phase, the electromagnetic field remains massless: after the determination of the Diracs observables we get that both the reduced physical Hamiltonian and Lagrangian are nonlocal. In the Higgs phase, the electromagnetic field becomes massive and in terms of Diracs observables we get a local, but nonanalytic in the electric charge (or equivalently in the sum of the electromagnetic mass and of the residual Higgs field), physical Hamiltonian; however the associated Lagrangian is nonlocal. Some comments on the R-gauge-fixing, the possible elimination of the residual Higgs field and on the Nielsen–Olesen vortex solution close the paper.

(2+1)-Gravity with moving particles in an instantaneous gauge

Abstract By defining a regular gauge which is conformal-like and provides instantaneous field propagation, we investigate classical solutions of (2+1)-Gravity coupled to arbitrarily moving point-like particles. We show how to separate field equations from self-consistent motion and we provide a solution for the metric and the motion in the two-body case with arbitrary speed, up to second order in the mass parameters.

Classical scattering in 2+1 gravity with N spinning sources

Abstract The classical dynamics of N spinning point sources in 2+1 Einstein-Cartan gravity is considered. It corresponds to the ISO (2, 1) Chern-Simons theory, in which the torsion source is restricted to its intrinsic spin part. A class of explicit solutions is found for the dreibein and the spin connection, which are torsionless in the spinless limit. By using the residual local Poincare invariance of the solutions, we fix the gauge so that the metric is smooth outside the particles and satisfies proper asymptotic conditions at space and time infinity. We recover previous results for test bodies and find new ones for the scattering of two dynamical particles in the massless limit.

DIRAC'S OBSERVABLES FOR THE SU(3)×SU(2)×U(1) STANDARD MODEL

The missing complete Hamiltonian treatment of the standard SU(3) × SU(2) × U(1) model with Grassmann-valued fermion fields in the Higgs phase is given. We bypass the complications of the Hamiltonian theory in the Higgs phase, resulting from the spontaneous symmetry breaking with the Higgs mechanism, by studying the Hamiltonian formulation of the Higgs phase for the gauge-equivalent Lagrangian in the unitary gauge. A canonical basis of Diracs observables is found and the reduced physical Hamiltonian is evaluated. Its self-energy part is nonlocal for the electromagnetic and strong interactions, but local for the weak ones. Therefore, the Fermi four-fermion interaction reappears at the nonperturbative level.

GRAVITY ON A FUZZY SPHERE

We propose an action for gravity on a fuzzy sphere, based on a matrix model. We find striking similarities with an analogous model of two-dimensional gravity on a noncommutative plane, i.e. the solution space of both models is spanned by pure U(2) gauge transformations acting on the background solution of the matrix model, and there exist deformations of the classical diffeomorphisms which preserve the two-dimensional noncommutative gravity actions.

Exact quasi-static particle scattering in 2+1 gravity

Abstract We find an exact solution for the interacting gravitational two-body system at low velocity in (2 + 1) dimensions.

Dirac's Observables for the Higgs Model II: The Non-Abelian SU(2) Case

We search a canonical basis of Diracs observables for the classical non-Abelian Higgs model with fermions in the case of a trivial SU(2) principal bundle with a complex doublet of Higgs fields and with the fermions in a given representation of SU(2). Since each one of the three Gauss law first class constraints can be solved either in the corresponding longitudinal electric field or in the corresponding Higgs momentum, we get a priori eight disjoint phases of solutions of the model. The only two phases with SU(2) covariance are the SU(2) phase with massless SU(2) fields and the Higgs phase with massive SU(2) fields. The Dirac observables and the reduced physical (local) Hamiltonian and (non-local) Lagrangian of the Higgs phase are evaluated:the main result is the nonanalyticity in the SU(2) coupling constant, or equivalently in the sum of the residual Higgs field and of the mass of the SU(2) fields. Some comments on the function spaces needed for the gauge fields are made.