Noncommutative Quantum Hall Effect and Aharonov-Bohm Effect

We study a system of electrons moving on a noncommutative plane in the presence of an external magnetic field which is perpendicular to this plane. For generality we assume that the coordinates and the momenta are both noncommutative. We make a transformation from the noncommutative coordinates to a set of commuting coordinates and then we write the Hamiltonian for this system. The energy spectrum and the expectation value of the current can then be calculated and the Hall conductivity can be extracted. We use the same method to calculate the phase shift for the Aharonov-Bohm effect. Precession measurements could allow strong upper limits to be imposed on the noncommutativity coordinate and momentum parameters $\Theta$ and $\Xi$.Comment: 9 pages, RevTeX4, references added, small changes in the tex

The Vacuum Structure and Spectrum of N=2 Supersymmetric SU(N) Gauge Theory

We present an exact description of the metric on the moduli space of vacua and the spectrum of massive states for four dimensional N=2 supersymmetric SU(n) gauge theories. The moduli space of quantum vacua is identified with the moduli space of a special set of genus n-1 hyperelliptic Riemann surfaces.Comment: 11 pages, Revtex, 2 figures. Reference adde

Varieties of vacua in classical supersymmetric gauge theories

We give a simple description of the classical moduli space of vacua for supersymmetric gauge theories with or without a superpotential. The key ingredient in our analysis is the observation that the lagrangian is invariant under the action of the complexified gauge group \Gc. From this point of view the usual $D$-flatness conditions are an artifact of Wess--Zumino gauge. By using a gauge that preserves \Gc invariance we show that every constant matter field configuration that extremizes the superpotential is \Gc gauge-equivalent (in a sense that we make precise) to a unique classical vacuum. This result is used to prove that in the absence of a superpotential the classical moduli space is the algebraic variety described by the set of all holomorphic gauge-invariant polynomials. When a superpotential is present, we show that the classical moduli space is a variety defined by imposing additional relations on the holomorphic polynomials. Many of these points are already contained in the existing literature. The main contribution of the present work is that we give a careful and self-contained treatment of limit points and singularities.Comment: 14 pages, LaTeX (uses revtex.sty

Supersymmetry and Mass Gap in 2+1 Dimensions: A Gauge Invariant Hamiltonian Analysis

A Hamiltonian formulation of Yang-Mills-Chern-Simons theories with $0\leq N\leq 4$ supersymmetry in terms of gauge-invariant variables is presented, generalizing earlier work on nonsupersymmetric gauge theories. Special attention is paid to the volume measure of integration (over the gauge orbit space of the fields) which occurs in the inner product for the wave functions and arguments relating it to the renormalization of the Chern-Simons level number and to mass-gaps in the spectrum of the Hamiltonians are presented. The expression for the integration measure is consistent with the absence of mass gap for theories with extended supersymmetry (in the absence of additional matter hypermultiplets and/or Chern-Simons couplings), while for the minimally supersymmetric case, there is a mass-gap, the scale of which is set by a renormalized level number, in agreement with indications from existing literature. The realization of the supersymmetry algebra and the Hamiltonian in terms of the gauge invariant variables is also presented.Comment: 31 pages, References added, typos correcte

Giant magnon bound states from strongly coupled N=4 SYM

We calculate in a very simple way the spectrum of giant magnon bound states at strong coupling in N=4 SYM, by utilizing the description of the field theory vacuum in terms of a condensate of eigenvalues of commuting matrices. We further show that these calculations can be understood in terms of the central charge extension that permits the calculation of BPS masses in the Coulomb branch of N=4 SYM. This paper shows further evidence that the strong coupling expansion of the maximally supersymmetric Yang-Mills theory in four dimensions can be done systematically from first principles, without the assumption of integrability.Comment: 19 pages, uses revte

Monopoles, Duality and Chiral Symmetry Breaking in N=2 Supersymmetric QCD

We study four dimensional $N=2$ supersymmetric gauge theories with matter multiplets. For all such models for which the gauge group is $SU(2)$, we derive the exact metric on the moduli space of quantum vacua and the exact spectrum of the stable massive states. A number of new physical phenomena occur, such as chiral symmetry breaking that is driven by the condensation of magnetic monopoles that carry global quantum numbers. For those cases in which conformal invariance is broken only by mass terms, the formalism automatically gives results that are invariant under electric-magnetic duality. In one instance, this duality is mixed in an interesting way with $SO(8)$ triality.Comment: 89 page

Horizons and plane waves: A review

We review the attempts to construct black hole/string solutions in asymptotically plane wave spacetimes. First, we demonstrate that geometries admitting a covariantly constant null Killing vector cannot admit event horizons, which implies that pp-waves can't describe black holes. However, relaxing the symmetry requirements allows us to generate solutions which do possess regular event horizons while retaining the requisite asymptotic properties. In particular, we present two solution generating techniques and use them to construct asymptotically plane wave black string/brane geometries.Comment: 15 pages, harvmac. Review to appear in Modern Physics Letters A v2: added reference

Symplectic SUSY Gauge Theories with Antisymmetric Matter

We investigate the confining phase vacua of supersymmetric Sp(2\NC) gauge theories that contain matter in both fundamental and antisymmetric representations. The moduli spaces of such models with \NF=3 quark flavors and \NA=1 antisymmetric field are analogous to that of SUSY QCD with \NF=\NC+1 flavors. In particular, the forms of their quantum superpotentials are fixed by classical constraints. When mass terms are coupled to W_{(\NF=3,\NA=1)} and heavy fields are integrated out, complete towers of dynamically generated superpotentials for low energy theories with fewer numbers of matter fields can be derived. Following this approach, we deduce exact superpotentials in $Sp(4)$ and $Sp(6)$ theories which cannot be determined by symmetry considerations or integrating in techniques. Building upon these simple symplectic group results, we also examine the ground state structures of several $Sp(4) \times Sp(4)$ and $Sp(6) \times Sp(2)$ models. We emphasize that the top-down approach may be used to methodically find dynamical superpotentials in many other confining supersymmetric gauge theories.Comment: 21 pages, Revte

Solving Four Dimensional Field Theories with the Dirichlet Fivebrane

The realization of ${\cal N}=2$ four dimensional super Yang-Mills theories in terms of a single Dirichlet fivebrane in type IIB string theory is considered. A classical brane computation reproduces the full quantum low energy effective action. This result has a simple explanation in terms of mirror symmetry.Comment: Final version to appear in Phys. Rev.

Emergent Geometry and Quantum Gravity

We explain how quantum gravity can be defined by quantizing spacetime itself. A pinpoint is that the gravitational constant G = L_P^2 whose physical dimension is of (length)^2 in natural unit introduces a symplectic structure of spacetime which causes a noncommutative spacetime at the Planck scale L_P. The symplectic structure of spacetime M leads to an isomorphism between symplectic geometry (M, \omega) and Riemannian geometry (M, g) where the deformations of symplectic structure \omega in terms of electromagnetic fields F=dA are transformed into those of Riemannian metric g. This approach for quantum gravity allows a background independent formulation where spacetime as well as matter fields is equally emergent from a universal vacuum of quantum gravity which is thus dubbed as the quantum equivalence principle.Comment: Invited Review for Mod. Phys. Lett. A, 17 page