A New Supersymmetric Extension of Conformal Mechanics

In this paper a new supersymmetric extension of conformal mechanics is put forward. The beauty of this extension is that all variables have a clear geometrical meaning and the super-Hamiltonian turns out to be the Lie-derivative of the Hamiltonian flow of standard conformal mechanics. In this paper we also provide a supersymmetric extension of the other conformal generators of the theory and find their "square-roots". The whole superalgebra of these charges is then analyzed in details. We conclude the paper by showing that, using superfields, a constraint can be built which provides the exact solution of the system.Comment: 11 pages, no figure

Wavefunctions and counting formulas for quasiholes of clustered quantum Hall states on a sphere

The quasiholes of the Read-Rezayi clustered quantum Hall states are considered, for any number of particles and quasiholes on a sphere, and for any degree k of clustering. A set of trial wavefunctions, that are zero-energy eigenstates of a k+1-body interaction, and so are symmetric polynomials that vanish when any k+1 particle coordinates are equal, is obtained explicitly and proved to be both complete and linearly independent. Formulas for the number of states are obtained, without the use of (but in agreement with) conformal field theory, and extended to give the number of states for each angular momentum. An interesting recursive structure emerges in the states that relates those for k to those for k-1. It is pointed out that the same numbers of zero-energy states can be proved to occur in certain one-dimensional models that have recently been obtained as limits of the two-dimensional k+1-body interaction Hamiltonians, using results from the combinatorial literature.Comment: 9 pages. v2: minor corrections; additional references; note added on connection with one-dimensional Hamiltonians of recent interes

Embedding of the Lie superalgebra D(2,1;α)D(2, 1 ; \alpha) into the Lie superalgebra of pseudodifferential symbols on S1∣2S^{1|2}

We obtain an embedding of a one-parameter family of exceptional simple Lie superalgebras $D(2, 1 ; \alpha)$ into the Lie superalgebra of pseudodifferential symbols on the supercircle $S^{1|2}$. Correspondingly, there is an embedding of $D(2, 1 ; \alpha)$ into a nontrivial central extension of the derived contact superconformal algebra $K'(4)$ realized in terms of $4\times 4$ matrices over a Weyl algebra.Comment: 19 pages, LaTex, to be published in J.Math. Phy

Highest weight representations of the quantum algebra U_h(gl_\infty)

A class of highest weight irreducible representations of the quantum algebra U_h(gl_\infty) is constructed. Within each module a basis is introduced and the transformation relations of the basis under the action of the Chevalley generators are explicitly written.Comment: 7 pages, PlainTe

Rigorous Proof of a Liquid-Vapor Phase Transition in a Continuum Particle System

We consider particles in ${\Bbb R}^d, d \geq 2$, interacting via attractive pair and repulsive four-body potentials of the Kac type. Perturbing about mean field theory, valid when the interaction range becomes infinite, we prove rigorously the existence of a liquid-gas phase transition when the interaction range is finite but long compared to the interparticle spacing.Comment: 11 pages, in ReVTeX, e-mail addresses: [email protected], [email protected], [email protected]

Finite energy shifts in SU(n) supersymmetric Yang-Mills theory on T^3xR at weak coupling

We consider a semi-classical treatment, in the regime of weak gauge coupling, of supersymmetric Yang-Mills theory in a space-time of the form T^3xR with SU(n)/Z_n gauge group and a non-trivial gauge bundle. More specifically, we consider the theories obtained as power series expansions around a certain class of normalizable vacua of the classical theory, corresponding to isolated points in the moduli space of flat connections, and the perturbative corrections to the free energy eigenstates and eigenvalues in the weakly interacting theory. The perturbation theory construction of the interacting Hilbert space is complicated by the divergence of the norm of the interacting states. Consequently, the free and interacting Hilbert furnish unitarily inequivalent representation of the algebra of creation and annihilation operators of the quantum theory. We discuss a consistent redefinition of the Hilbert space norm to obtain the interacting Hilbert space and the properties of the interacting representation. In particular, we consider the lowest non-vanishing corrections to the free energy spectrum and discuss the crucial importance of supersymmetry for these corrections to be finite.Comment: 31 pages, 1 figure, v4 Minor changes, references correcte

Weight Vectors of the Basic A_1^(1)-Module and the Littlewood-Richardson Rule

The basic representation of \A is studied. The weight vectors are represented in terms of Schur functions. A suitable base of any weight space is given. Littlewood-Richardson rule appears in the linear relations among weight vectors.Comment: February 1995, 7pages, Using AMS-Te

Quantum R-matrix and Intertwiners for the Kashiwara Algebra

We study the algebra $B_q(\ge)$ presented by Kashiwara and introduce intertwiners similar to $q$-vertex operators. We show that a matrix determined by 2-point functions of the intertwiners coincides with a quantum R-matrix (up to a diagonal matrix) and give the commutation relations of the intertwiners. We also introduce an analogue of the universal R-matrix for the Kashiwara algebra.Comment: 21 page

Instantons, supersymmetric vacua, and emergent geometries

We study instanton solutions and superpotentials for the large number of vacua of the plane-wave matrix model and a 2+1 dimensional Super Yang-Mills theory on $R\times S^2$ with sixteen supercharges. We get the superpotential in the weak coupling limit from the gauge theory description. We study the gravity description of these instantons. Perturbatively with respect to a background, they are Euclidean branes wrapping cycles in the dual gravity background. Moreover, the superpotential can be given by the energy of the electric charge system characterizing each vacuum. These charges are interpreted as the eigenvalues of matrices from a reduction for the 1/8 BPS sector of the gauge theories. We also discuss qualitatively the emergence of the extra spatial dimensions appeared on the gravity side.Comment: 29 pages, 3 figures, latex. v2: references added, comments added. v3: accepted version in PR