arXiv subjects
Publications and source records attributed to Jens Hoppe.
A formulation of U(1) - symmetric classical membrane motions (preserving one rotational symmetry) is given, and reductions to systems of ODE's, as well as some ideas concerning singularities and integrability.
A certain G_2 times U(1) invariant Hamiltonian arising from the standard membrane matrix model via conjugating any of the supercharges by a cubic, octonionic, exponential is proven to have a spectrum covering the whole positive half-axis. The model could be useful in determining a normalizable zero-energy state in the original SO(9) invariant SU(N) matrix model.
We explore the 1/2 BPS configurations in BMN matrix theory with SO(3) angular momentum of $SO(3)\times SO(6)$ symmetry. The fluctuation analysis of the BPS configurations near the abelian solutions and also the fuzzy two sphere vacua reveals how nonabelian BPS configurations emerge. Especially the irreducible nonabelian configurations seem to have the maximal angular momentum of order $N^3$, beyond which they collapse to abelian ones. We also find some new BPS configurations explicitly.
We introduce C-Algebras of compact Riemann surfaces $Σ$ as non-commutative analogues of the Poisson algebra of smooth functions on $Σ$. Representations of these algebras give rise to sequences of matrix-algebras for which matrix-commutators converge to Poisson-brackets as $N\to\infty$. For a particular class of surfaces, nicely interpolating between spheres and tori, we completely characterize (even for the intermediate singular surface) all finite dimensional representations of the corresponding C-algebras.
Simple recursion relations for zero energy states of supersymmetric matrix models are derived by using an unconventional reducible representation for the fermionic degrees of freedom.
We reveal a dynamical SU(2) symmetry in the asymptotic description of supersymmetric matrix models. We also consider a recursive approach for determining the ground state, and point out some additional properties of the model(s).
We relate free vector dynamics to the eigenvalue motion of a time-dependent real-symmetric NxN matrix, and give a geodesic interpretation to Ruijsenaars Schneider models.
A multiparameter class of integrable systems is introduced.
Several reductions of the bosonic BMN matrix model equations to ordinary point particle Hamiltonian dynamics in the plane (or R^3) are given - as well as a few explicit solutions (some of which, as N->infinity, correspond to membranes rotating with constant angular velocity, others to higher dimensional objects).
New solutions to the classical equations of motion of a bosonic matrix-membrane are given. Their continuum limit defines 3-manifolds (in Minkowski space) whose mean curvature vanishes. Part of the construction are minimal surfaces in S^7, and their discrete analogues.
A non-linear functional $Q[u,v]$ is given that governs the loss, respectively gain, of (doubly degenerate) eigenvalues of fourth order differential operators $L = \partial^4 + \partial u \partial + v$ on the line. Apart from factorizing $L$ as $A^{*}A + E_{0}$, providing several explicit examples, and deriving various relations between $u$, $v$ and eigenfunctions of $L$, we find $u$ and $v$ such that $L$ is isospectral to the free operator $L_{0} = \partial^{4}$ up to one (multiplicity 2) eigenvalue $E_{0} < 0$. Not unexpectedly, this choice of $u$, $v$ leads to exact solutions of the corresponding time-dependent PDE's.
We consider reduced Super Yang-Mills Theory in $d+1$ dimensions, where $d=2,3,5,9$. We present commutators to prove that for $d=3,5$ and 9 a possible ground state must be a $Spin(d)$ singlet. We also discuss the case $d=2$, where we give an upper bound on the total angular momentum and show that for odd dimensional gauge group no $Spin(d)$ invariant state exists in the Hilbert space.
Section I contains introductory remarks about surface motions. Section II gives a detailed derivation of $H=-Δ-Tr\sum_{i<j}[X_i,X_j]^2$ as describing a quantized discrete analogue of relativistically invariant membrane dynamics. Section III concerns the question of zero-energy bound-states in SU(N)-invariant supersymmetric matrix models. Section IV discusses the space of solutions of some differential matrix equations on $(-\infty,+\infty)$, interpolating between different representations of $su(2)$. Some exercises are added, and one remark/conjecture concerning 5-commutators.
The asymptotic form of a SU(3) matrix theory groundstate is found by showing that a recent ansatz for a supersymmetric wavefunction is non-trivial (i.e. non-zero).
We study the quantum mechanical model obtained as a dimensional reduction of N=1 super Yang-Mills theory to a periodic light-cone "time". After mapping the theory to a cohomological field theory, the partition function (with periodic boundary conditions) regularized by a massive term appears to be equal to the partition function of the twisted matrix oscillator. We show that this partition function perturbed by the operator of the holonomy around the time circle is a tau function of Toda hierarchy. We solve the model in the large N limit and study the universal properties of the solution in the scaling limit of vanishing perturbation. We find in this limit a phase transition of Gross-Witten type.
Some ideas are presented concerning the question which of the harmonic wavefunctions constructed in [hep-th/9909191] may be annihilated by all supercharges.
For the SU(N) invariant supersymmetric matrix model related to membranes in 4 space-time dimensions we argue that = 0 for the previously obtained solution of Q chi = 0, Q^{dagger} Psi = 0.
The method introduced in [hep-th/9805020] is simplified, and used to calculate the asymptotic form of all SU(2) \times SO(d=3, resp. 5) invariant wave functions satisfying $Q_{\hatβ} Ψ= 0, \hatβ = 1 ... 4$ resp. 8, where $Q_{\hatβ}$ are the supercharges of the SU(2) matrix model related to supermembranes in d+2=5 (resp. 7) space-time dimensions. For d=3, there exist 2 asymptotic solutions, both of which are constant (hence non-normalizable) in the flat directions, confirming previous arguments that gauge-invariant zero energy states should not exist for d<9. For d=5, however, out of 4 asymptotic singlet solutions (3 with orbital angular momentum $l=0$, one having $l=1$) the one with $l=1$ does fall off fast enough to be asymptotically normalizable, hence requiring further analysis to be excluded as being extendable to a global solution.