arXiv subjects
Publications and source records attributed to Jens Hoppe.
We show that the discrete Painlev\'e-type equations arising from quantum minimal surfaces are equations for recurrence coefficients of orthogonal polynomials for indefinite hermitian products. As a consequence, we obtain an explicit formula for the initial conditions leading to positive solutions.
Apart from relating interesting quantum mechanical systems to equations describing a parabolic discrete minimal surface, the quantization of a cubic minimal surface in $\mathbb{R}^4$ is considered.
Several ways to reduce to a first order ODE the non-linear PDE's governing the relativistic motion of an axially symmetric membrane in 4 space time dimensions, as well as examples for a previously found non-trivial transformation generating solutions, are given.
For the simplest membrane matrix model (corresponding to reduced 3 dimensional SU(2) Yang Mills theory) the form of the ground state wave function is given.
By direct, elementary, considerations it is shown that the SU(2) x SO(d=2,3) invariant sector of the bosonic membrane matrix model is governed by (two, resp. three-dimensional) x^2 y^2 models
Some explicit examples are given for gauge compensating transformations and explicit forms of axially symmetric membrane solutions
An exact GH membrane matrix model solution is given that corresponds to the world volume swept out by a fast moving axially symmetric drop with a sharp tip.
It is shown that, somewhat similar to the case of classical Baecklund transformations for surfaces of constant negative curvature, infinitely many axially symmetric minimal hypersurfaces in 4-dimensional Minkowski-space can be obtained, in a non-trivial way, from any given one by combining the scaling symmetries of the equations in light cone coordinates with a non-obvious symmetry (the analogue of Bianchis original transformation) - which can be shown to be involutive/correspond to a space-reflection.
We revisit the stability (instability) of the outer (inner) catenoid connecting two concentric circular rings and give an explicit new construction of the unstable mode of the inner catenoid by studying the spectrum of an exactly solvable one-dimensional Schrödinger operator with an asymmetric Darboux-Pöschl-Teller potential.
Bäcklund-type transformations in four-dimensional space-time and an intriguing reduced zero-curvature formulation for axially symmetric membranes, with diffeomorphism-, resp. Lorentz-, symmetries reappearing after orthonormal gauge-fixing, are found.
New classes of exact M(em)brane solutions in M+2 dimensional Minkowski space are presented (some describing non-trivial topology changes, while others explicitly avoid finite-time singularity formation)
A class of exact membrane solutions is quantized.
Three classes of new, algebraic, zero-mean-curvature hypersurfaces in pseudo-Euclidean spaces are given.
We consider epimorphisms from quantum minimal surface algebras onto involutroy subalgebras of split real simply-laced Kac-Moody algebras and provide examples of affine and finite type. We also provide epimorphisms onto such Kac-Moody algebras themselves, where reality of the construction is important. The results extend to the complex situation.
It is shown that the previously noticed internal dynamical $SO(D-1)$ symmetry arXiv:1003.5189 for relativistic M-branes moving in $D$-dimensional space-time is naturally realized in the (extended by powers of $\frac{1}{p_+}$) enveloping algebra of the Poincaré algebra.
Discrete minimal surface algebras and Yang Mills algebras may be related to (generalized) Kac Moody algebras, just as Membrane (matrix) models and the IKKT model - including a novel construction technique for minimal surfaces.
The $SU(N)$--invariant matrix model potential is written as a sum of squares with only four frequencies (whose multiplicities and simple $N$--dependence are calculated).
Supersymmetrizable theories, such as M(em)branes and associated matrix-models related to Yang-Mills theory, possess r-matrices