Quantization, Calibration and Planning for Euclidean Motions in Robotic Systems

The properties of Euclidean motions are fundamental in all areas of robotics research. Throughout the past several decades, investigations on some low-level tasks like parameterizing specific movements and generating effective motion plans have fostered high-level operations in an autonomous robotic system. In typical applications, before executing robot motions, a proper quantization of basic motion primitives could simplify online computations; a precise calibration of sensor readings could elevate the accuracy of the system controls. Of particular importance in the whole autonomous robotic task, a safe and efficient motion planning framework would make the whole system operate in a well-organized and effective way. All these modules encourage huge amounts of efforts in solving various fundamental problems, such as the uniformity of quantization in non-Euclidean manifolds, the calibration errors on unknown rigid transformations due to the lack of data correspondence and noise, the narrow passage and the curse of dimensionality bottlenecks in developing motion planning algorithms, etc. Therefore, the goal of this dissertation is to tackle these challenges in the topics of quantization, calibration and planning for Euclidean motions

Vector coherent state representations, induced representations, and geometric quantization: II. Vector coherent state representations

It is shown here and in the preceeding paper (quant-ph/0201129) that vector coherent state theory, the theory of induced representations, and geometric quantization provide alternative but equivalent quantizations of an algebraic model. The relationships are useful because some constructions are simpler and more natural from one perspective than another. More importantly, each approach suggests ways of generalizing its counterparts. In this paper, we focus on the construction of quantum models for algebraic systems with intrinsic degrees of freedom. Semi-classical partial quantizations, for which only the intrinsic degrees of freedom are quantized, arise naturally out of this construction. The quantization of the SU(3) and rigid rotor models are considered as examples.Comment: 31 pages, part 2 of two papers, published versio

Conformal Field Theory and Geometry of Strings

What is quantum geometry? This question is becoming a popular leitmotiv in theoretical physics and in mathematics. Conformal field theory may catch a glimpse of the right answer. We review global aspects of the geometry of conformal fields, such as duality and mirror symmetry, and interpret them within Connes' non-commutative geometry. Extended version of lectures given by the 2nd author at the Mathematical Quantum Theory Conference, Vancouver, Canada, August 4 to 8, 1993Comment: 44 pages, latex file, 5 references adde

Finite and infinite-dimensional symmetries of pure N=2 supergravity in D=4

We study the symmetries of pure N=2 supergravity in D=4. As is known, this theory reduced on one Killing vector is characterised by a non-linearly realised symmetry SU(2,1) which is a non-split real form of SL(3,C). We consider the BPS brane solutions of the theory preserving half of the supersymmetry and the action of SU(2,1) on them. Furthermore we provide evidence that the theory exhibits an underlying algebraic structure described by the Lorentzian Kac-Moody group SU(2,1)^{+++}. This evidence arises both from the correspondence between the bosonic space-time fields of N=2 supergravity in D=4 and a one-parameter sigma-model based on the hyperbolic group SU(2,1)^{++}, as well as from the fact that the structure of BPS brane solutions is neatly encoded in SU(2,1)^{+++}. As a nice by-product of our analysis, we obtain a regular embedding of the Kac-Moody algebra su(2,1)^{+++} in e_{11} based on brane physics.Comment: 70 pages, final version published in JHE

Solving 3d Gravity with Virasoro TQFT

We propose a precise reformulation of 3d quantum gravity with negative cosmological constant in terms of a topological quantum field theory based on the quantization of the Teichm\"uller space of Riemann surfaces that we refer to as ``Virasoro TQFT.'' This TQFT is similar, but importantly not equivalent, to $\text{SL}(2,\mathbb{R})$ Chern-Simons theory. This sharpens the folklore that 3d gravity is related to $\text{SL}(2,\mathbb{R})$ Chern-Simons theory into a precise correspondence, and resolves some well-known issues with this lore at the quantum level. Our proposal is computationally very useful and provides a powerful tool for the further study of 3d gravity. In particular, we explain how together with standard TQFT surgery techniques this leads to a fully algorithmic procedure for the computation of the gravity partition function on a fixed topology exactly in the central charge. Mathematically, the relation leads to many nontrivial conjectures for hyperbolic 3-manifolds, Virasoro conformal blocks and crossing kernels.Comment: 72 page

Soliton in AdS5xS5 SSSSG theory including Grassmann odd parameters

In this work, starting from the classical Type IIB Superstring in AdS5x S5 background theory, written as a 2d-Sigma model on PSU(2; 2|4) coset, we show its equivalence to SSSSG theory through the Pohlmeyer Reduction Procedure ([13],[9]). We explicitly construct a classical soliton solution for the generalized semi-symmetric space sine-Gordon theory (SSSSG) in AdS5X S5 background ([11]) including 2 Grassmann odd parameters. Alongside this, we present the purely bosonic case based on S5 symmetric space