Theta series, wall-crossing and quantum dilogarithm identities

Motivated by mathematical structures which arise in string vacua and gauge theories with N=2 supersymmetry, we study the properties of certain generalized theta series which appear as Fourier coefficients of functions on a twisted torus. In Calabi-Yau string vacua, such theta series encode instanton corrections from $k$ Neveu-Schwarz five-branes. The theta series are determined by vector-valued wave-functions, and in this work we obtain the transformation of these wave-functions induced by Kontsevich-Soibelman symplectomorphisms. This effectively provides a quantum version of these transformations, where the quantization parameter is inversely proportional to the five-brane charge $k$. Consistency with wall-crossing implies a new five-term relation for Faddeev's quantum dilogarithm $\Phi_b$ at $b=1$, which we prove. By allowing the torus to be non-commutative, we obtain a more general five-term relation valid for arbitrary $b$ and $k$, which may be relevant for the physics of five-branes at finite chemical potential for angular momentum.Comment: 26 pages; v2: added discussion on relation to complex Chern-Simons, misprints correcte

AFLOW-SYM: Platform for the complete, automatic and self-consistent symmetry analysis of crystals

Determination of the symmetry profile of structures is a persistent challenge in materials science. Results often vary amongst standard packages, hindering autonomous materials development by requiring continuous user attention and educated guesses. Here, we present a robust procedure for evaluating the complete suite of symmetry properties, featuring various representations for the point-, factor-, space groups, site symmetries, and Wyckoff positions. The protocol determines a system-specific mapping tolerance that yields symmetry operations entirely commensurate with fundamental crystallographic principles. The self consistent tolerance characterizes the effective spatial resolution of the reported atomic positions. The approach is compared with the most used programs and is successfully validated against the space group information provided for over 54,000 entries in the Inorganic Crystal Structure Database. Subsequently, a complete symmetry analysis is applied to all 1.7$+$ million entries of the AFLOW data repository. The AFLOW-SYM package has been implemented in, and made available for, public use through the automated, $\textit{ab-initio}$ framework AFLOW.Comment: 24 pages, 6 figure

Lagrangian dynamics for solid multi-body systems with the Moving Frame formalism

Classical mechanics is the study of the motion of particles, solid bodies or of systems of bodies, and the effects of forces and moments on them. The dynamical behavior may either be studied by the application of Newtonian mechanics, Lagrangian mechanics or Hamiltonian Mechanics. Either approach produces mathematical equations describing the time evolution of the multi-body system. The resulting equations of motion for many bodies are typically non-linear and of large scale. The Lagrangian approach is more abstract than the Newtonian approach in the sense that it employs more advanced theory from mathematics, and has therefore traditionally mostly been a topic studied in mathematics, physics or on master level engineering. The Moving Frame Method by H. Murakami et. Impelluso is a formalism which makes advanced results from classical mechanics and group theory available to bachelor level engineers. The method allows engineers without deep understanding of these fields to describe the dynamic behavior of systems that would otherwise be too complex to approach. The method is therefore a framework on which engineers may rely without having a background in group theory or calculus of variations. In this thesis the focus will be on further generalizing the multi-body Moving Frame Method. We will restrict ourselves to multi-body systems with generalized coordinates which are all free rotations. These systems may all be idealized as N-body three-dimensional pendulums (These are sometimes named open kinematic chains). The three main contributions of this thesis is a coordinate free formulation of the Euler-Lagrange equations under the Moving Frame Formalism. An algorithmic approach to generating the equations of motion. The application of quaternions as representation of rotations in numerical simulation of multi-body systems rotating in three dimensions.Masteroppgave i anvendt og beregningsorientert matematikkMAMN-MABMAB39

Rigid Body Dynamics, Inertial Reference Frames, and Graphics Coordinate Systems: A Resolution of Conflicting Conventions and Terminology

Prepared for: NPS MOVES Academic GroupApproved for public release; distribution is unlimited