Deformation quantization of gerbes

This is the first in a series of articles devoted to deformation quantization of gerbes. Here we give basic definitions and interpret deformations of a given gerbe as Maurer-Cartan elements of a differential graded Lie algebra (DGLA). We classify all deformations of a given gerbe on a symplectic manifold, as well as provide a deformation-theoretic interpretation of the first Rozansky-Witten class.Comment: Revised versio

A variant of the Mukai pairing via deformation quantization

We give a new method to prove a formula computing a variant of Caldararu's Mukai pairing \cite{Cal1}. Our method is based on some important results in the area of deformation quantization. In particular, part of the work of Kashiwara and Schapira in \cite{KS} as well as an algebraic index theorem of Bressler, Nest and Tsygan in \cite{BNT},\cite{BNT1} and \cite{BNT2} are used. It is hoped that our method is useful for generalization to settings involving certain singular varieties.Comment: 8 pages. Comments and suggestions welcom

Kikuchi ultrafast nanodiffraction in four-dimensional electron microscopy

Coherent atomic motions in materials can be revealed using time-resolved X-ray and electron Bragg diffraction. Because of the size of the beam used, typically on the micron scale, the detection of nanoscale propagating waves in extended structures hitherto has not been reported. For elastic waves of complex motions, Bragg intensities contain all polarizations and they are not straightforward to disentangle. Here, we introduce Kikuchi diffraction dynamics, using convergent-beam geometry in an ultrafast electron microscope, to selectively probe propagating transverse elastic waves with nanoscale resolution. It is shown that Kikuchi band shifts, which are sensitive only to the tilting of atomic planes, reveal the resonance oscillations, unit cell angular amplitudes, and the polarization directions. For silicon, the observed wave packet temporal envelope (resonance frequency of 33 GHz), the out-of-phase temporal behavior of Kikuchi's edges, and the magnitude of angular amplitude (0.3 mrad) and polarization [011] elucidate the nature of the motion: one that preserves the mass density (i.e., no compression or expansion) but leads to sliding of planes in the antisymmetric shear eigenmode of the elastic waveguide. As such, the method of Kikuchi diffraction dynamics, which is unique to electron imaging, can be used to characterize the atomic motions of propagating waves and their interactions with interfaces, defects, and grain boundaries at the nanoscale