Parabolic bundles and representations of the fundamental group

Let X be as smooth complex projective variety with Neron-Severi group isomorphic to Z, and D an irreducible divisor with normal crossing singularities. Assume r is equal to 2 or 3. We prove that if the fundamental group of X doesn't have irreducible PU(r) representations, then the fundamental group of X-D doesn't have irreducible U(r) representations. The proof uses the non-existence of certain stable parabolic bundles. We also obtain a similar result for GL(2) when D is smooth and X is a complex surface.Comment: 13 pages, 1 figure, LaTeX2

Symplectic Geometry of character varieties and SU(2)SU(2) Lattice Gauge Theory I

Given a finite connected graph $\Lambda$, the space of $SU(2)$ lattice gauge-fields on $\Lambda$, modulo gauge transformations, is a Lagrangian submanifold -- with mild singularities -- of the $SU(2)$ character variety (= phase-space of Chern-Simons theory) of an associated surface. We present evidence that, in the limit of large $\Lambda$, integration over the character variety with respect to the Liouville measure approximates lattice-theoretic integrals. By the works of W. Goldman, L. Jeffrey and J. Weitsman, the formalism of Duistermaat-Heckman applies to the relevant integrals over the character variety. A continuous version of the Verlinde algebra facilitates computations. In two dimensions we recover standard expressions. For the theory on a 3-dimensional periodic lattice $\Lambda$ with Migdal action we get a very pleasant expression for the symplectic partition function, and with the Wilson action a more elaborate one. Each is a sum of a series with positive terms. One can also write down expressions for plaquette-plaquette correlations and 't Hooft loops.Comment: 43 pages, typos correcte

Oded Schramm and the Schramm-Loewner evolution: in memoriam

This article does not have an abstract

Factorisation of generalised theta functions. I

We prove a version of "factorisation", relating the space of sections of theta bundles on the moduli spaces of (parabolic, rank 2) vector bundles on curves of genus g and g-1

Yang-Mills theory for bundle gerbes

Given a bundle gerbe with connection on an oriented Riemannian manifold of dimension at least equal to 3, we formulate and study the associated Yang-Mills equations. When the Riemannian manifold is compact and oriented, we prove the existence of instanton solutions to the equations and also determine the moduli space of instantons, thus giving a complete analysis in this case. We also discuss duality in this context.Comment: Latex2e, 7 pages, some typos corrected, to appear in J. Phys. A: Math. and Ge

Ultrasound sensing using the acousto-optic effect in polymer dispersed liquid crystals

Acousto-optic effects are demonstrated in polymer dispersed liquid crystal (PDLC) films, showing promise for applications in ultrasound sensing. The PDLC films are used to image two displacement profiles of an air-coupled flexural transducers resonant modes at 295 kHz and 730 kHz. Results are confirmed using laser vibrometry. The regions on the transducers with the largest displacements are clearly imaged by the PDLC films, with the resolution agreeing well with laser vibrometry scanning. Imaging takes significantly less time than a scanning system (switching time of a few seconds, as compared to 8 hours for laser vibrometry). Heating effects are carefully monitored using thermal imaging, and are found not to be the main cause of PDLC clearing