Geometric constructions on the algebra of densities

The algebra of densities \Den(M) is a commutative algebra canonically associated with a given manifold or supermanifold $M$. We introduced this algebra earlier in connection with our studies of Batalin--Vilkovisky geometry. The algebra \Den(M) is graded by real numbers and possesses a natural invariant scalar product. This leads to important geometric consequences and applications to geometric constructions on the original manifold. In particular, there is a classification theorem for derivations of the algebra \Den(M). It allows a natural definition of bracket operations on vector densities of various weights on a (super)manifold $M$, similar to how the classical Fr\"{o}licher--Nijenhuis theorem on derivations of the algebra of differential forms leads to the Nijenhuis bracket. It is possible to extend this classification from "vector fields" (derivations) on \Den(M) to "multivector fields". This leads to the striking result that an arbitrary even Poisson structure on $M$ possesses a canonical lifting to the algebra of densities. (The latter two statements were obtained by our student A.Biggs.) This is in sharp contrast with the previously studied case of an odd Poisson structure, where extra data are required for such a lifting.Comment: LaTeX, 23 p

The Lagrangian Loop Representation of Lattice U(1) Gauge Theory

It is showed how the Hamiltonian lattice $loop$ $representation$ can be cast straightforwardly in the Lagrangian formalism. The procedure is general and here we present the simplest case: pure compact QED. This connection has been shaded by the non canonical character of the algebra of the fundamental loop operators. The loops represent tubes of electric flux and can be considered the dual objects to the Nielsen-Olesen strings supported by the Higgs broken phase. The lattice loop classical action corresponding to the Villain form is proportional to the quadratic area of the loop world sheets and thus it is similar to the Nambu string action. This loop action is used in a Monte Carlo simulation and its appealing features are discussed.Comment: 13 pp, UAB-FT-341/9

Generalized inattentional blindness from a Global Workspace perspective

We apply Baars' Global Workspace model of consciousness to inattentional blindness, using the groupoid network method of Stewart et al. to explore modular structures defined by information measures associated with cognitive process. Internal cross-talk breaks the fundamental groupoid symmetry, and, if sufficiently strong, creates, in a highly punctuated manner, a linked, shifting, giant component which instantiates the global workspace of consciousness. Embedding, exterior, information sources act as an external field which breaks the groupoid symmetry in a somewhat different manner, definng the slowly-acting contexts of Baars' theory and providing topological constraints on the manifestations of consciousness. This analysis significantly extends recent mathematical treatments of the global workspace, and identifies a shifting, topologically-determined syntactical and grammatical 'bottleneck' as a tunable rate distortion manifold which constrains what sensory or other signals can be brought to conscious attention, typically in a punctuated manner. Sensations outside the limits of that filter's syntactic 'bandpass' have lower probability of detection, regardless of their structure, accounting for generalized forms of inattentional blindness