Knot polynomial states of quantum gravity in terms of loops and extended loops: Some remarks
In this paper we review the status of several solutions to all the constraints of quantum gravity that have been proposed in terms of loops and extended loops, based on knot polynomials. We discuss pitfalls of several of the results, and in particular the issues of covariance and regularization of the constraints in terms of extended loops. We also propose a formalism for ‘‘thickened out loops,’’ which does not face the covariance problems of extended loops and may allow to regularize expressions in a consistent manner.
We give a brief introduction to the study of the algebraic structures -- and their geometrical interpretations -- which arise in the BRST construction of a conformal string background. Starting from the chiral algebra $\cA$ of a string background, we consider a number of elementary but universal operations on the chiral algebra. From these operations we deduce a certain fundamental odd Poisson structure, known as a Gerstenhaber algebra, on the BRST cohomology of $\cA$. For the 2D string background, the correponding G-algebra can be partially described in term of a geometrical G-algebra of the affine plane $\bC^2$. This paper will appear in the proceedings of {\it Strings 95}.