Weighted Projective Lines and Rational Surface Singularities

In this paper we study rational surface singularities R with star shaped dual graphs, and under very mild assumptions on the self-intersection numbers we give an explicit description of all their special Cohen-Macaulay modules. We do this by realising R as a certain Z-graded Veronese subring S^x of the homogeneous coordinate ring S of the Geigle-Lenzing weighted projective line X, and we realise the special CM modules as explicitly described summands of the canonical tilting bundle on X. We then give a second proof that these are special CM modules by comparing qgr S^x and coh X, and we also give a necessary and sufficient combinatorial criterion for these to be equivalent categories. In turn, we show that qgr S^x is equivalent to qgr of the reconstruction algebra, and that the degree zero piece of the reconstruction algebra coincides with Ringel's canonical algebra. This implies that the reconstruction algebra contains the canonical algebra, and furthermore its qgr category is derived equivalent to the canonical algebra, thus linking the reconstruction algebra of rational surface singularities to the canonical algebra of representation theory. Comment: Final version

Kac Polynomials for Canonical Algebras

We prove that the number of geometrically indecomposable representations of fixed dimension vector $\mathbf{d}$ of a canonical algebra $C$ defined over a finite field $\mathbb{F}_q$ is given by a polynomial in $q$ (depending on $C$ and $\mathbf{d}$). We prove a similar result for squid algebras. Finally, we express the volume of the moduli stacks of representations of these algebras of a fixed dimension vector in terms of the corresponding Kac polynomials.

INFORMATION GEOMETRY FOR SOME LIE ALGEBRAS

For certain unitary representations of a Lie algebra [Formula: see text] we define the statistical manifold ℳ of states as the convex cone of [Formula: see text] for which the partition function Z= Tr exp {-X} is finite. The Hessian of Ψ= log Z defines a Riemannian metric g on [Formula: see text], (the Bogoliubov–Kubo–Mori metric); [Formula: see text] foliates into the union of coadjoint orbits, each of which can be given a complex structure (that of Kostant). The program is carried out for so(3), and for sl(2,R) in the discrete series. We show that ℳ=R+× CP 1 and R+×H respectively. We show that for the metaplectic representation of the quadratic canonical algebra, ℳ=R+× CP 2/Z2. Exactly solvable model dynamics is constructed in each case.

ELECTRIC-MAGNETIC DUALITY AND THE “LOOP REPRESENTATION” IN ABELIAN GAUGE THEORIES

Abelian gauge theories are quantized in a geometric representation that generalizes the loop representation and treats electric and magnetic operators on the same footing. The usual canonical algebra is turned into a topological algebra of nonlocal operators that resembles the order-disorder dual algebra of ’t Hooft. These dual operators provide a complete description of the physical phase space of the theories.

THE CANONICAL ALGEBRA OF SYMMETRIES FOR GAUGE THEORIES: THE EXAMPLE OF ANYONS

Some special features of the algebra of continuous symmetries for constrained systems are illustrated through the example of the nonrelativistic ideal gas of anyons. The relation between constants of motion and symmetry generators for this gauge system is completely worked out. The generators of both its rigid and gauge continuous symmetries are explicitly given in the canonical formalism. Also the interplay between these two types of symmetries within the symmetry algebra is exhibited.