Some Notes on Rational Spaces

Set constraints are inclusions between expressions denoting set of ground terms over a finitely ranked alphabet $\Sigma$. Rational spaces are topological spaces obtained as spaces of runs of topological $\Sigma$-hypergraphs. They were introduced by Kozen in \cite{K95a}, where the topological structure of the spaces of solutions to systems of set constraints was given in terms of rational spaces. In this paper we continue the investigation of rational spaces. We give a Myhill-Nerode like characterization of rational points, which in turn is used to re-derive results about the rational points of finitary rational spaces. We define congruences on $\Sigma$-hypergraphs, investigate their interplay with the Myhill-Nerode characterization, and finally we determine the computational complexity of some decision problems related to rational spaces

Homological Pisot Substitutions and Exact Regularity

We consider one-dimensional substitution tiling spaces where the dilatation (stretching factor) is a degree d Pisot number, and where the first rational Cech cohomology is d-dimensional. We construct examples of such "homological Pisot" substitutions that do not have pure discrete spectra. These examples are not unimodular, and we conjecture that the coincidence rank must always divide a power of the norm of the dilatation. To support this conjecture, we show that homological Pisot substitutions exhibit an Exact Regularity Property (ERP), in which the number of occurrences of a patch for a return length is governed strictly by the length. The ERP puts strong constraints on the measure of any cylinder set in the corresponding tiling space.Comment: 16 pages, LaTeX, no figure

On manifolds with multiple lens space filings

An irreducible 3--manifold with torus boundary either is a Seifert fibered space or admits at most three lens space fillings according to the Cyclic Surgery Theorem. We examine the sharpness of this theorem by classifying the non-hyperbolic manifolds with more than one lens space filling, classifying the hyperbolic manifolds obtained by filling of the Minimally Twisted 5 Chain complement that have three lens space fillings, showing that the doubly primitive knots in $S^3$ and $S^1 \times S^2$ have no unexpected extra lens space surgery, and showing that the Figure Eight Knot Sister Manifold is the only non-Seifert fibered manifold with a properly embedded essential once-punctured torus and three lens space fillings.Comment: 30 pages, 13 figure

Beta-gamma system, pure spinors and Hilbert series of arc spaces

Algorithms are presented for calculating the partition function of constrained beta-gamma systems in terms of the generating functions of the individual fields of the theory, the latter obtained as the Hilbert series of the arc space of the algebraic variety defined by the constraint. Examples of a beta-gamma system on a complex surface with an $A_1$ singularity and pure spinors are worked out and compared with existing results.Comment: 19 pages; sections 2.2, 4.2 and 5, revised for clarity; Macaulay2 code for pure spinors added in appendix A; acknowledgments added; version to appear in JHE

Hopf algebras and finite tensor categories in conformal field theory

In conformal field theory the understanding of correlation functions can be divided into two distinct conceptual levels: The analytic properties of the correlators endow the representation categories of the underlying chiral symmetry algebras with additional structure, which in suitable cases is the one of a finite tensor category. The problem of specifying the correlators can then be encoded in algebraic structure internal to those categories. After reviewing results for conformal field theories for which these representation categories are semisimple, we explain what is known about representation categories of chiral symmetry algebras that are not semisimple. We focus on generalizations of the Verlinde formula, for which certain finite-dimensional complex Hopf algebras are used as a tool, and on the structural importance of the presence of a Hopf algebra internal to finite tensor categories.Comment: 46 pages, several figures. v2: missing text added after (4.5), references added, and a few minor changes. v3: typos corrected, bibliography update