Khovanov's homology for tangles and cobordisms

We give a fresh introduction to the Khovanov Homology theory for knots and links, with special emphasis on its extension to tangles, cobordisms and 2-knots. By staying within a world of topological pictures a little longer than in other articles on the subject, the required extension becomes essentially tautological. And then a simple application of an appropriate functor (a `TQFT') to our pictures takes them to the familiar realm of complexes of (graded) vector spaces and ordinary homological invariants.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper33.abs.htm

Properties of the Volume Operator in Loop Quantum Gravity I: Results

We analyze the spectral properties of the volume operator of Ashtekar and Lewandowski in Loop Quantum Gravity, which is the quantum analogue of the classical volume expression for regions in three dimensional Riemannian space. Our analysis considers for the first time generic graph vertices of valence greater than four. Here we find that the geometry of the underlying vertex characterizes the spectral properties of the volume operator, in particular the presence of a `volume gap' (a smallest non-zero eigenvalue in the spectrum) is found to depend on the vertex embedding. We compute the set of all non-spatially diffeomorphic non-coplanar vertex embeddings for vertices of valence 5--7, and argue that these sets can be used to label spatial diffeomorphism invariant states. We observe how gauge invariance connects vertex geometry and representation properties of the underlying gauge group in a natural way. Analytical results on the spectrum on 4-valent vertices are included, for which the presence of a volume gap is proved. This paper presents our main results; details are provided by a companion paper arXiv:0706.0382v1.Comment: 36 pages, 7 figures, LaTeX. See also companion paper arXiv:0706.0382v1. Version as published in CQG in 2008. See arXiv:1003.2348 for important remarks regarding the sigma configurations. Subsequent computations have revealed some minor errors, which do not change the qualitative results but modify some of the numbers presented her

Representational information: a new general notion and measure\ud of information

In what follows, we introduce the notion of representational information (information conveyed by sets of dimensionally defined objects about their superset of origin) as well as an\ud original deterministic mathematical framework for its analysis and measurement. The framework, based in part on categorical invariance theory [Vigo, 2009], unifies three key constructsof universal science – invariance, complexity, and information. From this unification we define the amount of information that a well-defined set of objects R carries about its finite superset of origin S, as the rate of change in the structural complexity of S (as determined by its degree of categorical invariance), whenever the objects in R are removed from the set S. The measure captures deterministically the significant role that context and category structure play in determining the relative quantity and quality of subjective information conveyed by particular objects in multi-object stimuli

Criteria for Conformal Invariance of (0,2) Models

It is argued that many linear (0,2) models flow in the infrared to conformally invariant solutions of string theory. The strategy in the argument is to show that the effective space-time superpotential must vanish because there is no place where it can have a pole. This conclusion comes from either of two different analyses, in which the Kahler class or the complex structure of the gauge bundle is varied, while keeping everything else fixed. In the former case, we recover from the linear sigma model the usual simple pole in the ${\bf \bar {27}}^3$ Yukawa coupling but show that an analogous pole does not arise in the couplings of gauge singlet modes. In the latter case, a dimension count shows that the world-sheet instanton sum does not ``see'' the singularities of the gauge bundle and hence cannot have a pole.Comment: 36 pages, harvmac. Equation (3.14) and miscellaneous typos corrected. (Error in section 5.2 has been corrected in 1999 version.

Special geometry on the 101 dimesional moduli space of the quintic threefold

A new method for explicit computation of the CY moduli space metric was proposed by the authors recently. The method makes use of the connection of the moduli space with a certain Frobenius algebra. Here we clarify this approach and demonstrate its efficiency by computing the Special geometry of the 101-dimensional moduli space of the quintic threefold around the orbifold point.Comment: We made exposition more clear, in particular we explained how to generalize our idea