Candelas and Font introduced the notion of a 'top' as half of a three dimensional reflexive polytope and noticed that Dynkin diagrams of enhanced gauge groups in string theory can be read off from them. We classify all tops satisfying a generalized definition as a lattice polytope with one facet containing the origin and the other facets at distance one from the origin. These objects torically encode the local geometry of a degeneration of an elliptic fibration. We give a prescription for assigning an affine, possibly twisted Kac-Moody algebra to any such top (and more generally to any elliptic fibration structure) in a precise way that involves the lengths of simple roots and the coefficients of null roots. Tops related to twisted Kac-Moody algebras can be used to construct string compactifications with reduced rank of the gauge group.

Applications of Riemannian quantum geometry to cosmology have had notable successes. In particular, the fundamental discreteness underlying quantum geometry has led to a natural resolution of the big bang singularity. However, the precise mathematical structure underlying loop quantum cosmology and the sense in which it implements the full quantization program in a symmetry reduced model has not been made explicit. The purpose of this paper is to address these issues, thereby providing a firmer mathematical and conceptual foundation to the subject.

We use the Konishi anomaly equations to construct the exact effective superpotential of the glueball superfields in various N = 1 supersymmetric gauge theories. We use the superpotentials to study in detail the structure of the spaces of vacua of these theories. We consider chiral and non-chiral 1SU(N) models, the exceptional gauge group G₂ and models that break supersymmetry dynamically.

We give a new derivation of the quasinormal frequencies of Schwarzschild black holes in d greater than or equal to 4 and Reissner-Nordstrom black holes in d = 4, in the limit of infinite damping. For Schwarzschild in d greater than or equal to 4 we find that the asymptotic real part is T_Hawkinglog(3) for scalar perturbations and for some gravitational perturbations; this confirms a result previously obtained by other means in the case d = 4. For Reissner-Nordstrom in d = 4 we find a specific generally aperiodic behavior for the quasinormal frequencies, both for scalar perturbations and for electromagnetic-gravitational perturbations. The formulae are obtained by studying the monodromy of the perturbation analytically continued to the complex plane; the analysis depends essentially on the behavior of the potential in the 'unphysical' region near the black hole singularity.

A new approach is suggested to the problem of quantising causal sets, or topologies, or other such models for space-time (or space). The starting point is the observation that entities of this type can be regarded as objects in a category whose arrows are structure-preserving maps. This motivates investigating the general problem of quantising a system whose `configuration space' (or history-theory analogue) can be regarded as the set of objects Ob(Q) in a category Q. In this first of a series of papers, we study this question in general and develop a scheme based on constructing an analogue of the group that is used in the canonical quantisation of a system whose configuration space is a manifold Q is isomorphic to G/H where G and H are Lie groups. In particular, we choose as the analogue of G the monoid of 'arrow fields' on Q. Physically, this means that an arrow between two objects in the category is viewed as some sort of analogue of momentum. After finding the 'category quantisation monoid', we show how suitable representations can be constructed using a bundle of Hilbert spaces over Ob(Q).

Liouville theory with a negative norm boson and no screening charge corresponds to an exact classical solution of closed bosonic string theory describing time-dependent bulk tachyon condensation. A simple expression for the two point function is proposed based on renormalization/analytic continuation of the known results for the ordinary (positive-norm) Liouville theory. The expression agrees exactly with the minisuperspace result for the closed string pair-production rate (which diverges at finite time). Puzzles concerning the three-point function are presented and discussed.