Paul S. Aspinwall

Calabi-Yau moduli space, mirror manifolds and spacetime topology change in string theory☆

A permanent magnet motor is provided which has first and second U-shaped sections comprising the motor stator. Each section is constructed of low reluctance magnetic material. The motor armature shaft passes through a hole in the bight of each section and carries a commutator. An air space separates the bight and shaft to provide a high reluctance path to the shaft. The two parallel sides of the first section fit between the two sides of the second section in a contact relation; the combination defining a two piece magnetic circuit that passes around the motor from end to end. The first section contains a permanent magnet mounted upon each of the sides at diametrically opposite locations adjacent the armature. The flux coupling the two magnets passes through the armature core. The magnets define a pole pair and are spaced very close to the edge of the armature core to minimize the air gap between the armature and the magnet. The first section is unbroken and provides a homogeneous flux path between the magnets. The second section provides a parallel-secondary flux path from the magnets. Additional low reluctance material is included on the bight of each section to compensate for the increased reluctance resulting from the hole through which the armature passes. Positioning stops are provided on the second section, which contact the first section as it is positioned, to establish the proper distal relation between the bights to accommodate the armature and commutator.

Enhanced gauge symmetries and K3 surfaces

Abstract String-string duality dictates that type IIA strings compactified on a K3 surface acquire non-abelian gauge groups for certain values of the K3 moduli. We argue that, contrary to expectation, the theories for which such enhanced gauge symmetries appear are not orbifolds in the string sense. For a specific example we show that a theory with enhanced gauge symmetry and an orbifold theory have the same classical K3 surface as a target space but the value of the “ B -field” differs. This raises the possibility that the conformal field theory associated to a string theory with an enhanced gauge group is badly behaved in some way.

Topological field theory and rational curves

We analyze the quantum field theory corresponding to a string propagating on a Calabi-Yau threefold. This theory naturally leads to the consideration of Wittens topological non-linear σ-model and the structure of rational curves on the Calabi-Yau manifold. We study in detail the case of the world-sheet of the string being mapped to a multiple cover of an isolated rational curve and we show that a natural compactification of the moduli space of such a multiple cover leads to a formula in agreement with a conjecture by Candelas, de la Ossa, Green and Parkes.

Some relationships between dualities in string theory

Some relationships between string theories and eleven-dimensional supergravity are discussed and reviewed. We see how some relationships can be derived from others. The cases of N = 2 supersymmetry in nine dimensions and N = 4 supersymmetry in four dimensions are discussed in some detail. The latter case leads to consideration of quotients of a K3 surface times a torus and to a possible peculiar relationship between eleven-dimensional supergravity and the heterotic strings in ten dimensions.

Point - like instantons on K3 orbifolds

The map between the moduli space of F-theory (or type 11 string) compactifications and heterotic string compactifications can be considerably simplified by using “stable degenerations”. We discuss how this method applies to both the E8 × E8 and the Spin(32)/Z2 heterotic string. As a simple application of the method we derive some basic properties of the non-perturbative physics of collections of E8 or Spin(32)/Z2 point-like instantons sitting at ADE singularities on a K3 surface.

On the ubiquity of K3 fibrations in string duality

Abstract We consider the general case of N = 2 dual pairs of type IIA/heterotic string theories in four dimensions. We show that if the type IIA string in this pair can be viewed as having been compactified on a Calabi-Yau manifold in the usual way then this manifold must be of the form of a K3 fibration. We also see how the bound on the rank of the gauge group of the perturbative heterotic string has a natural interpretation on the type IIA side.

Derived Categories and Zero-Brane Stability

We define a particular class of topological field theories associated to open strings and prove the resulting D-branes and open strings form the bounded derived category of coherent sheaves. This derivation is a variant of some ideas proposed recently by Douglas. We then argue that any 0-brane on any Calabi-Yau threefold must become unstable along some path in the Kahler moduli space. As a byproduct of this analysis we see how the derived category can be invariant under a birational transformation between Calabi-Yaus.

Multiple mirror manifolds and topology change in string theory

Abstract We use mirror symmetry to establish the first concrete arena of spacetime topology change in string theory. In particular, we establish that the quantum theories based on certain nonlinear sigma models with topologically distinct target spaces can be smoothly connected even though classically a physical singularity would be encountered. We accomplish this by rephrasing the description of these nonlinear sigma models in terms of their mirror manifold partners - a description in which the full quantum theory can be described exactly using lowest order geometrical methods. We establish that, for the known class of mirror manifolds, the moduli space of the corresponding conformal field theory requires not just two but numerous topologically distinct Calabi-Yau manifolds for its geometric interpretation. A single family of continously connected conformal theories thereby probes a host of topologically distinct geometrical spaces giving rise to multiple mirror manifolds .

D-Brane Stability and Monodromy

We review the idea of �-stability for B-type D-branes on a Calabi–Yau manifold. It is shown that the octahedral axiom from the theory of derived categories is an essential ingredient in the study of stability. Various examples in the context of the quintic Calabi–Yau threefold are studied and we plot the lines of marginal stability in several cases. We derive the conjecture of Kontsevich, Horja and Morrison for the derived category version of monodromy around a “conifold” point. Finally, we propose an application of these ideas to the study of supersymmetry breaking.

The SO(32) heterotic string on a K3 surface

Abstract The SO(32) heterotic string on a K3 surface is analyzed in terms of the dual theory of a type II string (or F-theory) on an elliptically fibred Calabi-Yau manifold. The results are in beautiful agreement with earlier work by Witten using very different methods. In particular, we find gauge groups of SO(32) × Sp( k ) appearing at points in the moduli space identified with point-like instantons and see hypermultiplets in the ( 32, 2 k ) representation becoming massless at the same time. We also discuss some aspects of the E 8 × E 8 case.