Loop corrections and graceful exit in string cosmology

We examine the effect of perturbative string loops on the cosmological pre-big-bang evolution. We study loop corrections derived from heterotic string theory compactified on a $Z_N$ orbifold and we consider the effect of the all-order loop corrections to the Kahler potential and of the corrections to gravitational couplings, including both threshold corrections and corrections due to the mixed Kahler-gravitational anomaly. We find that string loops can drive the evolution into the region of the parameter space where a graceful exit is in principle possible, and we find solutions that, in the string frame, connect smoothly the superinflationary pre-big-bang evolution to a phase where the curvature and the derivative of the dilaton are decreasing. We also find that at a critical coupling the loop corrections to the Kahler potential induce a ghost-like instability, i.e. the kinetic term of the dilaton vanishes. This is similar to what happens in Seiberg-Witten theory and signals the transition to a new regime where the light modes in the effective action are different and are related to the original ones by S-duality. In a string context, this means that we enter a D-brane dominated phase.Comment: 39 pages, Latex, 17 eps figure

Enumerative geometry of Calabi-Yau 4-folds

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in CP5, are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.Comment: 44 page

The moduli space of stable quotients

A moduli space of stable quotients of the rank n trivial sheaf on stable curves is introduced. Over nonsingular curves, the moduli space is Grothendieck's Quot scheme. Over nodal curves, a relative construction is made to keep the torsion of the quotient away from the singularities. New compactifications of classical spaces arise naturally: a nonsingular and irreducible compactification of the moduli of maps from genus 1 curves to projective space is obtained. Localization on the moduli of stable quotients leads to new relations in the tautological ring generalizing Brill-Noether constructions. The moduli space of stable quotients is proven to carry a canonical 2-term obstruction theory and thus a virtual class. The resulting system of descendent invariants is proven to equal the Gromov-Witten theory of the Grassmannian in all genera. Stable quotients can also be used to study Calabi-Yau geometries. The conifold is calculated to agree with stable maps. Several questions about the behavior of stable quotients for arbitrary targets are raised.Comment: 50 page

Medial/skeletal linking structures for multi-region configurations

We consider a generic configuration of regions, consisting of a collection of distinct compact regions $\{\Omega_i\}$ in $\mathbb{R}^{n+1}$ which may be either smooth regions disjoint from the others or regions which meet on their piecewise smooth boundaries $\mathcal{B}_i$ in a generic way. We introduce a skeletal linking structure for the collection of regions which simultaneously captures the regions' individual shapes and geometric properties as well as the "positional geometry" of the collection. The linking structure extends in a minimal way the individual "skeletal structures" on each of the regions, allowing us to significantly extend the mathematical methods introduced for single regions to the configuration. We prove for a generic configuration of regions the existence of a special type of Blum linking structure which builds upon the Blum medial axes of the individual regions. This requires proving several transversality theorems for certain associated "multi-distance" and "height-distance" functions for such configurations. We show that by relaxing the conditions on the Blum linking structures we obtain the more general class of skeletal linking structures which still capture the geometric properties. In addition to yielding geometric invariants which capture the shapes and geometry of individual regions, the linking structures are used to define invariants which measure positional properties of the configuration such as: measures of relative closeness of neighboring regions and relative significance of the individual regions for the configuration. These invariants, which are computed by formulas involving "skeletal linking integrals" on the internal skeletal structures, are then used to construct a "tiered linking graph," which identifies subconfigurations and provides a hierarchical ordering of the regions.Comment: 135 pages, 36 figures. Version to appear in Memoirs of the Amer. Math. So

Difficulties of an Infrared Extension of Differential Renormalization

We investigate the possibility of generalizing differential renormalization of D.Z.Freedman, K.Johnson and J.I.Latorre in an invariant fashion to theories with infrared divergencies via an infrared $\tilde{R}$ operation. Two-dimensional $\sigma$ models and the four-dimensional $\phi^4$ theory diagrams with exceptional momenta are used as examples, while dimensional renormalization serves as a test scheme for comparison. We write the basic differential identities of the method simultaneously in co-ordinate and momentum space, introducing two scales which remove ultraviolet and infrared singularities. The consistent set of Fourier-transformation formulae is derived. However, the values for tadpole-type Feynman integrals in higher orders of perturbation theory prove to be ambiguous, depending on the order of evaluation of the subgraphs. In two dimensions, even earlier than this ambiguity manifests itself, renormalization-group calculations based on infrared extension of differential renormalization lead to incorrect results. We conclude that the extended differential renormalization procedure does not perform the infrared $\tilde{R}$ operation in a self-consistent way, as the original recipe does the ultraviolet $R$ operation.Comment: (minor changes have been made to make clear that no infrared problems occur in the original ultraviolet procedure of [1]; subsection 2.1 has been added to outline the ideas a simple example), 26 pages, LaTeX, JINR preprint E2-92-538, Dubna (Dec.1992

Linear optimization over homogeneous matrix cones

A convex cone is homogeneous if its automorphism group acts transitively on the interior of the cone, i.e., for every pair of points in the interior of the cone, there exists a cone automorphism that maps one point to the other. Cones that are homogeneous and self-dual are called symmetric. The symmetric cones include the positive semidefinite matrix cone and the second order cone as important practical examples. In this paper, we consider the less well-studied conic optimization problems over cones that are homogeneous but not necessarily self-dual. We start with cones of positive semidefinite symmetric matrices with a given sparsity pattern. Homogeneous cones in this class are characterized by nested block-arrow sparsity patterns, a subset of the chordal sparsity patterns. We describe transitive subsets of the automorphism groups of the cones and their duals, and important properties of the composition of log-det barrier functions with the automorphisms in this set. Next, we consider extensions to linear slices of the positive semidefinite cone, i.e., intersection of the positive semidefinite cone with a linear subspace, and review conditions that make the cone homogeneous. In the third part of the paper we give a high-level overview of the classical algebraic theory of homogeneous cones due to Vinberg and Rothaus. A fundamental consequence of this theory is that every homogeneous cone admits a spectrahedral (linear matrix inequality) representation. We conclude by discussing the role of homogeneous cone structure in primal-dual symmetric interior-point methods.Comment: 59 pages, 10 figures, to appear in Acta Numeric

Unitary local systems, multiplier ideals, and polynomial periodicity of Hodge numbers

The space of unitary local systems of rank one on the complement of an arbitrary divisor in a complex projective algebraic variety can be described in terms of parabolic line bundles. We show that multiplier ideals provide natural stratifications of this space. We prove a structure theorem for these stratifications in terms of complex tori and convex rational polytopes, generalizing to the quasi-projective case results of Green-Lazarsfeld and Simpson. As an application we show the polynomial periodicity of Hodge numbers of congruence covers in any dimension, generalizing results of E. Hironaka and Sakuma. We extend the structure theorem and polynomial periodicity to the setting of cohomology of unitary local systems. In particular, we obtain a generalization of the polynomial periodicity of Betti numbers of unbranched congruence covers due to Sarnak-Adams. We derive a geometric characterization of finite abelian covers, which recovers the classic one and the one of Pardini. We use this, for example, to prove a conjecture of Libgober about Hodge numbers of abelian covers.Comment: final version, to appear in Adv. Mat