An Abundance of K3 Fibrations from Polyhedra with Interchangeable Parts

Even a cursory inspection of the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties reveals striking structures. These patterns correspond to webs of elliptic-K3 fibrations whose mirror images are also elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fibers. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, explains much of the structure of the observed patterns.Comment: 30 pages, 15 colour figure

A critical look at strings

This is an invited contribution to the Special Issue of "Foundations of Physics" titled "Forty Years Of String Theory: Reflecting On the Foundations". I have been asked to assess string theory as an outsider, and to compare it with the theory, methods, and expectations in my own field.Comment: 7 page

Canonical Quantization Inside the Schwarzschild Black Hole

We propose a scheme for quantizing a scalar field over the Schwarzschild manifold including the interior of the horizon. On the exterior, the timelike Killing vector and on the horizon the isometry corresponding to restricted Lorentz boosts can be used to enforce the spectral condition. For the interior we appeal to the need for CPT invariance to construct an explicitly positive definite operator which allows identification of positive and negative frequencies. This operator is the translation operator corresponding to the inexorable propagation to smaller radii as expected from the classical metric. We also propose an expression for the propagator in the interior and express it as a mode sum.Comment: 8 pages, LaTex. Title altered. One reference added. A few typos esp. eq.(7),(38) corrected. To appear in Class.Q.Gra

The Rotating Quantum Thermal Distribution

We show that the rigidly rotating quantum thermal distribution on flat space-time suffers from a global pathology which can be cured by introducing a cylindrical mirror if and only if it has a radius smaller than that of the speed-of-light cylinder. When this condition is met, we demonstrate numerically that the renormalized expectation value of the energy-momentum stress tensor corresponds to a rigidly rotating thermal bath up to a finite correction except on the mirror where there are the usual Casimir divergences.Comment: 8 pages, 2 PostScript figure

Vacuum Polarization and Energy Conditions at a Planar Frequency Dependent Dielectric to Vacuum Interface

The form of the vacuum stress-tensor for the quantized scalar field at a dielectric to vacuum interface is studied. The dielectric is modeled to have an index of refraction that varies with frequency. We find that the stress-tensor components, derived from the mode function expansion of the Wightman function, are naturally regularized by the reflection and transmission coefficients of the mode at the boundary. Additionally, the divergence of the vacuum energy associated with a perfectly reflecting mirror is found to disappear for the dielectric mirror at the expense of introducing a new energy density near the surface which has the opposite sign. Thus the weak energy condition is always violated in some region of the spacetime. For the dielectric mirror, the mean vacuum energy density per unit plate area in a constant time hypersurface is always found to be positive (or zero) and the averaged weak energy condition is proven to hold for all observers with non-zero velocity along the normal direction to the boundary. Both results are found to be generic features of the vacuum stress-tensor and not necessarily dependent of the frequency dependence of the dielectric.Comment: 16 pages, 4 figures, Revtex style Minor typographic corrections to equations and tex

M-theory moduli spaces and torsion-free structures

Motivated by the description of $\mathcal{N}=1$ M-theory compactifications to four-dimensions given by Exceptional Generalized Geometry, we propose a way to geometrize the M-theory fluxes by appropriately relating the compactification space to a higher-dimensional manifold equipped with a torsion-free structure. As a non-trivial example of this proposal, we construct a bijection from the set of $Spin(7)$-structures on an eight-dimensional $S^{1}$-bundle to the set of $G_{2}$-structures on the base space, fully characterizing the $G_{2}$-torsion clases when the total space is equipped with a torsion-free $Spin(7)$-structure. Finally, we elaborate on how the higher-dimensional manifold and its moduli space of torsion-free structures can be used to obtain information about the moduli space of M-theory compactifications.Comment: 24 pages. Typos fixed. Minor clarifications adde

Some Navigation Rules for D-Brane Monodromy

We explore some aspects of monodromies of D-branes in the Kahler moduli space of Calabi-Yau compactifications. Here a D-brane is viewed as an object of the derived category of coherent sheaves. We compute all the interesting monodromies in some nontrivial examples and link our work to recent results and conjectures concerning helices and mutations. We note some particular properties of the 0-brane.Comment: LaTeX2e, 28 pages, 4 figures, some typos corrected and refs adde

A-Model Correlators from the Coulomb Branch

We compute the contribution of discrete Coulomb vacua to A-Model correlators in toric Gauged Linear Sigma Models. For models corresponding to a compact variety, this determines the correlators at arbitrary genus. For non-compact examples, our results imply the surprising conclusion that the quantum cohomology relations break down for a subset of the correlators.Comment: 27 pages, 1 xy-pic figur

Introduction to Arithmetic Mirror Symmetry

We describe how to find period integrals and Picard-Fuchs differential equations for certain one-parameter families of Calabi-Yau manifolds. These families can be seen as varieties over a finite field, in which case we show in an explicit example that the number of points of a generic element can be given in terms of p-adic period integrals. We also discuss several approaches to finding zeta functions of mirror manifolds and their factorizations. These notes are based on lectures given at the Fields Institute during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics

F-theory on Genus-One Fibrations

We argue that M-theory compactified on an arbitrary genus-one fibration, that is, an elliptic fibration which need not have a section, always has an F-theory limit when the area of the genus-one fiber approaches zero. Such genus-one fibrations can be easily constructed as toric hypersurfaces, and various $SU(5)\times U(1)^n$ and $E_6$ models are presented as examples. To each genus-one fibration one can associate a $\tau$-function on the base as well as an $SL(2,\mathbb{Z})$ representation which together define the IIB axio-dilaton and 7-brane content of the theory. The set of genus-one fibrations with the same $\tau$-function and $SL(2,\mathbb{Z})$ representation, known as the Tate-Shafarevich group, supplies an important degree of freedom in the corresponding F-theory model which has not been studied carefully until now. Six-dimensional anomaly cancellation as well as Witten's zero-mode count on wrapped branes both imply corrections to the usual F-theory dictionary for some of these models. In particular, neutral hypermultiplets which are localized at codimension-two fibers can arise. (All previous known examples of localized hypermultiplets were charged under the gauge group of the theory.) Finally, in the absence of a section some novel monodromies of Kodaira fibers are allowed which lead to new breaking patterns of non-Abelian gauge groups.Comment: 53 pages, 9 figures, 6 tables. v2: references adde