MathSciDoc: An Archive for Mathematician ∫

String theory has had a profound influence on research in Calabi-Yau spaces over the past twenty-five years. We first briefly mention some of the work in Khler Calabi-Yau manifolds that was influenced by the discovery of mirror symmetry in the late 1980s. We then discuss some of the mathematical motivations behind the recent work on non-Khler Calabi-Yau manifolds, which arise in string compactifications with fluxes. After extending mirror symmetry to non-Khler Calabi-Yau manifolds, we show how this leads to new cohomologies and invariants of non-Khler symplectic manifolds.

This paper gives a survey of wave dynamics in the Kerr space-time geometry, the mathematical model of a rotating black hole in equilibrium. After a brief introduction to the Kerr metric, we review the separability properties of linear wave equations for fields of general spin s= 0,rac {1}{2}, 1, 2, corresponding to scalar, Dirac, electromagnetic fields and linearized gravitational waves. We give results on the long-time dynamics of Dirac and scalar waves, including decay rates for massive Dirac fields. For scalar waves, we give a rigorous treatment of superradiance and describe rigorously a mechanism of energy extraction from a rotating black hole. Finally, we discuss the open problem of linear stability of the Kerr metric and present partial results.

We explain how Polchinski's work on D-branes re-read from a noncommutative version of Grothendieck's equivalence of local geometries and function rings gives rise to an intrinsic prototype definition of D-branes (of B-type) as an Azumaya-type noncommutative space. Several originally open-string induced properties of D-branes can be reproduced solely by this intrinsic definition. We study also the moduli space of D0-branes on a commutative target space in this setup. Some of its features resembles gas of D0-branes in Vafa's work.

In this note, we give a list of Calabi-Yau hypersurfaces in weighted projective 4-spaces with the property that a hypersurface contains naturally a pencil of K3 variety. For completeness we also obtain a similar list in the case K3 hypersurfaces in weighted projective 3-spaces. The first list significantly enlarges the list of K3-fibrations of\KlemmLercheMayr~ which has been obtained on some assumptions on the weights. Our lists are expected to correspond to examples of the so-called heterotic-type II duality\KachruVafa\KachruSilverstein.

A classical set of birational invariants of a variety are its spaces of pluricanonical forms and some of their canonically defined subspaces. Each of these vector spaces admits a typical metric structure which is also birationally invariant. These vector spaces so metrized will be referred to as the pseudonormed spaces of the original varieties. A fundamental question is the following: Given two mildly singular projective varieties with some of the first variety9s pseudonormed spaces being isometric to the corresponding ones of the second variety9s, can one construct a birational map between them that induces these isometries? In this work, a positive answer to this question is given for varieties of general type. This can be thought of as a theorem of Torelli type for birational equivalence.

We show how two topologically distinct spaces-the Kahler K3 x T^ 2 and the non-Kahler T^ 2 bundle over K3-can be smoothly connected in heterotic string theory. The transition occurs when the base K3 is deformed to the T^ 4/Z_2 orbifold limit. The orbifold theory can be mapped via duality to M-theory on K3 x K3 where the transition corresponds to an exchange of the two K3's.

We find a concise relation between the moduli , of a rational Narain lattice (,) and the corresponding momentum lattices of left and right chiral algebras via the Gauss product. As a byproduct, we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices.

We study a static, spherically symmetric system of (2j+ 1) massive Dirac particles, each having angular momentum j, j= 1, 2,..., in a classical gravitational and SU (2) Yang-Mills field. We show that for any black hole solution of the associated Einstein-Dirac-Yang/Mills equations, the spinors must vanish identically outside of the event horizon.

IN 1970 Lawson [22] proved that two embedded closed diffeomorphic minimal surfaces in the unit three-dimensional sphere S3 in lRJ are ambiently isotopic in S3. Lawson proved this theorem by first proving that an embedded orientable closed minimal surface of genus y in a closed orientable Riemannian three-manifold Mwith positive Ricci curvature disconnects M-into two genus-8 handlebodics. A result of Frankel [7] was used to prove this. Lawson then applied a deep result of Waldhauscn [33] that states that decompositions of SJ into two genus-cl handlebodies arc unique up to ambient isotopy. More prcciscly. Waldhauscns uniqucncss thcorcm states that whenever a closed surface of genus y in S separates SJ into handlcbodics, then the embedding of the surface is as simple as possible; in other words, the surface is obtained from a two-sphere Sz c Sby adding handles in an unknotted manner.

We explore how introducing a non-trivial Mordell-Weil group changes the structure of the Coulomb phases of a five-dimensional gauge theory from an M-theory compactified on an elliptically fibered Calabi-Yau threefolds with a I _2 + I _2 collision of singularities. The resulting gauge theory has a semi-simple Lie algebra _2 or _2 . We compute topological invariants relevant for the physics, such as the Euler characteristic, Hodge numbers, and triple intersection numbers. We determine the matter representation geometrically by computing weights via intersection of curves and fibral divisors. We fix the number of charged hypermultiplets transforming in each representations by comparing the triple intersection numbers and the one-loop prepotential. This condition is enough to fix the number of representation when the Mordell-Weil group is _2 but not when it is trivial. The vanishing of the fourth power of the curvature forms in the anomaly polynomial is enough to fix the number of representations. We discuss anomaly cancellations of the six-dimensional uplifted. In particular, the gravitational anomaly is also considered as the Hodge numbers are computed explicitly without counting the degrees of freedom of the Weierstrass equation.