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Holomorphic Parabolic Geometries and Calabi-Yau Manifolds
Benjamin McKay
Symmetry, Integrability and Geometry : Methods and Applications , 2011,
Abstract: We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact K hler manifolds.
Holomorphic Parabolic Geometries and Calabi-Yau Manifolds  [PDF]
Benjamin McKay
Mathematics , 2008, DOI: 10.3842/SIGMA.2011.090
Abstract: We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact K\"ahler manifolds.
Holomorphic Cartan geometries and Calabi--Yau manifolds  [PDF]
Indranil Biswas,Benjamin McKay
Mathematics , 2008, DOI: 10.1016/j.geomphys.2009.12.011
Abstract: We prove that the only Calabi--Yau projective manifolds which bear holomorphic Cartan geometries are precisely the abelian varieties. (Nous d\'emontrons que les seules vari\'et\'es projectives de Calabi--Yau qui poss\`edent des g\'eom\'etrie holomorphes de Cartan sont les vari\'et\'es ab\'eliennes.)
Holomorphic Cartan geometries, Calabi--Yau manifolds and rational curves  [PDF]
Indranil Biswas,Benjamin McKay
Mathematics , 2010, DOI: 10.1016/j.difgeo.2009.09.003
Abstract: We prove that if a Calabi--Yau manifold $M$ admits a holomorphic Cartan geometry, then $M$ is covered by a complex torus. This is done by establishing the Bogomolov inequality for semistable sheaves on compact K\"ahler manifolds. We also classify all holomorphic Cartan geometries on rationally connected complex projective manifolds.
Polynomial Roots and Calabi-Yau Geometries  [PDF]
Yang-Hui He
Advances in High Energy Physics , 2011, DOI: 10.1155/2011/719672
Abstract: The examination of roots of constrained polynomials dates back at least to Waring and to Littlewood. However, such delicate structures as fractals and holes have only recently been found. We study the space of roots to certain integer polynomials arising naturally in the context of Calabi-Yau spaces, notably Poincaré and Newton polynomials, and observe various salient features and geometrical patterns. 1. Introduction and Summary The subject of roots of monovariate polynomials is, without doubt, an antiquate one and has germinated an abundance of fruitful research over the ages. It is, therefore, perhaps surprising that any new statements could at all be made regarding such roots. The advent of computer algebra, chaotic phenomena, and random ensembles has, however, indeed shed new light upon so ancient a metier. Polynomials with constrained coefficients and form, though permitted to vary randomly, have constituted a vast field itself. As far back as 1782, Edward Waring, in relation to his famous problem on power summands, had shown that for cubic polynomials with random real coefficients, the ratio of the probability of finding nonreal zeros versus that of not finding non-real zeros is less than or equal to 2. Constraining the coefficients to be integers within a fixed range has, too, its own history. It was realised in [1] that a degree random polynomial with distributed evenly, the expected number of real roots is of order asymptotically in . This was furthered by [2] to be essentially independent of the statistics, in that has the same asymptotics (cf. also [3, 4]), as much for being evenly distributed real numbers, in , or as Gaussian distributed in . Continual development ensued (q.v. also [5]), notably by Littlewood [6], Erd?s and Turán [7], Hammersley [8], and Kac [9]. Indeed, a polynomial with coefficients only taking values as has come to be known as a Littlewood polynomial, and the Littlewood Problem asks for the the precise asymptotics, in the degree, of such polynomials taking values, with complex arguments, on the unit circle. The classic work of Montgomery [10] and Odlyzko [11], constituting one of the most famous computer experiments in mathematics (q. v. Section?? 3.1 of [12] for some recent remarks on the distributions), empirically showed that the distribution of the (normalized) spacings between successive critical zeros of the Riemann zeta function is the same as that of a Gaussian unitary ensemble of random matrices, whereby infusing our subject with issues of uttermost importance. Subsequently, combining the investigation of zeros
Heterotic Non-Kahler Geometries via Polystable Bundles on Calabi-Yau Threefolds  [PDF]
Bjorn Andreas,Mario Garcia-Fernandez
Mathematics , 2010, DOI: 10.1016/j.geomphys.2011.10.013
Abstract: In arXiv:1008.1018 it is shown that a given stable vector bundle $V$ on a Calabi-Yau threefold $X$ which satisfies $c_2(X)=c_2(V)$ can be deformed to a solution of the Strominger system and the equations of motion of heterotic string theory. In this note we extend this result to the polystable case and construct explicit examples of polystable bundles on elliptically fibered Calabi-Yau threefolds where it applies. The polystable bundle is given by a spectral cover bundle, for the visible sector, and a suitably chosen bundle, for the hidden sector. This provides a new class of heterotic flux compactifications via non-Kahler deformation of Calabi-Yau geometries with polystable bundles. As an application, we obtain examples of non-Kahler deformations of some three generation GUT models.
Effective superpotentials for compact D5-brane Calabi-Yau geometries  [PDF]
Hans Jockers,Masoud Soroush
Mathematics , 2008, DOI: 10.1007/s00220-008-0727-7
Abstract: For compact Calabi-Yau geometries with D5-branes we study N=1 effective superpotentials depending on both open- and closed-string fields. We develop methods to derive the open/closed Picard-Fuchs differential equations, which control D5-brane deformations as well as complex structure deformations of the compact Calabi-Yau space. Their solutions encode the flat open/closed coordinates and the effective superpotential. For two explicit examples of compact D5-brane Calabi-Yau hypersurface geometries we apply our techniques and express the calculated superpotentials in terms of flat open/closed coordinates. By evaluating these superpotentials at their critical points we reproduce the domain wall tensions that have recently appeared in the literature. Finally we extract orbifold disk invariants from the superpotentials, which, up to overall numerical normalizations, correspond to orbifold disk Gromov-Witten invariants in the mirror geometry.
Nonabelian 2D Gauge Theories for Determinantal Calabi-Yau Varieties  [PDF]
Hans Jockers,Vijay Kumar,Joshua M. Lapan,David R. Morrison,Mauricio Romo
Mathematics , 2012, DOI: 10.1007/JHEP11(2012)166
Abstract: The two-dimensional supersymmetric gauged linear sigma model (GLSM) with abelian gauge groups and matter fields has provided many insights into string theory on Calabi--Yau manifolds of a certain type: complete intersections in toric varieties. In this paper, we consider two GLSM constructions with nonabelian gauge groups and charged matter whose infrared CFTs correspond to string propagation on determinantal Calabi-Yau varieties, furnishing another broad class of Calabi-Yau geometries in addition to complete intersections. We show that these two models -- which we refer to as the PAX and the PAXY model -- are dual descriptions of the same low-energy physics. Using GLSM techniques, we determine the quantum K\"ahler moduli space of these varieties and find no disagreement with existing results in the literature.
Topological Transitions and Enhancon-like Geometries in Calabi-Yau Compactifications of M-Theory  [PDF]
Thomas Mohaupt
Physics , 2002, DOI: 10.1002/prop.200310099
Abstract: We study the impact of topological phase transitions of the internal Calabi-Yau threefold on the space-time geometry of five-dimensional extremal black holes and black strings. For flop transitions and SU(2) gauge symmetry enhancement we show that solutions can always be continued and that the behaviour of metric, gauge fields and scalars can be characterized in a model independent way. Then we look at supersymmetric solutions which describe naked singularities rather than geometries with a horizon. For black strings we show that the solution cannot become singular as long as the scalar fields take values inside the Kahler cone. For black holes we establish the same result for the elliptic fibrations over the Hirzebruch surfaces F_0, F_1, F_2. These three models exhibit a behaviour similar to the enhancon, since one runs into SU(2) enhancement before reaching the apparent singularity. Using the proper continuation inside the enhancon radius one finds that the solution is regular.
A New Construction of Calabi-Yau Manifolds: Generalized CICYs  [PDF]
Lara B. Anderson,Fabio Apruzzi,Xin Gao,James Gray,Seung-Joo Lee
Mathematics , 2015,
Abstract: We present a generalization of the complete intersection in products of projective space (CICY) construction of Calabi-Yau manifolds. CICY three-folds and four-folds have been studied extensively in the physics literature. Their utility stems from the fact that they can be simply described in terms of a `configuration matrix', a matrix of integers from which many of the details of the geometries can be easily extracted. The generalization we present is to allow negative integers in the configuration matrices which were previously taken to have positive semi-definite entries. This broadening of the complete intersection construction leads to a larger class of Calabi-Yau manifolds than that considered in previous work, which nevertheless enjoys much of the same degree of calculational control. These new Calabi-Yau manifolds are complete intersections in (not necessarily Fano) ambient spaces with an effective anticanonical class. We find examples with topology distinct from any that has appeared in the literature to date. The new manifolds thus obtained have many interesting features. For example, they can have smaller Hodge numbers than ordinary CICYs and lead to many examples with elliptic and K3-fibration structures relevant to F-theory and string dualities.
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