June 30, 2020
Calabi-Yau spaces, or Kahler spaces admitting zero Ricci curvature, have played a pivotal role in theoretical physics and pure mathematics for the last half-century. In physics, they constituted the first and natural solution to compactification of superstring theory to our 4-dimensional universe, primarily due to one of their equivalent definitions being the admittance of covariantly constant spinors. Since the mid-1980s, physicists and mathematicians have joined forces in creating explicit examples of Calabi-Yau spaces, compiling databases of formidable size, including the complete intersecion (CICY) dataset, the weighted hypersurfaces dataset, the elliptic-fibration dataset, the Kreuzer-Skarke toric hypersurface dataset, generalized CICYs etc., totaling at least on the order of 10^10 manifolds. These all contribute to the vast string landscape, the multitude of possible vacuum solutions to string compactification. More recently, this collaboration has been enriched by computer science and data science, the former, in bench-marking the complexity of the algorithms in computing geometric quantities and the latter, in applying techniques such as machine-learning in extracting unexpected information. These endeavours, inspired by the physics of the string landscape, have rendered the investigation of Calabi-Yau spaces one of the most exciting and inter-disciplinary fields. Invited contribution to the Oxford Research Encyclopedia of Physics, B.~Foster Ed., OUP, 2020
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December 7, 2018
We present a pedagogical introduction to the recent advances in the computational geometry, physical implications, and data science of Calabi-Yau manifolds. Aimed at the beginning research student and using Calabi-Yau spaces as an exciting play-ground, we intend to teach some mathematics to the budding physicist, some physics to the budding mathematician, and some machine-learning to both. Based on various lecture series, colloquia and seminars given by the author in the past...
January 5, 2020
In these lecture notes, we survey the landscape of Calabi-Yau threefolds, and the use of machine learning to explore it. We begin with the compact portion of the landscape, focusing in particular on complete intersection Calabi-Yau varieties (CICYs) and elliptic fibrations. Non-compact Calabi-Yau manifolds are manifest in Type II superstring theories, they arise as representation varieties of quivers, used to describe gauge theories in the bulk familiar four dimensions. Final...
August 1, 2013
With a bird's-eye view, we survey the landscape of Calabi-Yau threefolds, compact and non-compact, smooth and singular. Emphasis will be placed on the algorithms and databases which have been established over the years, and how they have been useful in the interaction between the physics and the mathematics, especially in string and gauge theories. A skein which runs through this review will be algorithmic and computational algebraic geometry and how, implementing its princip...
November 29, 2020
We briefly overview how, historically, string theory led theoretical physics first to precise problems in algebraic and differential geometry, and thence to computational geometry in the last decade or so, and now, in the last few years, to data science. Using the Calabi-Yau landscape -- accumulated by the collaboration of physicists, mathematicians and computer scientists over the last 4 decades -- as a starting-point and concrete playground, we review some recent progress i...
December 28, 2023
Calabi-Yau (CY) manifolds play a ubiquitous role in string theory. As a supersymmetry-preserving choice for the 6 extra compact dimensions of superstring compactifications, these spaces provide an arena in which to explore the rich interplay between physics and geometry. These lectures will focus on compact CY manifolds and the long standing problem of determining their Ricci flat metrics. Despite powerful existence theorems, no analytic expressions for these metrics are know...
March 10, 2015
Yau proved an existence theorem for Ricci-flat K\"ahler metrics in the 1970's, but we still have no closed form expressions for them. Nevertheless there are several ways to get approximate expressions, both numerical and analytical. We survey some of this work and explain how it can be used to obtain physical predictions from superstring theory.
April 28, 2009
While Calabi-Yau hypersurfaces in toric ambient spaces provide a huge number of examples, theoretical considerations as well as applications to string phenomenology often suggest a broader perspective. With even the question of finiteness of diffeomorphism types of CY 3-folds unsettled, an important idea is Reid's conjecture that the moduli spaces are connected by certain singular transitions. We summarize the results of our recent construction of a large class of new CY spac...
August 19, 2012
This is a somewhat personal account of the contributions of Max Kreuzer to the study of Calabi-Yau manifolds and has been prepared as a contribution to the Memorial Volume: Strings, Gauge Fields, and the Geometry Behind - The Legacy of Maximilian Kreuzer, to be published by World Scientific.
November 5, 2014
Kreuzer and Skarke famously produced the largest known database of Calabi-Yau threefolds by providing a complete construction of all 473,800,776 reflexive polyhedra that exist in four dimensions. These polyhedra describe the singular limits of ambient toric varieties in which Calabi-Yau threefolds can exist as hypersurfaces. In this paper, we review how to extract topological and geometric information about Calabi-Yau threefolds using the toric construction, and we provide, i...
February 15, 2022
We review briefly the characteristic topological data of Calabi--Yau threefolds and focus on the question of when two threefolds are equivalent through related topological data. This provides an interesting test case for machine learning methodology in discrete mathematics problems motivated by physics.