arXiv subjects

Lara B. Anderson

Publications and source records attributed to Lara B. Anderson.

At least 19 recordsLinked to original sources

arXiv2026

CIPro Package: Complete Intersections in Products of Projective Spaces and Line Bundles

CIPro is a Mathematica package for constructing and analyzing complete intersections in products of projective spaces and line bundles over such varieties. It computes properties of complete intersections, such as Chern classes, Hilbert series, GV invariants and symmetries, as well as properties of line bundles on complete intersections, including their cohomology groups. The package also consolidates a number of data sets available in the literature into a single system, including the lists of complete intersection Calabi-Yau three- and four-folds. This short tutorial introduces the package, provides a brief discussion of some of the mathematical background underlying its computations, and gives a series of examples to illustrate its use. These tools are of utility for many computations in string compactifications, especially for Calabi-Yau geometries appearing in Heterotic and Type II constructions. Many tools apply beyond the Calabi-Yau context, including for example, almost Fano bases in F-theory.

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arXiv2025

Mapping moduli across heterotic conifolds

In this work, we provide evidence for a duality between 4-dimensional Calabi-Yau compactifications of the heterotic string, in which the base manifolds are linked by a conifold transition. In recent work, a geometric proposal was put forward for how 5-branes and gauge bundles are carried across such transitions. It was observed that compactifications connected in this way lead to 4-dimensional effective theories with the same massless spectrum. Here we provide much stronger evidence that these heterotic conifold transitions do indeed lead to dual theories. We construct a duality map between the field spaces of the two compactifications and use it to demonstrate the agreement of large numbers of holomorphic functions appearing in the definition of the effective theories. In an example, we show that 147,440 independent superpotential Yukawa couplings agree across the duality as holomorphic functions of the moduli. In certain special cases, the putative duality studied here reduces to the target space duality of (0,2) gauged linear sigma models.

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arXiv2023

Lectures on Numerical and Machine Learning Methods for Approximating Ricci-flat Calabi-Yau Metrics

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 known. In this lecture series we review numerical approximation methods for Ricci flat CY metrics. Our first aim is to give a brief overview of the mathematical framework underlying CY geometry, and the various metrics that CY manifolds admit. We will then discuss the three types of numerical methods that have been developed to compute Ricci-flat CY metrics: Donaldson's algorithm, functional minimization methods, and machine learning methods. Due to the limited time/space we have, this will not be a comprehensive review, but instead we hope to give a brief survey and illustrate the essential tools, key ideas, and implementations of this rapidly advancing field.

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arXiv2023

Calabi-Yau Genus-One Fibrations and Twisted Dimensional Reductions of F-theory

In this brief note we explore the space of genus one and elliptic fibrations within CY manifolds, their organizing principles, and how they relate to the set of all CY manifolds. We provide examples of genus one fibered manifolds that exhibit different Hodge numbers -- and physically lead to different gauge groups - than their Jacobian fibrations. We suggest a physical mechanism for understanding this difference in twisted circle reductions of 6-dimensional compactifications of F-theory.

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arXiv2023

Twisted Fibrations in M/F-theory

In this work we investigate 5-dimensional theories obtained from M-theory on genus one fibered threefolds which exhibit twisted algebras in their fibers. We provide a base-independent algebraic description of the threefolds and compute light 5D BPS states charged under finite sub-algebras of the twisted algebras. We further construct the Jacobian fibrations that are associated to 6-dimensional F-theory lifts, where the twisted algebra is absent. These 6/5-dimensional theories are compared via twisted circle reductions of F-theory to M-theory. In the 5-dimensional theories we discuss several geometric transitions that connect twisted with untwisted fibrations. We present detailed discussions of $\mathfrak{e}_6^{(2)}, \mathfrak{so}_8^{(3)}$ and $\mathfrak{su}_3^{(2)}$ twisted fibers and provide several explicit example threefolds via toric constructions. Finally, limits are considered in which gravity is decoupled, including Little String Theories for which we match 2-group symmetries across twisted T-dual theories.

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arXiv2022

Branes and Bundles through Conifold Transitions and Dualities in Heterotic String Theory

Geometric transitions between Calabi-Yau manifolds have proven to be a powerful tool in exploring the intricate and interconnected vacuum structure of string compactifications. However, their role in N=1, 4-dimensional string compactifications remains relatively unexplored. In this work we present a novel proposal for transitioning the background geometry (including NS5-branes and holomorphic, slope-stable vector bundles) of 4-dimensional, N=1 heterotic string compactifications through a conifold transition connecting Calabi-Yau threefolds. Our proposal is geometric in nature but informed by the heterotic effective theory. Central to this study is a description of how the cotangent bundles of the deformation and resolution manifolds in the conifold can be connected by an apparent small instanton transition with a 5-brane wrapping the small resolution curves. We show that by a "pair creation" process 5-branes can be generated simultaneously in the gauge and gravitational sectors and used to describe a coupled minimal change in the manifold and gauge sector. This observation leads us to propose dualities for 5-branes and gauge bundles in heterotic conifolds which we then confirm at the level of spectrum in large classes of examples. While the 5-brane duality is novel, we observe that the bundle correspondence has appeared before in the Target Space Duality exhibited by (0,2) GLSMs. Thus our work provides a geometric explanation of (0,2) Target Space Duality.

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arXiv2022

Vanishing Yukawa Couplings and the Geometry of String Theory Models

We provide an overview of recent work which aims to understand patterns of vanishing Yukawa couplings that arise in models of particle physics derived from string theory. These patterns are seemingly linked to a plethora of different geometrical structures and our understanding of the subject has yet to be consolidated in a unified framework. This short note is based upon a talk that was given by one of the authors at the Nankai Symposium on Mathematical Dialogues. Therefore it is aimed at a mathematical audience of mixed academic background.

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arXiv2021

$\mathbb{P}^1$-fibrations in F-theory and String Dualities

In this work we study F-theory compactifications on elliptically fibered Calabi-Yau n-folds which have $\mathbb{P}^1$-fibered base manifolds. Such geometries, which we study in both 4- and 6-dimensions, are both ubiquitous within the set of Calabi-Yau manifolds and play a crucial role in heterotic/F-theory duality. We discuss the most general formulation of $\mathbb{P}^1$-bundles of this type, as well as fibrations which degenerate at higher codimension loci. In the course of this study, we find a number of new phenomena. For example, in both 4- and 6-dimensions we find transitions whereby the base of a $\mathbb{P}^1$-bundle can change nature, or "jump", at certain loci in complex structure moduli space. We discuss the implications of this jumping for the associated heterotic duals. We argue that $\mathbb{P}^1$-bundles with only rational sections lead to heterotic duals where the Calabi-Yau manifold is elliptically fibered over the section of the $\mathbb{P}^1$- bundle, and not its base. As expected, we see that degenerations of the $\mathbb{P}^1$-fibration of the F-theory base correspond to 5-branes in the dual heterotic physics, with the exception of cases in which the fiber degenerations exhibit monodromy. Along the way, we discuss a set of useful formulae and tools for describing F-theory compactifications on this class of Calabi-Yau manifolds.

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arXiv2021

Moduli-dependent Calabi-Yau and SU(3)-structure metrics from Machine Learning

We use machine learning to approximate Calabi-Yau and SU(3)-structure metrics, including for the first time complex structure moduli dependence. Our new methods furthermore improve existing numerical approximations in terms of accuracy and speed. Knowing these metrics has numerous applications, ranging from computations of crucial aspects of the effective field theory of string compactifications such as the canonical normalizations for Yukawa couplings, and the massive string spectrum which plays a crucial role in swampland conjectures, to mirror symmetry and the SYZ conjecture. In the case of SU(3) structure, our machine learning approach allows us to engineer metrics with certain torsion properties. Our methods are demonstrated for Calabi-Yau and SU(3)-structure manifolds based on a one-parameter family of quintic hypersurfaces in $\mathbb{P}^4.$

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arXiv2021

Generalized Vanishing Theorems for Yukawa Couplings in Heterotic Compactifications

Heterotic compactifications on Calabi-Yau threefolds frequently exhibit textures of vanishing Yukawa couplings in their low energy description. The vanishing of these couplings is often not enforced by any obvious symmetry and appears to be topological in nature. Recent results in the literature used differential geometric methods to explain the origin of some of this structure. A vanishing theorem was given which showed that the effect could be attributed, in part, to the embedding of the Calabi-Yau manifolds of interest inside higher dimensional ambient spaces, if the gauge bundles involved descended from vector bundles on those larger manifolds. In this paper, we utilize an algebro-geometric approach to provide an alternative derivation of some of these results, and are thus able to generalize them to a much wider arena than has been considered before. For example, we consider cases where the vector bundles of interest do not descend from bundles on the ambient space. In such a manner we are able to highlight the ubiquity with which textures of vanishing Yukawa couplings can be expected to arise in heterotic compactifications, with multiple different constraints arising from a plethora of different geometric features associated to the gauge bundle.

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arXiv2020

Chern-Simons Invariants and Heterotic Superpotentials

The superpotential in four-dimensional heterotic effective theories contains terms arising from holomorphic Chern-Simons invariants associated to the gauge and tangent bundles of the compactification geometry. These effects are crucial for a number of key features of the theory, including vacuum stability and moduli stabilization. Despite their importance, few tools exist in the literature to compute such effects in a given heterotic vacuum. In this work we present new techniques to explicitly determine holomorphic Chern-Simons invariants in heterotic string compactifications. The key technical ingredient in our computations are real bundle morphisms between the gauge and tangent bundles. We find that there are large classes of examples, beyond the standard embedding, where the Chern-Simons superpotential vanishes. We also provide explicit examples for non-flat bundles where it is non-vanishing and fractionally quantized, generalizing previous results for Wilson lines.

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arXiv2019

Extending the Geometry of Heterotic Spectral Cover Constructions

In this work we extend the well-known spectral cover construction first developed by Friedman, Morgan, and Witten to describe more general vector bundles on elliptically fibered Calabi-Yau geometries. In particular, we consider the case in which the Calabi-Yau fibration is not in Weierstrass form, but can rather contain fibral divisors or multiple sections (i.e. a higher rank Mordell-Weil group). In these cases, general vector bundles defined over such Calabi-Yau manifolds cannot be described by ordinary spectral data. To accomplish this we employ well established tools from the mathematics literature of Fourier-Mukai functors. We also generalize existing tools for explicitly computing Fourier-Mukai transforms of stable bundles on elliptic Calabi-Yau manifolds. As an example of these new tools we produce novel examples of chirality changing small instanton transitions. The goal of this work is to provide a geometric formalism that can substantially increase the understood regimes of heterotic/F-theory duality.

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arXiv2019

Heterotic/Heterotic and Heterotic/F-theory Duality

We consider heterotic target space dual (0,2) GLSMs on elliptically fibered Calabi-Yau manifolds. In this context, each half of the "dual" heterotic theories must in turn have an F-theory dual. Moreover, the apparent relationship between two heterotic compactifications seen in (0,2) heterotic target space dual pairs should, in principle, induce some putative correspondence between the dual F-theory geometries. It has previously been conjectured in the literature that (0,2) target space duality might manifest in F-theory as multiple K3-fibrations of the same elliptically fibered Calabi-Yau manifold. In this work we investigate this conjecture in the context of both 6-dimensional and 4-dimensional effective theories and demonstrate that in general, (0,2) target space duality cannot be explained by such a simple phenomenon alone. In all cases, we provide evidence that non-geometric data in F-theory must play at least some role in the induced F-theory correspondence, while leaving the full determination of the putative new F-theory duality to future work.

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arXiv2019

F-Theory on Quotients of Elliptic Calabi-Yau Threefolds

In this work we consider quotients of elliptically fibered Calabi-Yau threefolds by freely acting discrete groups and the associated physics of F-theory compactifications on such backgrounds. The process of quotienting a Calabi-Yau geometry produces not only new genus one fibered manifolds, but also new effective 6-dimensional physics. These theories can be uniquely characterized by the much simpler covering space geometry and the symmetry action on it. We use this method to construct examples of F-theory models with an array of discrete gauge groups and non-trivial monodromies, including an example with Z6 discrete symmetry.

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arXiv2018

F-theory on Quotient Threefolds with (2,0) Discrete Superconformal Matter

We explore 6-dimensional compactifications of F-theory exhibiting (2,0) superconformal theories coupled to gravity that include discretely charged superconformal matter. Beginning with F-theory geometries with Abelian gauge fields and superconformal sectors, we provide examples of Higgsing transitions which break the $U(1)$ gauge symmetry to a discrete remnant in which the matter fields are also non-trivially coupled to a (2,0) SCFT. In the compactification background this corresponds to a geometric transition linking two fibered Calabi-Yau geometries defined over a singular base complex surface. An elliptically fibered Calabi-Yau threefold with non-zero Mordell-Weil rank can be connected to a smooth non-simply connected genus one fibered geometry constructed as a Calabi-Yau quotient. These hyperconifold transitions exhibit multiple fibers in co-dimension 2 over the base.

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arXiv2018

TASI Lectures on Geometric Tools for String Compactifications

In this work we provide a self-contained and modern introduction to some of the tools, obstacles and open questions arising in string compactifications. Techniques and current progress are illustrated in the context of smooth heterotic string compactifications to 4-dimensions. Progress is described on bounding and enumerating possible string backgrounds and their properties. We provide an overview of constructions, partial classifications, and moduli problems associated to Calabi-Yau manifolds and holomorphic bundles over them.

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arXiv2018

Singular Geometry and Higgs Bundles in String Theory

This brief survey aims to set the stage and summarize some of the ideas under discussion at the Workshop on Singular Geometry and Higgs Bundles in String Theory, to be held at the American Institute of Mathematics from October 30th to November 3rd, 2017. One of the most interesting aspects of the duality revolution in string theory is the understanding that gauge fields and matter representations can be described by intersection of branes. Since gauge theory is at the heart of our description of physical interactions, it has opened the door to the geometric engineering of many physical systems, and in particular those involving Higgs bundles. This note presents a curated overview of some current advances and open problems in the area, with no intention of being a complete review of the whole subject.

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arXiv2017

T-Branes at the Limits of Geometry

Singular limits of 6D F-theory compactifications are often captured by T-branes, namely a non-abelian configuration of intersecting 7-branes with a nilpotent matrix of normal deformations. The long distance approximation of such 7-branes is a Hitchin-like system in which simple and irregular poles emerge at marked points of the geometry. When multiple matter fields localize at the same point in the geometry, the associated Higgs field can exhibit irregular behavior, namely poles of order greater than one. This provides a geometric mechanism to engineer wild Higgs bundles. Physical constraints such as anomaly cancellation and consistent coupling to gravity also limit the order of such poles. Using this geometric formulation, we unify seemingly different wild Hitchin systems in a single framework in which orders of poles become adjustable parameters dictated by tuning gauge singlet moduli of the F-theory model.

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