arXiv subjects

Brent D. Nelson

Publications and source records attributed to Brent D. Nelson.

At least 19 recordsLinked to original sources

arXiv2025

Vacuum Geometry of the Standard Model

Vacuum structure of a quantum field theory is a crucial property. In theories with extended symmetries, such as supersymmetric gauge theories, the vacuum is typically a continuous manifold, called the vacuum moduli space, parametrized by the expectation values of scalar fields. Starting from the R-parity preserving superpotential at renormalizable order, we use Gröbner bases to determine the explicit structure, as an algebraic variety, of the vacuum geometry of the minimal supersymmetric extension of the Standard Model. Gröbner bases have doubly exponential computational complexity (for this case, $7^{2^{1023}}$ operations); we exploit symmetry and multigrading to render the computation tractable. This geometry has three irreducible components of complex dimensions $1$, $15$, and $29$, each being a so-called rational variety. The defining equations of the components express the solutions to F-terms and D-terms in terms of the gauge invariant operators and are interpreted in terms of classical geometric constructions.

hep-th↗

arXiv2025

The Vacuum Moduli Space of the Minimal Supersymmetric Standard Model

A starting point in the study of the minimal supersymmetric Standard Model (MSSM) is the vacuum moduli space, which is a highly complicated algebraic variety: it is the image of an affine variety $X \subset \mathbb{C}^{49}$ under a symplectic quotient map $ϕ$ to $\mathbb{C}^{973}$. Previous work computed the vacuum moduli space of the electroweak sector; geometrically this corresponds to studying a restriction of $ϕ$: $\mathbb{C}^{13} \stackrel{ϕ^{\texttt{res}}}{\longrightarrow} \mathbb{C}^{22}$. We analyze the geometry of the full vacuum moduli space for superpotentials $W_{\rm minimal}$ (without neutrinos) and $W_{\rm MSSM}$ (with neutrinos) in $\mathbb{C}^{973}$. In both cases, we prove that $X$ consists of three irreducible components $X_1$, $X_2$, and $X_3$, and determine the images $M_i$ of the $X_i$ under $ϕ$. For $W_{\rm minimal}$ we show they have, respectively, dimensions $1$, $15$, and $29$, and prove that each of the $M_i$ is a rational variety, while for $W_{\rm MSSM}$ we show that $M_3$ is the only component. Restricting the $M_i$ to the electroweak sector, we recover known results. We describe the components of the vacuum moduli space geometrically in terms of incidence varieties to a product of Segre varieties.

hep-th↗

arXiv2018

Estimating Calabi-Yau Hypersurface and Triangulation Counts with Equation Learners

We provide the first estimate of the number of fine, regular, star triangulations of the four-dimensional reflexive polytopes, as classified by Kreuzer and Skarke (KS). This provides an upper bound on the number of Calabi-Yau threefold hypersurfaces in toric varieties. The estimate is performed with deep learning, specifically the novel equation learner (EQL) architecture. We demonstrate that EQL networks accurately predict numbers of triangulations far beyond the $h^{1,1}$ training region, allowing for reliable extrapolation. We estimate that number of triangulations in the KS dataset is $10^{10,505}$, dominated by the polytope with the highest $h^{1,1}$ value.

hep-th↗

arXiv2018

Vacuum Selection from Cosmology on Networks of String Geometries

We introduce network science as a framework for studying the string landscape. Two large networks of string geometries are constructed, where nodes are extra-dimensional six-manifolds and edges represent topological transitions between them. We show that a standard bubble cosmology model on the networks has late-time behavior determined by the largest eigenvector of $-(\mathbf{L} + \mathbf{D})$, where $\mathbf{L}$ and $\mathbf{D}$ are the Laplacian and degree matrices of the networks, which provides a dynamical mechanism for vacuum selection in the string landscape.

hep-th↗

arXiv2018

Dark Glueballs and their Ultralight Axions

Dark gauge sectors and axions are well-motivated in string theory. We demonstrate that if a confining gauge sector gives rise to dark glueballs that are a fraction of the dark matter, and the associated axion has a decay constant near the string scale, then this axion is ultralight and naturally realizes the fuzzy dark matter scenario with a modest tuning of a temperature ratio. Astrophysical observations constrain the size of the glueball component relative to the axionic component, while electric dipole moments constrain mixing with the QCD axion.

hep-ph↗

arXiv2018

hep-th

We apply techniques in natural language processing, computational linguistics, and machine-learning to investigate papers in hep-th and four related sections of the arXiv: hep-ph, hep-lat, gr-qc, and math-ph. All of the titles of papers in each of these sections, from the inception of the arXiv until the end of 2017, are extracted and treated as a corpus which we use to train the neural network Word2Vec. A comparative study of common n-grams, linear syntactical identities, word cloud and word similarities is carried out. We find notable scientific and sociological differences between the fields. In conjunction with support vector machines, we also show that the syntactic structure of the titles in different sub-fields of high energy and mathematical physics are sufficiently different that a neural network can perform a binary classification of formal versus phenomenological sections with 87.1% accuracy, and can perform a finer five-fold classification across all sections with 65.1% accuracy.

cs.CL↗

arXiv2017

Categorisation and Detection of Dark Matter Candidates from String/M-theory Hidden Sectors

We study well-motivated dark matter candidates arising from weakly-coupled hidden sectors in compactified string/$M$-theory. Imposing generic top-down constraints greatly restricts allowed candidates. By considering the possible mechanisms for achieving the correct dark matter relic density, we compile categories of viable dark matter candidates and annihilation mediators. We consider the case where supersymmetry breaking occurs via moduli stabilisation and is gravitationally mediated to the visible and other hidden sectors, without assuming sequestering of the sector in which supersymmetry is broken. We find that in this case, weakly-coupled hidden sectors only allow for fermionic dark matter. Additionally, most of the mechanisms for obtaining the full relic density only allow for a gauge boson mediator, such as a dark $Z'$. Given these considerations, we study the potential for discovering or constraining the allowed parameter space given current and future direct detection experiments, and direct production at the LHC. We also present a model of a hidden sector which would contain a satisfactory dark matter candidate.

hep-ph↗

arXiv2017

Machine Learning in the String Landscape

We utilize machine learning to study the string landscape. Deep data dives and conjecture generation are proposed as useful frameworks for utilizing machine learning in the landscape, and examples of each are presented. A decision tree accurately predicts the number of weak Fano toric threefolds arising from reflexive polytopes, each of which determines a smooth F-theory compactification, and linear regression generates a previously proven conjecture for the gauge group rank in an ensemble of $\frac43 \times 2.96 \times 10^{755}$ F-theory compactifications. Logistic regression generates a new conjecture for when $E_6$ arises in the large ensemble of F-theory compactifications, which is then rigorously proven. This result may be relevant for the appearance of visible sectors in the ensemble. Through conjecture generation, machine learning is useful not only for numerics, but also for rigorous results.

hep-th↗

arXiv2017

New Large Volume Solutions

In previous work, we have commenced the task of unpacking the $473,800,776$ reflexive polyhedra by Kreuzer and Skarke into a database of Calabi-Yau threefolds (see http://www.rossealtman.com). In this paper, following a pedagogical introduction, we present a new algorithm to isolate Swiss cheese solutions characterized by "holes," or small 4-cycles, descending from the toric divisors inherent to the original four dimensional reflexive polyhedra. Implementing these methods, we find $2,268$ explicit Swiss cheese manifolds, over half of which have $h^{1,1}=6$. Many of our solutions have multiple large cycles. Such Swiss cheese geometries facilitate moduli stabilization in string compactifications and provide flat directions for cosmological inflation.

hep-th↗

arXiv2016

String Theory and the Dark Glueball Problem

We study cosmological constraints on dark pure Yang-Mills sectors. Dark glueballs are overproduced for large regions of ultraviolet parameter space. The problem may be alleviated in two ways: via a large preferential reheating into the visible sector, motivating certain inflation or modulus decay models, or via decays into axions or moduli, which are strongly constrained by nucleosynthesis and $ΔN_{\text{eff}}$ bounds. String models frequently have multiple hidden Yang-Mills sectors, which are subject to even stronger constraints due to the existence of multiple dark glueballs.

hep-ph↗

arXiv2016

The lightest visible-sector supersymmetric particle is likely to be unstable

We argue, based on typical properties of known solutions of string/$M$-theory, that the lightest supersymmetric particle of the visible sector will not be stable. In other words, dark matter is {\em not} a particle with Standard Model quantum numbers, such as a WIMP. The argument is simple and based on the typical occurrence of a) hidden sectors, b) interactions between the Standard Model (visible) sector and these hidden sectors, and c) the lack of an argument against massive neutral hidden sector particles being lighter than the lightest visible supersymmetric particle. These conclusions do not rely on arguments such as R-parity violation.

hep-ph↗

arXiv2015

Testing R-parity with Geometry

We present a complete classification of the vacuum geometries of all renormalizable superpotentials built from the fields of the electroweak sector of the MSSM. In addition to the Severi and affine Calabi-Yau varieties previously found, new vacuum manifolds are identified; we thereby investigate the geometrical implication of theories which display a manifest matter parity (or R-parity) via the distinction between leptonic and Higgs doublets, and of the lepton number assignment of the right-handed neutrino fields. We find that the traditional R-parity assignments of the MSSM more readily accommodate the neutrino see-saw mechanism with non-trivial geometry than those superpotentials that violate R-parity. However there appears to be no geometrical preference for a fundamental Higgs bilinear in the superpotential, with operators that violate lepton number, such as νH\bar{H}, generating vacuum moduli spaces equivalent to those with a fundamental bilinear.

hep-th↗

arXiv2015

Mirage Models Confront the LHC: III. Deflected Mirage Mediation

We complete the study of a class of string-motivated effective supergravity theories in which modulus-induced soft supersymmetry breaking is sufficiently suppressed in the observable sector so as to be competitive with anomaly-mediated supersymmetry breaking. Here we consider deflected mirage mediation (DMM), where contributions from gauge mediation are added to those arising from gravity mediation and anomaly mediation. We update previous work that surveyed the rich parameter space of such theories, in light of data from the CERN Large Hadron Collider (LHC) and recent dark matter detection experiments. Constraints arising from LHC superpartner searches at $\sqrt{s} = 8\,{\rm TeV}$ are considered, and discovery prospects at $\sqrt{s} = 14\,{\rm TeV}$ are evaluated. We find that deflected mirage mediation generally allows for SU(3)-charged superpartners of significantly lower mass (given current knowledge of the Higgs mass and neutralino relic density) than was found for the `pure' mirage mediation models of Kachru et al. Consequently, discovery prospects are enhanced for many combinations of matter multiplet modular weights. We examine the experimental challenges that will arise due to the prospect of highly compressed spectra in DMM, and the correlation between accessibility at the LHC and discovery prospects at large-scale liquid xenon dark matter detectors.

hep-ph↗

arXiv2015

Dark Matter Production Mechanisms with a Non-Thermal Cosmological History - A Classification

We perform a comprehensive study of models of dark matter (DM) in a Universe with a non-thermal cosmological history, i.e with a phase of pressure-less matter domination before the onset of big-bang nucleosynethesis (BBN). Such cosmological histories are generically predicted by UV completions that contain gravitationally coupled scalar fields (moduli). We classify the different production mechanisms for DM in this framework, generalizing previous works by considering a wide range of DM masses/couplings and allowing for DM to be in equilibrium with a "dark" sector. We identify four distinct parametric regimes for the production of relic DM, and derive accurate semi-analytic approximations for the DM relic abundance. Our results are particularly relevant for supersymmetric theories, in which the standard non-thermally produced DM candidates are disfavored by indirect detection constraints. We also comment on experimental signals in this framework, focusing on novel effects involving the power spectrum of DM density perturbations. In particular, we identify a class of models where the spectrum of DM density perturbations is sensitive to the pressure-less matter dominated era before BBN, giving rise to interesting astrophysical signatures to be looked for in the future. A worthwhile future direction would be to study well-motivated theoretical models within this framework and carry out detailed studies of the pattern of expected experimental signals.

hep-ph↗

arXiv2014

A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List

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, in a companion online database (see http://nuweb1.neu.edu/cydatabase), a detailed inventory of these quantities which are of interest to physicists. Many of the singular ambient spaces described by the Kreuzer-Skarke list can be smoothed out into multiple distinct toric ambient spaces describing different Calabi-Yau threefolds. We provide a list of the different Calabi-Yau threefolds which can be obtained from each polytope, up to current computational limits. We then give the details of a variety of quantities associated to each of these Calabi-Yau such as Chern classes, intersection numbers, and the Kähler and Mori cones, in addition to the Hodge data. This data forms a useful starting point for a number of physical applications of the Kreuzer-Skarke list.

hep-th↗

arXiv2014

The Geometry of Generations

We present an intriguing and precise interplay between algebraic geometry and the phenomenology of generations of particles. Using the electroweak sector of the MSSM as a testing ground, we compute the moduli space of vacua as an algebraic variety for multiple generations of Standard Model matter and Higgs doublets. The space is shown to have Calabi-Yau, Grassmannian, and toric signatures which sensitively depend on the number of generations of leptons, as well as inclusion of Majorana mass terms for right-handed neutrinos. We speculate as to why three generations is special.

hep-th↗

arXiv2014

Veronese Geometry and the Electroweak Vacuum Moduli Space

We explain the origin of the Veronese surface in the vacuum moduli space geometry of the MSSM electroweak sector. While this result appeared many years ago using techniques of computational algebraic geometry, it has never been demonstrated analytically. Here, we present an analytical derivation of the vacuum geometry of the electroweak theory by understanding how the F- and D-term relations lead to the Veronese surface. We moreover give a detailed description of this geometry, realising an extra branch as a zero-dimensional point when quadratic Higgs lifting deformations are incorporated into the superpotential.

hep-th↗

arXiv2013

Mirage Models Confront the LHC: II. Flux-Stabilized Type IIB String Theory

We continue the study of a class of string-motivated effective supergravity theories in light of current data from the CERN Large Hadron Collider (LHC). In this installment we consider Type IIB string theory compactified on a Calabi-Yau orientifold in the presence of fluxes, in the manner originally formulated by Kachru, et al. We allow for a variety of potential uplift mechanisms and embeddings of the Standard Model field content into D3 and D7 brane configurations. We find that an uplift sector independent of the Kahler moduli, as is the case with anti-D3 branes, is inconsistent with data unless the matter and Higgs sectors are localized on D7 branes exclusively, or are confined to twisted sectors between D3 and D7 branes. We identify regions of parameter space for all possible D-brane configurations that remain consistent with PLANCK observations on the dark matter relic density and measurements of the CP-even Higgs mass at the LHC. Constraints arising from LHC searches at 8 TeV center-of-mass energies, and the LUX dark matter detection experiment, are discussed. The discovery prospects for the remaining parameter space at dark matter direct detection experiments are described, and signatures for detection of superpartners at the LHC, with center-of-mass energy of 14 TeV, are analyzed.

hep-ph↗