Numerical Algebraic Geometry: A New Perspective on String and Gauge Theories

Abstract

The interplay rich between algebraic geometry and string and gauge theories has recently been immensely aided by advances in computational algebra. However, these symbolic (Gröbner) methods are severely limited by algorithmic issues such as exponential space complexity and being highly sequential. In this paper, we introduce a novel paradigm of numerical algebraic geometry which in a plethora of situations overcomes these short-comings. Its so-called 'embarrassing parallelizability' allows us to solve many problems and extract physical information which elude the symbolic methods. We describe the method and then use it to solve various problems arising from physics which could not be otherwise solved.

36 pages

References in corpus (19)

• The Master Space of N=1 Gauge Theories
• De Sitter hunting in a classical landscape
• SQCD: A Geometric Apercu
• Heterotic Compactification, An Algorithmic Approach
• On Slow-roll Moduli Inflation in Massive IIA Supergravity with Metric Fluxes
• On the Cosmology of Type IIA Compactifications on SU(3)-structure Manifolds
• Finding All the Stationary Points of a Potential Energy Landscape via Numerical Polynomial Homotopy Continuation Method
• STRINGVACUA: A Mathematica Package for Studying Vacuum Configurations in String Phenomenology
• Phase transitions detached from stationary points of the energy landscape
• Exploring the Vacuum Geometry of N=1 Gauge Theories
• Mastering the Master Space
• Involutive Algorithms for Computing Groebner Bases
• Energy landscape analysis of the two-dimensional nearest-neighbor ϕ^4 model
• Enumerating Gribov copies on the lattice
• M-theory on seven-dimensional manifolds with SU(3) structure
• Numerical Polynomial Homotopy Continuation Method and String Vacua
• Stationary point analysis of the one-dimensional lattice Landau gauge fixing functional, aka random phase XY Hamiltonian
• Lectures on D-branes, Gauge Theories and Calabi-Yau Singularities
• alphaCertified: certifying solutions to polynomial systems

Cited by in corpus (38)

• Vevacious: A Tool For Finding The Global Minima Of One-Loop Effective Potentials With Many Scalars
• Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods
• Building an explicit de Sitter
• On the completeness of solutions of Bethe's equations
• A note on obstinate tachyons in classical dS solutions
• Algebraic Geometrization of the Kuramoto Model: Equilibria and Stability Analysis
• Tumbling through a landscape: Evidence of instabilities in high-dimensional moduli spaces
• Finding all flux vacua in an explicit example
• Enumerating Gribov copies on the lattice
• The Calabi-Yau Landscape: from Geometry, to Physics, to Machine-Learning
• The classification of two-loop integrand basis in pure four-dimension
• Coulomb, Landau and Maximally Abelian Gauge Fixing in Lattice QCD with Multi-GPUs
• Energy Landscape of the Finite-Size Mean-field 3-Spin Spherical Model
• Lecture Notes on Multi-loop Integral Reduction and Applied Algebraic Geometry
• Exploring the energy landscape of XY models
• Stability of R parity in supersymmetric models extended by U(1)_{B-L}
• Hilbert Series, Machine Learning, and Applications to Physics
• A Collection of Challenging Optimization Problems in Science, Engineering and Economics
• Communication: Newton homotopies for sampling stationary points of potential energy landscapes
• Numerical Analyses on Moduli Space of Vacua
• Certifying the Potential Energy Landscape
• Global Structure of Curves from Generalized Unitarity Cut of Three-loop Diagrams
• Enumerating Copies in the First Gribov Region on the Lattice in up to four Dimensions
• Calabi-Yau Varieties: from Quiver Representations to Dessins d'Enfants
• Solving Parameterized Polynomial Systems with Decomposable Projections
• Potential Energy Landscapes for the 2D XY Model: Minima, Transition States and Pathways
• Potential Energy Landscape of the Two-Dimensional XY Model: Higher-Index Stationary Points
• Veronese Geometry and the Electroweak Vacuum Moduli Space
• An Inversion-Relaxation Approach for Sampling Stationary Points of Spin Model Hamiltonians
• Certification and the Potential Energy Landscape
• Statistics of Stationary Points of Random Finite Polynomial Potentials
• Testing R-parity with Geometry
• Gauge-fixing on the Lattice via Orbifolding
• Experiments on the zeros of harmonic polynomials using certified counting
• Certified counting of roots of random univariate polynomials
• The Statistics of Vacuum Geometry
• Parallel degree computation for solution space of binomial systems with an application to the master space of $\mathcal{N}=1$ gauge theories
• Exploring the Potential Energy Landscape Over a Large Parameter-Space