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Results for “differential and algebraic geometry”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

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At least 55 records · Page 3
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DOE OSTI2022

Landau discriminants

Scattering amplitudes in quantum field theories have intricate analytic properties as functions of the energies and momenta of the scattered particles. In perturbation theory, their singularities are governed by a set of nonlinear polynomial equations, known as Landau equations, for each individual Feynman diagram. The singularity locus of the associated Feynman integral is made precise with the notion of the Landau discriminant, which characterizes when the Landau equations admit a solution. In order to compute this discriminant, we present approaches from classical elimination theory, as well as a numerical algorithm based on homotopy continuation. These methods allow us to compute Landau discriminants of various Feynman diagrams up to 3 loops, which were previously out of reach. For instance, the Landau discriminant of the envelope diagram is a reducible surface of degree 45 in the three-dimensional space of kinematic invariants. We investigate geometric properties of the Landau discriminant, such as irreducibility, dimension and degree. In particular, we find simple examples in which the Landau discriminant has codimension greater than one. Furthermore, we describe a numerical procedure for determining which parts of the Landau discriminant lie in the physical regions. In order to study degenerate limits of Landau equations and bounds on the degree of the Landau discriminant, we introduce Landau polytopes and study their facet structure. Finally, we provide an efficient numerical algorithm for the computation of the number of master integrals based on the connection to algebraic statistics.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2023

Decorated TQFTs and their Hilbert spaces

We discuss topological quantum field theories that compute topological invariants which depend on additional structures (or decorations) on three-manifolds. The q-series invariant $\hat{Z}$(q) proposed by Gukov, Pei, Putrov, and Vafa is an example of such an invariant. We describe how to obtain these decorated invariants by cutting and gluing and make a proposal for Hilbert spaces that are assigned to two-dimensional surfaces in the $\hat{Z}$-TQFT.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2021

Twistor Actions for Integrable Systems

Many integrable systems can be reformulated as holomorphic vector bundles on twistor space. This is a powerful organizing principle in the theory of integrable systems. One shortcoming is that it is formulated at the level of the equations of motion. From this perspective, it is mysterious that integrable systems have Lagrangians. In this paper, we study a Chern-Simons action on twistor space and use it to derive the Lagrangians of some integrable sigma models. Our focus is on examples that come from dimensionally reduced gravity and supergravity. The dimensional reduction of general relativity to two spacetime dimensions is an integrable coset sigma model coupled to a dilaton and 2d gravity. The dimensional reduction of supergravity to two spacetime dimensions is an integrable coset sigma model coupled to matter fermions, a dilaton, and 2d supergravity. We derive Lax operators and Lagrangians for these 2d integrable systems using the Chern-Simons theory on twistor space. In the supergravity example, we use an extended setup in which twistor Chern-Simons theory is coupled to a pair of matter fermions.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Differential equations for cosmological correlators

Cosmological fluctuations retain a memory of the physics that generated them in their spatial correlations. The strength of correlations varies smoothly as a function of external kinematics, which is encoded in differential equations satisfied by cosmological correlation functions. In this work, we provide a broader perspective on the origin and structure of these differential equations. As a concrete example, we study conformally coupled scalar fields in a power-law cosmology. The wavefunction coefficients in this model have integral representations, with the integrands being the product of the corresponding flat-space results and “twist factors” that depend on the cosmological evolution. Similar twisted integrals arise for loop amplitudes in dimensional regularization, and their recent study has led to the discovery of rich mathematical structures and powerful new tools for computing multi-loop Feynman integrals in quantum field theory. The integrals of interest in cosmology are also part of a finite-dimensional basis of master integrals, which satisfy a system of first-order differential equations. We develop a formalism to derive these differential equations for arbitrary tree graphs. The results can be represented in graphical form by associating the singularities of the differential equations with a set of graph tubings. Upon differentiation, these tubings grow in a local and predictive fashion. In fact, a few remarkably simple rules allow us to predict — by hand — the equations for all tree graphs. While the rules of this “kinematic flow” are defined purely in terms of data on the boundary of the spacetime, they reflect the physics of bulk time evolution. We also study the analogous structures in tr ϕ 3 theory, and see some glimpses of hidden structure in the sum over planar graphs. This suggests that there is an autonomous combinatorial or geometric construction from which cosmological correlations, and the associated spacetime, emerge.

Cosmological models↗
DOE OSTI2026

Cluster bootstrap for cosmological correlators

We show that cosmological wavefunction coefficients associated with n-site chain and loop graphs for a cubic scalar theory in de Sitter spacetime have symbol alphabets given by subsets of A 2n−2 and B 2n−1 cluster variables, respectively, and satisfy the associated cluster adjacency properties. The key step in proving this is identifying a precise connection between graph “tubings” that appear in the kinematic flow equation and polygon “triangulations” that encode the combinatorics of cluster compatibility. Our results imply that cosmological wavefunction coefficients in a general power-law FRW cosmology satisfy cluster adjacency to all orders in the ϵ expansion around the de Sitter limit. We use this information as bootstrap input to show that de Sitter symbols for n ≤ 4 are uniquely determined by simple physical constraints.

differential and algebraic geometry↗
DOE OSTI2020

Bootstrap bounds on closed Einstein manifolds

A compact Riemannian manifold is associated with geometric data given by the eigenvalues of various Laplacian operators on the manifold and the triple overlap integrals of the corresponding eigenmodes. This geometric data must satisfy certain consistency conditions that follow from associativity and the completeness of eigenmodes. We show that it is possible to obtain nontrivial bounds on the geometric data of closed Einstein manifolds by using semidefinite programming to study these consistency conditions, in analogy to the conformal bootstrap bounds on conformal field theories. These bootstrap bounds translate to constraints on the tree-level masses and cubic couplings of Kaluza-Klein modes in theories with compact extra dimensions. We show that in some cases the bounds are saturated by known manifolds.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2020

Symbol alphabets from plabic graphs

Symbol alphabets of n-particle amplitudes in N = 4 super-Yang-Mills theory are known to contain certain cluster variables of Gr(4, n) as well as certain algebraic functions of cluster variables. In this paper we suggest an algorithm for computing these symbol alphabets from plabic graphs by solving matrix equations of the form C ∙ Z = 0 to associate functions on Gr(m, n) to parameterizations of certain cells of Gr(k, n) indexed by plabic graphs. For m = 4 and n = 8 we show that this association precisely reproduces the 18 algebraic symbol letters of the two-loop NMHV eight-particle amplitude from four plabic graphs.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2022

Causal diamonds, cluster polytopes and scattering amplitudes

The “amplituhedron” for tree-level scattering amplitudes in the bi-adjoint φ 3 theory is given by the ABHY associahedron in kinematic space, which has been generalized to give a realization for all finite-type cluster algebra polytopes, labelled by Dynkin diagrams. In this letter we identify a simple physical origin for these polytopes, associated with an interesting (1 + 1)-dimensional causal structure in kinematic space, along with solutions to the wave equation in this kinematic “spacetime” with a natural positivity property. The notion of time evolution in this kinematic spacetime can be abstracted away to a certain “walk”, associated with any acyclic quiver, remarkably yielding a finite cluster polytope for the case of Dynkin quivers. The A n–3 , B n–1 /C n–1 and D n polytopes are the amplituhedra for n-point tree amplitudes, one-loop tadpole diagrams, and full integrand of one-loop amplitudes. We also introduce a polytope D¯ n , which chops the D n polytope in half along a symmetry plane, capturing one-loop amplitudes in a more efficient way.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2022

The stratification of rigidity

We show that a master integrand basis exists for all planar, two-loop amplitudes in massless four-dimensional theories which is fully stratified by rigidity — with each integrand being either pure and strictly polylogarithmic or (pure and) strictly elliptic-polylogarithmic, with each of the later involving a single elliptic curve. Such integrands can be said to have definite rigidity.

1/N Expansion↗
DOE OSTI2024

Graviton topology

Over the past three decades, it has been shown that discrete and continuous media can support topologically nontrivial waves. Recently, it was shown that the same is true of the vacuum, in particular, right (R) and left (L) circularly polarized photons are topologically nontrivial. Here, we study the topology of another class of massless particles, namely gravitons. We show that the collection of all gravitons forms a topologically trivial vector bundle over the lightcone, allowing us to construct a globally smooth basis for gravitons. The graviton bundle also has a natural geometric splitting into two topologically nontrivial subbundles, consisting of the R and L gravitons. The R and L gravitons are unitary irreducible bundle representations of the Poincaré group, and are thus elementary particles; their topology is characterized by the Chern numbers ∓4. This nontrivial topology obstructs the splitting of graviton angular momentum into spin and orbital angular momentum.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Riemann-Hilbert correspondence and blown up surface defects

The relationship of two dimensional quantum field theory and isomonodromic deformations of Fuchsian systems has a long history. Recently four-dimensional N = 2 gauge theories joined the party in a multitude of roles. In this paper we study the vacuum expectation values of intersecting half-BPS surface defects in SU(2) theory with N f = 4 fundamental hypermultiplets. We show they form a horizontal section of a Fuchsian system on a sphere with 5 regular singularities, calculate the monodromy, and define the associated isomonodromic tau-function. Using the blowup formula in the presence of half-BPS surface defects, initiated in the companion paper, we obtain the GIL formula, establishing an unexpected relation of the topological string/free fermion regime of supersymmetric gauge theory to classical integrability.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2025

Renormalizing two-fermion operators in the SMEFT via supergeometry

We extend the geometric framework of field-space covariance for loop computations, thereby unifying the treatment of scalars, fermions, and gauge bosons in effective field theories. This allows us to derive a manifestly covariant formula for one-loop UV divergences that includes contributions from mixed boson-fermion graphs. The result is expressed in terms of geometric invariants of the field-space supermanifold. As a demonstration of this formula, we compute the renormalization group equations for two-fermion operators at the dimension-eight level in the Standard Model Effective Field Theory.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Deformation of Lie–Poisson algebras and chirality

Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian symmetric spectra: If λ is an eigenvalue of the linearized generator, -λ and $\barλ$ (hence, -$\barλ$) are also eigenvalues—the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical Hamiltonian systems) works out differently, resulting in breaking of the Hamiltonian symmetry of spectra; time-reversal asymmetry causes chirality. This interesting phenomenon was first found in analyzing the chiral motion of the rattleback, a boat-shaped top having misaligned axes of inertia and geometry [Z. Yoshida et al., Phys. Lett. A 381, 2772–2777 (2017)]. To elucidate how chiral spectra are generated, we study the three-dimensional Lie–Poisson systems and classify the prototypes of singularities that cause symmetry breaking. The central idea is the deformation of the underlying Lie algebra; invoking Bianchi’s list of all three-dimensional Lie algebras, we show that the so-called class-B algebras, which are produced by asymmetric deformations of the simple algebra s o ( 3 ) , yield chiral spectra when linearized around their singularities. The theory of deformation is generalized to higher dimensions, including the infinite-dimensional Poisson manifolds relevant to fluid mechanics.

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSIC↗
DOE OSTI2021

Integrability, normal forms, and magnetic axis coordinates

Integrable or near-integrable magnetic fields are prominent in the design of plasma confinement devices. Such a field is characterized by the existence of a singular foliation entirely consisting of invariant submanifolds. A compact regular leaf (a flux surface) of this foliation must be diffeomorphic to the two-torus. In a neighborhood of a flux surface, it is known that the magnetic field admits several exact smooth normal forms in which the field lines are straight. However, these normal forms break down near singular leaves, including elliptic and hyperbolic magnetic axes. In this work, the existence of exact smooth normal forms for integrable magnetic fields near elliptic and hyperbolic magnetic axes is established. In the elliptic case, smooth near-axis Hamada and Boozer coordinates are defined and constructed. Ultimately, these results establish previously conjectured smoothness properties for smooth solutions of the magnetohydrodynamic equilibrium equations. The key arguments are a consequence of a geometric reframing of integrability and magnetic fields: they are presymplectic systems.

97 MATHEMATICS AND COMPUTING↗
DOE OSTI2023

Four lectures on Euler integrals

These lecture notes provide a self-contained introduction to Euler integrals, which are frequently encountered in applications. In particle physics, they arise as Feynman integrals or string amplitudes. Our four selected topics demonstrate the diverse mathematical techniques involved in the study of Euler integrals, including polyhedral geometry, very affine varieties, differential equations, and computational algebra.

Matsubara-Heo, Saiei-Jaeyeong↗
DOE OSTI2023

Loop-by-loop differential equations for dual (elliptic) Feynman integrals

We present a loop-by-loop method for computing the differential equations of Feynman integrals using the recently developed dual form formalism. We give explicit prescriptions for the loop-by-loop fibration of multi-loop dual forms. Then, we test our formalism on a simple, but non-trivial, example: the two-loop three-mass elliptic sunrise family of integrals. We obtain an ε-form differential equation within the correct function space in a sequence of relatively simple algebraic steps. In particular, none of these steps relies on the analysis of q-series. Then, we discuss interesting properties satisfied by our dual basis as well as its simple relation to the known ε-form basis of Feynman integrands. The underlying K3-geometry of the three-loop four-mass sunrise integral is also discussed. Finally, we speculate on how to construct a “good” loop-by-loop basis at three-loop.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2020

Generalized Grad–Shafranov equation for non-axisymmetric MHD equilibria

The structure of static MHD equilibria that admit continuous families of Euclidean symmetries is well understood. Such field configurations are governed by the classical Grad–Shafranov equation, which is a single elliptic partial differential equation in two space dimensions. By revealing a hidden symmetry, here we show that in fact all smooth solutions of the equilibrium equations with non-vanishing pressure gradients away from the magnetic axis satisfy a generalization of the Grad–Shafranov equation. In contrast to solutions of the classical Grad–Shafranov equation, solutions of the generalized equation are not automatically equilibria, but instead only satisfy force balance averaged over the one-parameter hidden symmetry. We then explain how the generalized Grad–Shafranov equation can be used to reformulate the problem of finding exact three-dimensional smooth solutions of the equilibrium equations as finding an optimal volume-preserving symmetry.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
DOE OSTI2024

Beam Dynamics of the Muon $g\textrm{-}2$ Experiment

The Muon $g\textrm{-}2$ Experiment (E989) at Fermilab aims to measure the muon anomalous magnetic moment $a_{\mu}$ with unprecedented precision, potentially uncovering physics beyond the Standard Model of particle physics. The result based on Runs 1-3, released in 2023, achieved a precision of 0.20 ppm. The experiment circulates muons in a storage ring, measuring $a_{\mu}$ from decay positron time and energy measurements collected with calorimeters. To achieve the required accuracy, it is crucial to measure and control the magnetic field in the ring with high precision. Beam dynamics corrections are necessary for muons not orbiting exactly in the midplane, for their oscillations, and for electric field effects. Highly accurate beam dynamics simulations are instrumental for quantifying and validating the beam dynamics corrections, ultimately improving the precision of the $a_{\mu}$ measurement and facilitating the achievement of the ambitious $70\:\mathrm{ppb}$ systematic uncertainty goal. The measured field data was incorporated into models for simulations using three codes: \texttt{gm2ringsim} (an internal Geant4-based code), \textit{COSY INFINITY}, and \textit{BMAD}. The advantages of \texttt{gm2ringsim} include using CAD-based geometry and modelling the detector effects. \textit{COSY INFINITY} is a highly accurate and efficient code that uses high-order differential-algebraic transfer maps, precise fringe field calculations, and advanced symplectification methods. Symplectification is important for maintaining the physical correctness of the muon beam behaviour with high precision over the storage time, ensuring conservation of phase space volume and preventing artificial damping or excitation of particle motion. The experiment completed its final Run 6 in July 2023, collecting 21 times more data than the previous BNL experiment. Analyses of data from Runs 4-6 are ongoing, with results planned for release in 2025, potentially resolving the current tension between experiment and theory.

43 PARTICLE ACCELERATORS↗