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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 19 records
All sourcesNASA NTRSDOE OSTI
DOE OSTI2025

Hidden zeros of the cosmological wavefunction

Motivated by the recent discovery of hidden zeros in particle and string amplitudes, we characterize zeros of individual graph contributions to the cosmological wavefunction of a scalar field theory. We demonstrate that these contributions factorize near these zeros for all tree graphs and provide evidence that this extends to loop graphs as well. We explicitly construct polytopal realizations of the relevant graph associahedra and show that the cosmological zeros have natural geometric and physical interpretations. As a byproduct, we establish an equivalence between the wavefunction coefficients of chain graphs and flat-space Tr(ϕ 3 ) amplitudes, enabling us to leverage the cosmological zeros to uncover the recently discovered hidden zeros of colored amplitudes.

Cosmological models↗
DOE OSTI2024

Dualities among massive, partially massless and shift symmetric fields on (A)dS

We catalog all the electromagnetic-like dualities that exist between free dynamical bosonic fields of arbitrary symmetry type and mass on (anti-) de Sitter space in all dimensions, including dualities among the partially massless and shift symmetric fields. This generalizes to all these field types the well known fact that a massless p-form is dual to a massless (D − p − 2)-form in D spacetime dimensions. In the process, we describe the structure of the Weyl modules (the spaces of local operators linear in the fields and their derivative relations) for all the massive, partially massless and shift symmetric fields.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2024

Proximal Galerkin: A Structure-Preserving Finite Element Method for Pointwise Bound Constraints

The proximal Galerkin finite element method is a high-order, low iteration complexity, nonlinear numerical method that preserves the geometric and algebraic structure of pointwise bound constraints in infinite-dimensional function spaces. This paper introduces the proximal Galerkin method and applies it to solve free boundary problems, enforce discrete maximum principles, and develop a scalable, mesh-independent algorithm for optimal design with pointwise bound constraints. This paper also introduces the latent variable proximal point (LVPP) algorithm, from which the proximal Galerkin method derives. When analyzing the classical obstacle problem, we discover that the underlying variational inequality can be replaced by a sequence of second-order partial differential equations (PDEs) that are readily discretized and solved with, e.g., the proximal Galerkin method. Throughout this work, we arrive at several contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between high-order positivity-preserving discretizations and certain infinite-dimensional Lie groups; and (3) a gradient-based, bound-preserving algorithm for two-field, density-based topology optimization. The complete proximal Galerkin methodology combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis. Open-source implementations of our methods accompany this work to facilitate reproduction and broader adoption.

97 MATHEMATICS AND COMPUTING↗
NASA NTRS1983

On families of systems and deformations

Until recently, there has not been any systematic effort towards a theory of systems with parameter variations. It is presently suggested that the concept of families of systems is basic to such an effort, and entails the techniques of Lie theory, differential geometry, and algebraic geometry. Attention is given to the geometric characterization of certain families of systems that appear in control and identification problems. The ways in which families of systems degenerate as parameter variations become large are isolated, using the topological rather than algebraic geometric case.

Krishnaprasad, P. S.↗
DOE OSTI2024

Hadamard products and BPS networks

We study examples of fourth-order Picard-Fuchs operators that are Hadamard products of two second-order Picard-Fuchs operators. Each second-order Picard-Fuchs operator is associated with a family of elliptic curves, and the Hadamard product computes period integrals on the fibred product of the two elliptic surfaces. We construct 3-cycles on this geometry as the union of 2-cycles in the fibre over contours on the base. We then use the special Lagrangian condition to constrain the contours on the base. This leads to a construction that is reminiscent of spectral networks and exponential networks that have previously appeared in string theory literature.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

cymyc: $\underline{C}$alabi-$\underline{Y}$au $\underline{M}$etrics, $\underline{Y}$ukawas, and $\underline{C}$urvature

We introduce cymyc, a high-performance Python library for numerical investigation of the geometry of a large class of string compactification manifolds and their associated moduli spaces. We develop a well-defined geometric ansatz to numerically model tensor fields of arbitrary degree on a large class of Calabi-Yau manifolds. cymyc includes a machine learning component which incorporates this ansatz to model tensor fields of interest on these spaces by finding an approximate solution to the system of partial differential equations they should satisfy.

differential and algebraic geometry↗
DOE OSTI2025

Geometry of soft scalars at one loop

We extend the soft theorems for scattering amplitudes of scalar effective field theories to one-loop order. Our analysis requires carefully accounting for the fact that the soft limit is not guaranteed to commute with evaluating IR-divergent loop integrals; new results for the soft limit of general scalar one-loop integrals are presented. The geometric soft theorem remains unmodified for any derivatively-coupled scalar effective field theory, and we conjecture that this statement holds to all orders. In contrast, the soft theorem receives nontrivial corrections in the presence of potential interactions, analogous to the case of non-Abelian gauge theories. We derive the universal leading-order correction to the scalar soft theorem arising from potential interactions at one loop. Explicit examples are provided that illustrate the general results.

differential and algebraic geometry↗
DOE OSTI2026

Tracking discontinuities in parameter space

We develop a geometric framework in Feynman-parameter space to determine constraints on the sequential discontinuities of Feynman integrals. Our method is based on tracking the deformation of the integration contour as external kinematics are analytically continued. This procedure imposes powerful constraints on the analytic structure of Feynman integrals, providing crucial inputs for their bootstrap. We demonstrate the usefulness of this framework by applying it to integrals in dimensional regularization, with higher propagator powers, and to examples with non-uniform transcendental weight. The method is illustrated with several one- and two-loop calculations.

Differential and Algebraic Geometry↗
DOE OSTI2025

A physical basis for cosmological correlators from cuts

Significant progress has been made in our understanding of the analytic structure of FRW wavefunction coefficients, facilitated by the development of efficient algorithms to derive the differential equations they satisfy. Moreover, recent findings indicate that the twisted cohomology of the associated hyperplane arrangement defining FRW integrals overestimates the number of integrals required to define differential equations for the wave-function coefficient. We demonstrate that the associated dual cohomology is automatically organized in a way that is ideal for understanding and exploiting the cut/residue structure of FRW integrals. Utilizing this understanding, we develop a systematic approach to organize compatible sequential residues, which dictates the physical subspace of FRW integrals for any n -site, ℓ-loop graph. In particular, the physical subspace of tree-level FRW wavefunction coefficients is populated by differential forms associated to cuts/residues that factorize the integrand of the wavefunction coefficient into only flat space amplitudes. After demonstrating the validity of our construction using intersection theory, we develop simple graphical rules for cut tubings that enumerate the space of physical cuts and, consequently, differential forms without any calculation.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2026

Statistics of base polytopes in F-theory

We propose a new statistical ensemble of toric bases for elliptic Calabi-Yaus used in F-theory models, by focusing on only the convex hull of the base, i.e., the base polytope. This physically motivated coarse-graining greatly simplifies the combinatorial complexity of the part of the 4d F-theory landscape with toric bases. We develop a Monte Carlo approach that randomly samples the base polytopes within fixed boxes, with proper statistical weights. We first apply the algorithm to the set of 2d base polytopes, generating an enlarged set of toric 2d bases that include certain types of codimension-two (4,6) points, and we validate our approach against exact numbers. We then explore the set of 3d base polytopes which fit in a set of “maximal” 3d boxes, and estimate the total number of inequivalent 3d base polytopes to be 10 85 –10 90 . We provide statistical data such as the distribution of non-Higgsable gauge groups on these bases. Amusingly, a similar method can also be applied to generate reflexive polytopes in various dimensions. In both the reflexive and base polytope cases, the number of relevant polytopes obeys a Gaussian distribution as a function of the number of vertices, which can be understood in terms of other results on random polytopes in the math literature.

Differential and algebraic geometry↗
DOE OSTI2024

The soaring kite: a tale of two punctured tori

We consider the 5-mass kite family of self-energy Feynman integrals and present a systematic approach for constructing an ε-form basis, along with its differential equation pulled back onto the moduli space of two tori. Each torus is associated with one of the two distinct elliptic curves this family depends on. We demonstrate how the locations of relevant punctures, which are required to parametrize the full image of the kinematic space onto this moduli space, can be extracted from integrals over maximal cuts. A boundary value is provided such that the differential equation is systematically solved in terms of iterated integrals over g-kernels and modular forms. Then, the numerical evaluation of the master integrals is discussed, and important challenges in that regard are emphasized. In an appendix, we introduce new relations between g-kernels.

72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS↗
DOE OSTI2025

Kinematic flow for cosmological loop integrands

Recently, an interesting pattern was found in the differential equations satisfied by the Feynman integrals describing tree-level correlators of conformally coupled scalars in a power-law FRW cosmology [1, 2]. It was proven that simple and universal graphical rules predict the equations for arbitrary graphs as a flow in kinematic space. In this note, we show that the same rules — with one small addition — also determine the differential equations for loop integrands. We explain that both the basis of master integrals and the singularities of the differential equations can be represented by tubings of marked graphs. An important novelty in the case of loops is that some basis functions can vanish, and we present a graphical rule to identify these vanishing functions. Taking this into account, we then demonstrate that the kinematic flow correctly predicts the differential equations for all loop integrands.

Cosmological models↗
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 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 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↗
NASA NTRS1982

Algebraic grid generation

The numerical solution of partial differential equations about irregular geometries and with varying characteristic scales has created the need for coordinate systems and associated transformations which reflect both geometric and physical requirements. The process of finding coordinate transformations in discrete representations is called 'grid generation'. The present investigation is concerned with three algebraic grid generation methods. The methods include transfinite interpolation, the multisurface method, and the two-boundary technique. Interpolation formulas, in terms of homotopic mappings and constraints in terms of point positions and/or derivatives are the essential elements of the techniques. The methods are relatively simple to understand, they are explicit and do not require extensive computational effort, and they have a high degree of generality.

Smith, R. E.↗
NASA NTRSDate not supplied

Formal Verification of the Interaction Between Semi-Algebraic Sets and Real Analytic Functions

Semi-algebraic sets and real analytic functions are fundamental concepts in Real Algebraic Geometry and Real Analysis, respectively. These concepts interact in the study of Differential Equations, where the real analytic solution to a differential equation is known to enter or exit a semi-algebraic set in a predicable way. Motivated to enhance the capability to reason about differential equations in the Prototype Verification System (PVS), a formalization of multivariate polynomials, semi-algebraic sets, and real analytic functions is developed. The favorable way that a real analytic function enters and exits a semi-algebraic set is proven. It is further shown that if the function is assumed to be smooth, a slightly weaker assumption than real analytic, these favorable interactions with semi-algebraic sets may fail.

Real analytic functions↗