De Rham Research Articles

Simple vertex algebras arising from congruence subgroups

For any congruence subgroup Γ, we consider the vertex algebra of Γ-invariant global sections of chiral de Rham complex on the upper half plane that are meromorphic at the cusps. We give a description of the linear structure of the Γ-invariant vertex algebra by exhibiting a linear basis determined by meromorphic modular forms, and generalize the Rankin-Cohen bracket of modular forms to meromorphic modular forms. We also show that the Γ-invariant vertex algebra is simple.

$q$–de Rham complexes of higher level

In this article, we construct two kinds of de Rham–like complexes which compute the cohomology of complete crystals on the higher-level q -crystalline site, which was introduced in a previous article by the author. One complex is the q -analog of the higher de Rham complex constructed by Miyatani, and another complex is the q -analog of the jet complex constructed by Le Stum–Quirós. The complexes we construct can also be regarded as the higher-level analogs of the q –de Rham complex.

The action of mapping class groups on de Rham quasimorphisms

We study the action of the mapping class group on the subspace of de Rham classes in the degree-two bounded cohomology of a hyperbolic surface. In particular, we show that the only fixed nontrivial finite-dimensional subspace is the one generated by the Euler class. As a consequence, we get that the action of the mapping class group on the space of de Rham quasimorphisms has no fixed points.

The Motivic Anabelian Geometry of Local Heights on Abelian Varieties

We study the problem of describing local components of Néron–Tate height functions on abelian varieties over characteristic 0 0 local fields as functions on spaces of torsors under various realisations of the 2 2 -step unipotent motivic fundamental group of the G m \mathbb G_m -torsor corresponding to the defining line bundle. To this end, we present three main theorems giving such a description in terms of the Q ℓ \mathbb Q_\ell - and Q p \mathbb Q_p -pro-unipotent étale realisations when the base field is p p -adic, and in terms of the R \mathbb R -pro-unipotent Betti–de Rham realization when the base field is archimedean. In the course of proving the p p -adic instance of these theorems, we develop a new technique for studying local nonabelian Bloch–Kato Selmer sets, working with certain explicit cosimplicial group models for these sets and using methods from homotopical algebra. Among other uses, these models enable us to construct a pro-unipotent generalization of the Bloch–Kato exponential sequence under minimal assumptions.

On the periods of twisted moments of the Kloosterman connection

AbstractThis paper aims to study the Betti homology and de Rham cohomology of twisted symmetric powers of the Kloosterman connection of rank two on the torus. We compute the period pairing and, with respect to certain bases, interpret these associated period numbers in terms of the Bessel moments. Via the rational structures on Betti homology and de Rham cohomology, we prove the $$\mathbb {Q}$$ Q -linear and quadratic relations among these Bessel moments.

The characteristic cycle of a non-confluent ℓ-adic GKZ hypergeometric sheaf

An ℓ-adic GKZ hypergeometric sheaf is defined analogously to a GKZ hypergeometric D-module. We introduce an algorithm of computing the characteristic cycle of an ℓ-adic GKZ hypergeometric sheaf of certain type. Our strategy is to apply a formula of the characteristic cycle of the direct image of an ℓ-adic sheaf. We verify the requirements for the formula to hold by calculating the dimension of the direct image of a certain closed conical subset of cotangent bundle. We also define an ℓ-adic GKZ-type sheaf whose specialization tensored with a constant sheaf is isomorphic to an ℓ-adic non-confluent GKZ hypergeometric sheaf. On the other hand, the topological model of an ℓ-adic GKZ-type sheaf is isomorphic to the image by the de Rham functor of a non-confluent GKZ hypergeometric D-module whose characteristic cycle has been calculated. This gives an easier way to determine the characteristic cycle of an ℓ-adic non-confluent GKZ hypergeometric sheaf of certain type.

Double copy of 3D Chern-Simons theory and 6D Kodaira-Spencer gravity

We apply an algebraic double copy construction of gravity from gauge theory to three-dimensional (3D) Chern-Simons theory. The kinematic algebra K is the 3D de Rham complex of forms equipped, for a choice of metric, with a graded Lie algebra that is equivalent to the Schouten-Nijenhuis bracket on polyvector fields. The double copied gravity is defined on a subspace of K⊗K¯ and yields a topological double field theory for a generalized metric perturbation and two 2-forms. This local and gauge invariant theory is non-Lagrangian but can be rendered Lagrangian by abandoning locality. Upon fixing a gauge this reduces to the double copy of Chern-Simons theory previously proposed by Ben-Shahar and Johansson. Furthermore, using complex coordinates in C3 this theory is related to six-dimensional (6D) Kodaira-Spencer gravity in that truncating the two 2-forms and one equation yields the Kodaira-Spencer equations on a 3D real slice of C3. The full 6D Kodaira-Spencer theory can instead be obtained as a consistent truncation of a chiral double copy. Published by the American Physical Society 2024

Intersection de rham theory on spaces with isolated singularities

Intersection de rham theory on spaces with isolated singularities

Arbitrary order spline representation of cohomology generators for isogeometric analysis of eddy current problems

Common formulations of the eddy current problem involve either vector or scalar potentials, each with its own advantages and disadvantages. An impasse arises when using scalar potential-based formulations in the presence of conductors with non-trivial topology. A remedy is to augment the approximation spaces with generators of the first cohomology group. Most existing algorithms for this require a special, e.g., hierarchical, finite element basis construction. Using insights from de Rham complex approximation with splines, we show that additional conditions are here unnecessary. Spanning tree techniques can be adapted to operate on a hexahedral mesh resulting from isomorphisms between spline spaces of differential forms and de Rham complexes on an auxiliary control mesh.

Mass, momentum and energy preserving FEEC and broken-FEEC schemes for the incompressible Navier–Stokes equations

Abstract In this article we propose two finite-element schemes for the Navier–Stokes equations, based on a reformulation that involves differential operators from the de Rham sequence and an advection operator with explicit skew-symmetry in weak form. Our first scheme is obtained by discretizing this formulation with conforming FEEC (Finite Element Exterior Calculus) spaces: it preserves the point-wise divergence free constraint of the velocity, its total momentum and its energy, in addition to being pressure robust. Following the broken-FEEC approach, our second scheme uses fully discontinuous spaces and local conforming projections to define the discrete differential operators. It preserves the same invariants up to a dissipation of energy to stabilize numerical discontinuities. For both schemes we use a middle point time discretization that preserve these invariants at the fully discrete level and we analyze its well-posedness in terms of a CFL condition. While our theoretical results hold for general finite elements preserving the de Rham structure, we consider one application to tensor-product spline spaces. Specifically, we conduct several numerical test cases to verify the high order accuracy of the resulting numerical methods, as well as their ability to handle general boundary conditions.

Vanishing of local cohomology with applications to Hodge theory

Let H=((H,F•),L) be a polarized variation of Hodge structure on a smooth quasi-projective variety U. By M. Saito's theory of mixed Hodge modules, the variation of Hodge structure H can be viewed as a polarized Hodge module M∈HM(U). Let X be a compactification of U, and j:U↪X is the natural map. In this paper, we use local cohomology with mixed Hodge module theory to study j+M∈DbMHM(X). In particular, we study the graded pieces of the de Rham complex GrpFDR(j+M)∈Dcohb(X), and the Hodge structure of Hi(U,L) for i in sufficiently low degrees.

A 3D field-theoretic example for Hodge theory

We focus on the continuous symmetry transformations for the three (2 + 1)-dimensional (3D) system of a combination of the free Abelian 1-form and 2-form gauge theories within the framework of Becchi-Rouet-Stora-Tyutin (BRST) formalism. We establish that this combined system is a tractable field-theoretic model of Hodge theory. The symmetry operators of our present system provide the physical realizations of the de Rham cohomological operators of differential geometry at the algebraic level. Our present investigation is important in the sense that, for the first time, we are able to establish an odd dimensional (i.e., D = 3) field-theoretic system to be an example for Hodge theory (besides earlier works on a few interesting (0 + 1)-dimensional (1D) toy models as well as a set of well-known SUSY quantum mechanical systems of physical interest). For the sake of brevity, we have purposely not taken into account the 3D Chern-Simon term for the Abelian 1-form gauge field in our theory which allows the mass as well as the gauge invariance to co-exist together.

A motivic proof of the finiteness of the relative de Rham cohomology

A motivic proof of the finiteness of the relative de Rham cohomology

A remark on the paper of Deninger and Murre

We show that the results proven by Deninger and Murre in their paper (Deninger and Murre, 1991) directly imply that the Chern classes of the de Rham bundle of an abelian scheme are torsion elements in the Chow ring, a result that was later proven by van der Geer. We also discuss several results about the orders of these classes.

A discrete three-dimensional divdiv complex on polyhedral meshes with application to a mixed formulation of the biharmonic problem

In this work, following the Discrete de Rham (DDR) paradigm, we develop an arbitrary-order discrete divdiv complex on general polyhedral meshes. The construction rests on (1) discrete spaces that are spanned by vectors of polynomials whose components are attached to mesh entities and (2) discrete operators obtained mimicking integration by parts formulas. We provide an in-depth study of the algebraic properties of the local complex, showing that it is exact on mesh elements with trivial topology. The new DDR complex is used to design a numerical scheme for the approximation of biharmonic problems, for which we provide detailed stability and convergence analyses. Numerical experiments complete the theoretical results.

Representations Of Lie Algebras of Vector Fields on Pairing Between Gauge Modules and Rudakov Modules

For an irreducible affine variety X over an algebraically closed field of characteristic zero, we define two new classes of modules over the Lie algebra of vector fields on X: gauge modules and Rudakov modules. These modules admit a compatible action of the algebra of functions on X. We prove general simplicity theorems for these two types of modules, demonstrating their irreducibility under specific conditions. Additionally, we establish a pairing between gauge modules and Rudakov modules, highlighting the connections and interactions between these two classes of modules. We have established that gauge modules and Rudakov modules, corresponding to simple glN -modules, remain irreducible as modules over the Lie algebra of vector fields unless they appear in the de Rham complex. Additionally, we have studied the irreducibility of tensor products of Rudakov modules, providing a comprehensive understanding of these module structures and their applications.

On the Kinematics of Growth of Regular Boundaries of Bodies into Fractals

Generalizing smooth volumetric growth to the singular case, using de Rham currents and flat chains, we demonstrate how regular boundaries of bodies may evolve to fractals.

Homotopy BV-algebras in Hermitian geometry

We show that the de Rham complex of any almost Hermitian manifold carries a natural commutative BV∞-algebra structure satisfying the degeneration property. In the almost Kähler case, this recovers Koszul's BV-algebra, defined for any Poisson manifold. As a consequence, both the Dolbeault and the de Rham cohomologies of any compact Hermitian manifold are canonically endowed with homotopy hypercommutative algebra structures, also known as formal homotopy Frobenius manifolds. Similar results are developed for (almost) symplectic manifolds with Lagrangian subbundles.

Yang–Mills models on Noncommutative background

In our previous publications we have developed some elements of Noncommutative calculus on the enveloping algebras of Am type, in particular, analogs of the partial derivatives and de Rham complex were defined. Also, we introduced the notion of quantization with Noncommutative configuration space and quantized a few dynamical models in this sense. In the current paper we propose a method of quantizing the Yang–Mills models in the same sense.

Applications of Differential Geometry Linking Topological Bifurcations to Chaotic Flow Fields

At every point p on a smooth n-manifold M there exist n+1 skew-symmetric tensor spaces spanning differential r-forms ω with r=0,1,⋯,n. Because d∘d is always zero where d is the exterior differential, it follows that every exact r-form (i.e., ω=dλ where λ is an r−1-form) is closed (i.e., dω=0) but not every closed r-form is exact. This implies the existence of a third type of differential r-form that is closed but not exact. Such forms are called harmonic forms. Every smooth n-manifold has an underlying topological structure. Many different possible topological structures exist. What distinguishes one topological structure from another is the number of holes of various dimensions it possesses. De Rham’s theory of differential forms relates the presence of r-dimensional holes in the underlying topology of a smooth n-manifold M to the presence of harmonic r-form fields on the smooth manifold. A large amount of theory is required to understand de Rham’s theorem. In this paper we summarize the differential geometry that links holes in the underlying topology of a smooth manifold with harmonic fields on the manifold. We explore the application of de Rham’s theory to (i) visual, (ii) mechanical, (iii) electrical and (iv) fluid flow systems. In particular, we consider harmonic flow fields in the intracellular aqueous solution of biological cells and we propose, on mathematical grounds, a possible role of harmonic flow fields in the folding of protein polypeptide chains.