Space Of Differential Forms Research Articles

Necessity of quantizable geometry for quantum gravity

In this paper, Dirac Quantization of 3D gravity in the first-order formalism is attempted where instead of quantizing the connection and triad fields, the connection and the triad 1-forms themselves are quantized. The exterior derivative operator on the space of differential forms is treated as the ‘time’ derivative to compute the momenta conjugate to these 1-forms. This manner of quantization allows one to compute the transition amplitude in 3D gravity which has a close, but not exact, match with the transition amplitude computed via LQG techniques. This inconsistency is interpreted as being due to the non-quantizable nature of differential geometry.

Differential forms on orbifolds with corners

Motivated by symplectic geometry, we give a detailed account of differential forms and currents on orbifolds with corners, the pull-back and push-forward operations, and their fundamental properties. We work within the formalism where the category of orbifolds with corners is obtained as a localization of the category of étale proper groupoids with corners. Constructions and proofs are formulated in terms of the structure maps of the groupoids, avoiding the use of orbifold charts. The Fréchet space of differential forms on an orbifold and the dual space of currents are shown to be independent of which étale proper groupoid is chosen to represent the orbifold.

Geometry of Tangent Poisson–Lie Groups

Let G be a Poisson–Lie group equipped with a left invariant contravariant pseudo-Riemannian metric. There are many ways to lift the Poisson structure on G to the tangent bundle TG of G. In this paper, we induce a left invariant contravariant pseudo-Riemannian metric on the tangent bundle TG, and we express in different cases the contravariant Levi-Civita connection and curvature of TG in terms of the contravariant Levi-Civita connection and the curvature of G. We prove that the space of differential forms Ω*(G) on G is a differential graded Poisson algebra if, and only if, Ω*(TG) is a differential graded Poisson algebra. Moreover, we show that G is a pseudo-Riemannian Poisson–Lie group if, and only if, the Sanchez de Alvarez tangent Poisson–Lie group TG is also a pseudo-Riemannian Poisson–Lie group. Finally, some examples of pseudo-Riemannian tangent Poisson–Lie groups are given.

A Rigorous Construction of the Supersymmetric Path Integral Associated to a Compact Spin Manifold

We give a rigorous construction of the path integral in {mathcal {N}}=1/2 supersymmetry as an integral map for differential forms on the loop space of a compact spin manifold. It is defined on the space of differential forms which can be represented by extended iterated integrals in the sense of Chen and Getzler–Jones–Petrack. Via the iterated integral map, we compare our path integral to the non-commutative loop space Chern character of Güneysu and the second author. Our theory provides a rigorous background to various formal proofs of the Atiyah–Singer index theorem for twisted Dirac operators using supersymmetric path integrals, as investigated by Alvarez-Gaumé, Atiyah, Bismut and Witten.

Duality in Finite Element Exterior Calculus and Hodge Duality on the Sphere

Finite element exterior calculus refers to the development of finite element methods for differential forms, generalizing several earlier finite element spaces of scalar fields and vector fields to arbitrary dimension n, arbitrary polynomial degree r, and arbitrary differential form degree k. The study of finite element exterior calculus began with the $${\mathcal {P}}_r\varLambda ^k$$ P r Λ k and $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k families of finite element spaces on simplicial triangulations. In their development of these spaces, Arnold, Falk, and Winther rely on a duality relationship between $${\mathcal {P}}_r\varLambda ^k$$ P r Λ k and $$\mathring{{\mathcal {P}}}_{r+k+1}^-\varLambda ^{n-k}$$ P ˚ r + k + 1 - Λ n - k and between $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k and $$\mathring{{\mathcal {P}}}_{r+k}\varLambda ^{n-k}$$ P ˚ r + k Λ n - k . In this article, we show that this duality relationship is, in essence, Hodge duality of differential forms on the standard n-sphere, disguised by a change of coordinates. We remove the disguise, giving explicit correspondences between the $${\mathcal {P}}_r\varLambda ^k$$ P r Λ k , $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k , $$\mathring{{\mathcal {P}}}_r\varLambda ^k$$ P ˚ r Λ k and $$\mathring{{\mathcal {P}}}_r^-\varLambda ^k$$ P ˚ r - Λ k spaces and spaces of differential forms on the sphere. As a direct corollary, we obtain new pointwise duality isomorphisms between $${\mathcal {P}}_r \varLambda ^k$$ P r Λ k and $$\mathring{{\mathcal {P}}}_{r+k+1}^-\varLambda ^{n-k}$$ P ˚ r + k + 1 - Λ n - k and between $${\mathcal {P}}_r^-\varLambda ^k$$ P r - Λ k and $$\mathring{{\mathcal {P}}}_{r+k} \varLambda ^{n-k}$$ P ˚ r + k Λ n - k . These isomorphisms can be implemented via a simple computation, which we illustrate with examples.

The de Rham Cohomology through Hilbert Space Methods

We discuss canonical representations of the de Rham cohomology on a compact manifold with boundary. They are obtained by minimising the energy integral in a Hilbert space of differential forms that belong along with the exterior derivative to the domain of the adjoint operator. The corresponding Euler- Lagrange equations reduce to an elliptic boundary value problem on the manifold, which is usually referred to as the Neumann problem after Spencer

Numerical Solution of the Barenblatt - Zheltov - Kochina Equation with Additive "White Noise" in Spaces of Differential Forms on a Torus

The paper is devoted to the search for numerical solutions to the Cauchy problem for the linear stochastic Barenblatt-Zheltov-Kochina equation in space of smooth differential forms on a torus. Based on the previously obtained results on the type of analytical solution to the stochastic version of the Barenblatt-Zheltov-Kochina equation in spaces of smooth differential forms on smooth compact Riemannian manifolds without boundary, we choose several terms from the analytical solution in order to construct graphs of the numerical solution for various values of the coefficients and the inhomogeneous term. Since these equations are Sobolev type equations with a degenerate operator at the derivative, we can solve various initial-boundary value problems using the theory of degenerate analytic groups and semigroups of resolving operators. In the deterministic case, the solution is based on the phase subspace of the original space. In spaces of differential forms, we use the invariant form of the Laplacian, i.e. the Laplace- Beltrami operator. The phase space method is also used in non-deterministic case, but we use the Nelson- Gliklikh derivative due to the non-differentiability of "white noise" in the usual sense. In this paper, a two-dimensional torus plays the role of a smooth compact oriented Riemannian manifold without boundary. Numerical solutions are found using the Galerkin method and are presented for several fixed time points as graphs of the coefficients of differential forms obtained in Maple.

The Graf Product: A Clifford Structure Framework on the Exterior Bundle

The geometric product, defined by Graf on the space of differential forms, endows the sections of the exterior bundle by a structure that is necessary to construct a Clifford algebra. The Graf product is introduced and revisited with a suitable underlying framework that naturally encompasses a coframe in the cotangent bundle, besides the volume element centrality, the Hodge operator and the so called truncated subalgebra as well.

Natural operations on differential forms on contact manifolds

We characterize all natural linear operations between spaces of differential forms on contact manifolds. Our main theorem says roughly that such operations are built from some algebraic operators which we introduce and the exterior derivative.

Constructions of some minimal finite element systems

Within the framework of finite element systems, we show how spaces of differential forms may be constructed, in such a way that they are equipped with commuting interpolators and contain prescribed functions, and are minimal under these constraints. We show how various known mixed finite element spaces fulfill such a design principle, including trimmed polynomial differential forms, serendipity elements and TNT elements. We also comment on virtual element methods and provide a dimension formula for minimal compatible finite element systems containing polynomials of a given degree on hypercubes.

Imbedding inequalities for the composite operator in the Sobolev spaces of differential forms

We establish the imbedding inequalities for the composition of the homotopy operator and Green’s operator in the weighted Sobolev spaces and Orlicz-Sobolev spaces of differential forms. First we prove both the local and global $L^{p}$ estimates for the composite operator acting on differential forms and obtain the boundedness of the composite operator in the weighted $L^{p}$ spaces. Then, we further study the local and global $L^{\phi}$ -norm inequalities for the composite operator. As a consequence we obtain the imbedding inequalities in the Orlicz-Sobolev spaces.

Clifford analysis approach to a self-conjugate Cauchy type integral on Ahlfors regular surfaces

In this note, based on a natural isomorphism between the spaces of differential forms and Clifford algebra-valued multi-vector functions, the Cauchy type integral for self-conjugate differential forms in $\mathbb R^n$ is considered.

Scalar supersymmetry

We describe a new realization of supersymmetry, called scalar supersymmetry, acting in spaces of differential forms (bi-spinors), where transformation parameters are Lorentz scalars instead of spinors. The realization is related but is not reducible to the standard supersymmetry. Explicit construction of chiral multiplets that do not require doubling of the spectrum of a gauge theory is presented. A bi-spinor s-supersymmetric string action is described.

Minimal models, formality, and hard Lefschetz properties of solvmanifolds with local systems

For a simply connected solvable Lie group $G$ with a cocompact discrete subgroup $\Gamma$, we consider the space of differential forms on the solvmanifold $G/\Gamma$ with values in a certain flat bundle so that this space has a structure of a differential graded algebra (DGA). We construct Sullivan’s minimal model of this DGA. This result is an extension of Nomizu’s theorem for ordinary coefficients in the nilpotent case. By using this result, we refine Hasegawa’s result of formality of nilmanifolds and Benson-Gordon’s result of hard Lefschetz properties of nilmanifolds.

Arc spaces and DAHA representations

A theorem of Y. Berest, P. Etingof and V. Ginzburg states that finite-dimensional irreducible representations of a type A rational Cherednik algebra are classified by one rational number m/n. Every such representation is a representation of the symmetric group Sn. We compare certain multiplicity spaces in its decomposition into irreducible representations of Sn with the spaces of differential forms on a zero-dimensional moduli space associated with the plane curve singularity xm = yn.

Heat kernel estimates for the $${{\bar{\partial}}}$$ -Neumann problem on G-manifolds

We prove heat kernel estimates for the $${\bar{\partial}}$$ -Neumann Laplacian $${\square}$$ acting in spaces of differential forms over noncompact manifolds with a Lie group symmetry and compact quotient. We also relate our results to those for an associated Laplace-Beltrami operator on functions.

Caratheodory operator of differential forms

This article is devoted to extensions of some existing results about the Caratheodory operator from the function sense to the differential form situation. Similarly as the function sense, we obtain the convergence of sequences of differential forms defined by the Caratheodory operator. The main result in this article is the continuity and mapping property from one space of differential forms to another under some dominated conditions.

Spaces of differential forms and maps with controlled distortion

We study necessary and sufficient conditions for an approximately differentiable map between Riemannian manifolds to induce a bounded transfer operator of differential forms with respect to the norms of Lebesgue spaces. As a corollary, we see that every homeomorphism of class whose transfer operator of differential forms with norm in is an isomorphism must necessarily be either quasi-conformal or quasi-isometric. We give some applications of our results to the study of the functoriality of cohomology in Lebesgue spaces.

Spaces of differential forms and mappings with controlled distortion

Spaces of differential forms and mappings with controlled distortion

On the spectrum of geometric operators on Kähler manifolds

On a compact K\"ahler manifold there is a canonical action of a Lie-superalgebra on the space of differential forms. It is generated by the differentials, the Lefschetz operator and the adjoints of these operators. We determine the asymptotic distribution of irreducible representations of this Lie-superalgebra on the eigenspaces of the Laplace-Beltrami operator. Because of the high degree of symmetry the Laplace-Beltrami operator on forms can not be quantum ergodic. We show that after taking these symmetries into account quantum ergodicity holds for the Laplace-Beltrami operator and for the Spin^c-Dirac operators if the unitary frame flow is ergodic. The assumptions for our theorem are known to be satisfied for instance for negatively curved K\"ahler manifolds of odd complex dimension.