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This paper concerns Hodge-Dirac operators D = d + δ acting in L p (Ω, Λ) where Ω is a bounded open subset of R n satisfying some kind of Lipschitz condition, Λ is the exterior algebra of R n , d is the exterior derivative acting on the de Rham complex of differential forms on Ω, and δ is the interior derivative with tangential boundary conditions. In L 2 (Ω, Λ), δ = d * and D is self-adjoint, thus having bounded resolvents (I + itD) −1 t∈R as well as a bounded functional calculus in L 2 (Ω, Λ). We investigate the range of values p H < p < p H about p = 2 for which D has bounded resolvents and a bounded holomorphic functional calculus in L p (Ω, Λ). On domains which we call very weakly Lipschitz, we show that this is the same range of values as for which L p (Ω, Λ) has a Hodge (or Helmholz) decomposition, being an open interval that includes 2. The Hodge-Laplacian ∆ is the square of the Hodge-Dirac operator, i.e. −∆ = D 2 , so it also has a bounded functional calculus in L p (Ω, Λ) when p H < p < p H. But the Stokes operator with Hodge boundary conditions, which is the restriction of −∆ to the subspace of divergence free vector fields in L p (Ω, Λ 1) with tangential boundary conditions , has a bounded holomorphic functional calculus for further values of p, namely for max{1, p H S } < p < p H where p H S is the Sobolev exponent below p H , given by 1/p H S = 1/p H + 1/n, so that p H S < 2n/(n + 2). In 3 dimensions, p H S < 6/5. We show also that for bounded strongly Lipschitz domains Ω, p H < 2n/(n + 1) < 2n/(n − 1) < p H , in agreement with the known results that p H < 4/3 < 4 < p H in dimension 2, and p H < 3/2 < 3 < p H in dimension 3. In both dimensions 2 and 3, p H S < 1 , implying that the Stokes operator has a bounded functional calculus in L p (Ω, Λ 1) when Ω is strongly Lipschitz and 1 < p < p H .

Discrete exterior calculus (DEC) offers a coordinate–free discretization of exterior calculus especially suited for computations on curved spaces. In this work, we present an extended version of DEC on surface meshes formed by general polygons that bypasses the need for combinatorial subdivision and does not involve any dual mesh. At its core, our approach introduces a new polygonal wedge product that is compatible with the discrete exterior derivative in the sense that it satisfies the Leibniz product rule. Based on the discrete wedge product, we then derive a novel primal–to–primal Hodge star operator. Combining these three ‘basic operators’ we then define new discrete versions of the contraction operator and Lie derivative, codifferential and Laplace operator. We discuss the numerical convergence of each one of these proposed operators and compare them to existing DEC methods. Finally, we show simple applications of our operators on Helmholtz–Hodge decomposition, Laplacian surface fairing, and Lie advection of functions and vector fields on meshes formed by general polygons.

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., Comment: 21 pages

In this paper we prove a variant of the Stokes formula for differential forms of a finite codimension in a locally convex space (LCS). The main tool used by us for proving the mentioned formula is the surface layer theorem for surfaces of codimension 1 in a locally convex space which was proved earlier by the first author. Moreover, on some subspace of differential forms of the Sobolev type with respect to a differentiable measure we establish a formula expressing the operator adjoint to the exterior differential via standard operations of the calculus of differential forms and the logarithmic derivative. This connection was established earlier under stronger constraints imposed either on the LCS or on the measure, or on differential forms (the smoothness condition).

An affinor structure is a generalization of the notion of an almost complex structure associated with a symplectic form on a manifold of even dimension for vector bundles of arbitrary rank. An affinor structure is the field of the automorphisms of the vector bundle preserving the exterior derivative of some 1-form with radical of arbitrary dimension. The exterior derivative can be always defined on Lie algebroids, a special class of vector bundles. Therefore, the theory of affinor structures is considered on Lie algebroids. We show that the classical objects, such as a symplectic structure, a contact structure, and a Kahler structure, are particular cases of the general theory of affinor metric structures.

We construct projections from H Λ k ( Ω ) H \Lambda ^k(\Omega ) , the space of differential k k forms on Ω \Omega which belong to L 2 ( Ω ) L^2(\Omega ) and whose exterior derivative also belongs to L 2 ( Ω ) L^2(\Omega ) , to finite dimensional subspaces of H Λ k ( Ω ) H \Lambda ^k(\Omega ) consisting of piecewise polynomial differential forms defined on a simplicial mesh of Ω \Omega . Thus, their definition requires less smoothness than assumed for the definition of the canonical interpolants based on the degrees of freedom. Moreover, these projections have the properties that they commute with the exterior derivative and are bounded in the H Λ k ( Ω ) H \Lambda ^k(\Omega ) norm independent of the mesh size h h . Unlike some other recent work in this direction, the projections are also locally defined in the sense that they are defined by local operators on overlapping macroelements, in the spirit of the Clément interpolant. A double complex structure is introduced as a key tool to carry out the construction.

We prove the triviality of the first L 2 cohomology class of based path spaces of Riemannian manifolds furnished with Brownian motion measure, and the consequent vanishing of L 2 harmonic one-forms. We give explicit formulae for closed and co-closed one-forms expressed as differentials of functions and co-differentials of L 2 two-forms, respectively; these are considered as extended Clark–Ocone formulae. A feature of the proof is the use of the temporal structure of path spaces to relate a rough exterior derivative operator on one-forms to the exterior differentiation operator used to construct the de Rham complex and the self-adjoint Laplacian on L 2 one-forms. This Laplacian is shown to have a spectral gap.

This paper addresses the issue of structure-preserving discretization of open distributed-parameter systems with Hamiltonian dynamics. Employing the formalism of discrete exterior calculus, we introduce a simplicial Dirac structure as a discrete analogue of the Stokes-Dirac structure and demonstrate that it provides a natural framework for deriving finite-dimensional port-Hamiltonian systems that emulate their infinite-dimensional counterparts. The spatial domain, in the continuous theory represented by a finite-dimensional smooth manifold with boundary, is replaced by a homological manifold-like simplicial complex and its augmented circumcentric dual. The smooth differential forms, in discrete setting, are mirrored by cochains on the primal and dual complexes, while the discrete exterior derivative is defined to be the coboundary operator. This approach of discrete differential geometry, rather than discretizing the partial differential equations, allows to first discretize the underlying Stokes-Dirac structure and then to impose the corresponding finite-dimensional port-Hamiltonian dynamics. In this manner, a number of important intrinsically topological and geometrical properties of the system are preserved., Comment: Journal of Geometry and Physics

Given two $k-$forms $\alpha$ and $\beta$ we derive an identity relating $$ %TCIMACRO{\dint _{\Omega}} %BeginExpansion {\displaystyle\int_{\Omega}} %EndExpansion \left( \langle d\alpha;d\beta\rangle+\langle\delta\alpha;\delta\beta \rangle-\langle\nabla\alpha;\nabla\beta\rangle\right) $$ to an integral on the boundary of the domain and involving only the tangential and the normal components of $\alpha$ and $\beta.$ We use this identity to deduce in a very simple way the classical Gaffney inequality and a generalization of it.

We construct realizations of the generators of the κ-Minkowski space and κ-Poincaré algebra as formal power series in the h-adic extension of the Weyl algebra. The Hopf algebra structure of the κ-Poincaré algebra related to different realizations is given. We construct realizations of the exterior derivative and one-forms, and define a differential calculus on κ-Minkowski space which is compatible with the action of the Lorentz algebra. In contrast to the conventional bicovariant calculus, the space of one-forms has the same dimension as the κ-Minkowski space.

The Hodge–de Rham Theorem is introduced and discussed. This result has implications for the general study of several partial differential equations. Some propositions which have applications to the proof of this theorem are used to study some related results concerning a class of partial differential equation in a novel way.

The purpose of this paper is to extend to spinor electromagnetism the difierential forms, based on the Cartan exterior derivative and originally developed for tensor flelds, in a very compact way. To this end, difierential electromagnetic forms are flrst compared to conventional tensors. Then, using the local isomorphism between the O(3,C) and SL(2,C) groups supplying the well known connection between complex vectors and traceless second rank spinors, they are generalized to spinor electromagnetism and to Proca flelds. These difierential forms are flnally expressed in terms of Hertz potentials.

In a 1991 paper by Buttig and Eichhorn, the existence and uniqueness of a differential forms heat kernel on open manifolds of bounded geometry was proven. In that paper, it was shown that the heat kernel obeyed certain properties, one of which was a relationship between the derivative of heat kernel of different degrees. We will give a proof of this condition for complete manifolds with Ricci curvature bounded below, and then use it to give an integral representation of the heat kernel of degree $k$.

The aim of the present paper is to introduce the notion of first order (supersymmetric) Dirac operators on discrete and metric (``quantum'') graphs. In order to cover all self-adjoint boundary conditions for the associated metric graph Laplacian, we develop systematically a new type of discrete graph operators acting on a decorated graph. The decoration at each vertex of degree-d is given by a subspace of $\C^d$, generalising the fact that a function on the standard vertex space has only a scalar value. We develop the notion of exterior derivative, differential forms, Dirac and Laplace operators in the discrete and metric case, using a supersymmetric framework. We calculate the (supersymmetric) index of the discrete Dirac operator generalising the standard index formula involving the Euler characteristic of a graph. Finally, we show that the corresponding index for the metric Dirac operator agrees with the discrete one., Comment: 36 pages, some references added

Given a cellular complex, we construct spaces of differential forms which form a complex under the exterior derivative, which is isomorphic to the cochain complex of the cellular complex. The construction applies in particular to subsets of Euclidean space divided into polyhedra, for which it provides, for each k, a space of k-forms with a basis indexed by the set of k-dimensional cells. In the framework of mimetic finite differences, the construction provides a conforming reconstruction operator. The construction requires auxiliary spaces of differential forms on each cell, for which we provide two examples. When the cells are simplexes, the construction can be used to recover the standard mixed finite element spaces also called Whitney forms. We can also recover the dual finite elements previously constructed by A. Buffa and the author on the barycentric refinement of a two-dimensional mesh.

This paper studies the operator d d ∗ + d ∗ d acting on q-forms on an unbounded domain with smooth boundary, where d is the exterior derivative and d ∗ is the adjoint of d calculated using the Sobolev space topology. The domain of d ∗ is determined and an expression for d ∗ is obtained. The operator d d ∗ + d ∗ d gives rise to a boundary value problem. Global regularity is obtained using weighted norms and global existence is obtained by using the theory of compact operators.

The development of smoothed projections, constructed by combining the canonical interpolation operators defined from the degrees of freedom with a smoothing operator, have proved to be an eective tool in finite element exterior calculus. The advan- tage of these operators is that they are L 2 bounded projections, and still they commute with the exterior derivative. In the present paper we generalize the construction of these smoothed projections, such that also non quasi-uniform meshes and essential boundary conditions are covered. The new tool introduced here is a space dependent smoothing operator which commutes with the exterior derivative.

Based on the exterior differential calculus, we define the exterior difference system on the hypercubic lattice. The pairing formula, bracket, contract operator and Lie derivative operator for this system are also constructed. The discrete H. Cartan formulas are obtained too. After those preparations, we prove the discrete Frobenius Theorem, discrete retraction theorem and discrete Cartan–Kahler theorem.

Vertex- and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied. In this paper, we interpret these schemes as defining bases for discrete differential 0- resp. 2-forms , and complete the picture by introducing edge-based subdivision schemes to construct the missing bases for discrete differential 1-forms. Such subdivision schemes map scalar coefficients on edges from the coarse to the refined mesh and are intrinsic to the surface. Our construction is based on treating vertex-, edge-, and face-based subdivision schemes as a joint triple and enforcing that subdivision commutes with the topological exterior derivative. We demonstrate our construction for the case of arbitrary topology triangle meshes. Using Loop's scheme for 0-forms and generalized half-box splines for 2-forms results in a unique generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes. We also provide corresponding boundary stencils. Once a metric is supplied, the scalar 1-form coefficients define a smooth tangent vector field on the underlying subdivision surface. Design of tangent vector fields is made particularly easy with this machinery as we demonstrate.

On conformal manifolds of even dimension n ≥ 4 we construct a family of new conformally invariant differential complexes, each containing one coboundary operator of order greater than 1. Each bundle in each of these complexes appears either in the de Rham complex or in its dual (which is a different complex in the non-orientable case). Each of the new complexes is elliptic in case the conformal structure has Riemannian signature. We also construct gauge companion operators which (for differential forms of order k ≤ n/2) complete the exterior derivative to a conformally invariant and (in the case of Riemannian signature) elliptically coercive system. These (operator, gauge) pairs are used to define finite dimensional conformally stable form subspaces which are are candidates for spaces of conformal harmonics. This generalizes the n/2-form and 0-form cases, in which the harmonics are given by conformally invariant systems. These constructions are based on a family of operators on closed forms which generali...

We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kahler form exp(iJ) and the holomorphic form Omega. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: exp(iJ) is closed under the action of the twisted exterior derivative in IIA theory, and similarly Omega is closed in IIB. Modulo a different action of the B-field, this means that supersymmetric SU(3)-structure manifolds are all generalized Calabi-Yau manifolds, as defined by Hitchin. An equivalent, and somewhat more conventional, description is given as a set of relations between the components of intrinsic torsions modified by the NS flux and the Clifford products of RR fluxes with pure spinors, allowing for a classification of type II supersymmetric vacua on six-manifolds. We find in particular that supersymmetric six-manifolds are always complex for IIB backgrounds while they are twisted symplectic for IIA., 35 pages, LaTeX; v2: Corrections in sec. 4, references added, version to appear in JHEP

We give an elementary introduction to exterior differential systems and the Cartan–Kahler theorem. No proofs are given, but the results are illustrated by means of examples.

The symmetry group of an exterior differential system is used to simplify finding integral manifolds in the case of Pfaffian systems. This leads to the definition of a moment map for Pfaffian systems.

A Stein covering of a complex manifold may be used to realize its analytic cohomology in accordance with the Cech theory. If however, the Stein covering is parameterized by a smooth manifold rather than just a discrete set, then we construct a cohomology theory in which an exterior derivative replaces the usual combinatorial Cech differential. Our construction is motivated by integral geometry and the representation theory of Lie groups.

We show that the supersymmetry transformations for type II string theories on six-manifolds can be written as differential conditions on a pair of pure spinors, the exponentiated Kahler form e i J and the holomorphic form Ω. The equations are explicitly symmetric under exchange of the two pure spinors and a choice of even or odd-rank RR field. This is mirror symmetry for manifolds with torsion. Moreover, RR fluxes affect only one of the two equations: e i J is closed under the action of the twisted exterior derivative in IIA theory, and similarly Ω is closed in IIB. This means that supersymmetric SU(3)-structure manifolds are always complex in IIB while they are twisted symplectic in IIA. Modulo a different action of the B-field, these are all generalized Calabi–Yau manifolds, as defined by Hitchin. To cite this article: M. Grana et al., C. R. Physique 5 (2004).

A generally covariant and gauge invariant description of physical optics, the Sagnac effect, and the Aharonov-Bohm (AB) effect is developed using the appropriate phase factor for electrodynamics. The latter is a generally covariant development of the Dirac-Wu-Yang phase factor based on the generally covariant Stokes theorem. The Maxwell-Heaviside (MH) field theory fails to describe physical optics, interferometry, and topological phase effects in general because the phase factor in that theory is under-determined. A random number from a U(1) gauge transformations can be added to the MH phase factor. The generally covariant phase factor of the Evans unified field theory is gauge invariant and has the correct property under parity inversion to produce observables such as reflection, interferograms, the Sagnac and AB effects, the Tomita-Chao effect and topological phase effects in general. These effects are described simply and self consistently with the generally covariant Stokes theorem in which ordinary exterior derivative is replaced by the covariant exterior derivative of differential geometry.

The so-called covariant Poincare lemma on the induced cohomology of the spacetime exterior derivative in the cohomology of the gauge part of the BRST differential is extended to cover the case of arbitrary, non reductive Lie algebras. As a consequence, the general solution of the Wess-Zumino consistency condition with a non trivial descent can, for arbitrary (super) Lie algebras, be computed in the small algebra of the 1 form potentials, the ghosts and their exterior derivatives. For particular Lie algebras that are the semidirect sum of a semisimple Lie subalgebra with an ideal, a theorem by Hochschild and Serre is used to characterize more precisely the cohomology of the gauge part of the BRST differential in the small algebra. In the case of an abelian ideal, this leads to a complete solution of the Wess-Zumino consistency condition in this space. As an application, the consistent deformations of 2+1 dimensional Chern-Simons theory based on iso(2,1) are rediscussed., Comment: 39 pages Latex file, 1 eps figure, typos and proof of lemma 5 corrected

In this paper, a general Hamiltonian theory for Lagrangian systems on fibred manifolds is proposed. The concept of a Lepagean ( n +1)- form is defined (where n is the dimension of the base manifold), generalizing Krupka’s concept of a Lepagean n-form. Lepagean ( n +1)-forms are used to study Lagrangian and Hamiltonian systems. Innovations and new results concern the following: a Lagrangian system is considered as an equivalence class of local Lagrangians (of all orders starting from a minimal one); a Hamiltonian system is associated with an Euler–Lagrange form (not with a particular Lagrangian); Hamilton equations are based upon a Lepagean ( n +1)-form, and cover Hamilton–De Donder equations (which are based upon the exterior derivative of the Poincare–Cartan form) as a special case. First-order Hamiltonian systems, namely those carying higher-degree contact components of the corresponding Lepagean forms, are studied in detail. The presented geometric setting leads to a new (more general than the standard one) understanding of the concepts of regularity and Legendre transformation in the calculus of variations, relating them directly to the properties of the arising exterior differential systems . In this way, new regularity conditions and Legendre transformation formulas are obtained, depending on a Lepagean ( n +1)-form, i.e., related with the corresponding Euler–Lagrange form .

Let ρ:G↪Gl(n,\(\mathbb{F}\)) be a representation of a finite groupG over a field\(\mathbb{F}\) such that the ring of invariants\(\mathbb{F}\left[ V \right]^G \) is a polynomial algebra\(\mathbb{F}\left[ {f_1 ,... ,f_n } \right]\). It is known that in the nonmodular case (i.e., when the order of the group is not divisible by the characteristic of\(\mathbb{F}\)), the invariants ofG acting on the tensor product\(\mathbb{F}\left[ V \right] \otimes E\left[ V \right]\) of a polynomial and an exterior algebra are given by\(\mathbb{F}\left[ {f_1 ,... ,f_n } \right] \otimes E\left[ {df_1 ,... ,df_n } \right]\),d denoting the exterior derivative. We show that in the modular case, the ring of invariants in\(\mathbb{F}\left[ V \right] \otimes E\left[ V \right]\) is of this form if and only if\(\mathbb{F}\left[ V \right]^G \) is a polynomial algebra and all pseudoreflections in ϱ(G) are diagonalizable.

In this paper we consider a neutral spinning particle in interaction with a linear increasing rotating magnetic field and a scalar harmonic potential using the path integral formalism. The Pauli matrices which describe the spin dynamics are replaced by two fermionic oscillators via the Schwinger’s model. The calculations are carried out explicitly using fermionic exterior current sources. The problem is then reduced to that of a spinning forced harmonic particle whose spin is coupled to exterior derivative current sources. The result of the propagator is given as a series which is exactly summed up by means of the Laplace transformation and the use of some recurrence formula of the oscillator wave functions. The energy spectrum and the corresponding wave functions are also deduced.

The mixed variational formulation of many elliptic boundary value problems involves vector valued function spaces, like, in three dimensions, H ( curl ; Ω ) \boldsymbol H(\mathbf {\operatorname {curl}};\Omega ) and H ( Div ; Ω ) {{\boldsymbol H}(\operatorname {Div};\Omega )} . Thus finite element subspaces of these function spaces are indispensable for effective finite element discretization schemes. Given a simplicial triangulation of the computational domain Ω \Omega , among others, Raviart, Thomas and Nédélec have found suitable conforming finite elements for H ( Div ; Ω ) \boldsymbol H(\operatorname {Div};\Omega ) and H ( curl ; Ω ) \boldsymbol H(\mathbf {\operatorname {curl}};\Omega ) . At first glance, it is hard to detect a common guiding principle behind these approaches. We take a fresh look at the construction of the finite spaces, viewing them from the angle of differential forms. This is motivated by the well-known relationships between differential forms and differential operators: div \operatorname {div} , c u r l \operatorname {\mathbf {curl}} and g r a d \operatorname {\mathbf {grad}} can all be regarded as special incarnations of the exterior derivative of a differential form. Moreover, in the realm of differential forms most concepts are basically dimension-independent. Thus, we arrive at a fairly canonical procedure to construct conforming finite element subspaces of function spaces related to differential forms. In any dimension we can give a simple characterization of the local polynomial spaces and degrees of freedom underlying the definition of the finite element spaces. With unprecedented ease we can recover the familiar H ( Div ; Ω ) \boldsymbol H (\operatorname {Div};\Omega ) - and H ( curl ; Ω ) \boldsymbol H(\mathbf {\operatorname {curl}};\Omega ) -conforming finite elements, and establish the unisolvence of degrees of freedom. In addition, the use of differential forms makes it possible to establish crucial algebraic properties of the canonical interpolation operators and representation theorems in a single sweep for all kinds of spaces.

A Radon-Nikodým theorem for Banach valued polymeasures is proved.

This paper proves an isomorphism theorem for cochains and differential forms, before passing to cohomology. De Rham’s theorem is a consequence. This leads to an extension of much of calculus and homology theory to nonsmooth domains, called chainlets,and makes available combinatorial techniques for smooth domains that limit to the classic analytic methods. We find maximal subspaces of L 1 forms that satisfy Stokes’s theorem for domains of chainlets giving a measurable, as well as optimal, extension of the theory.

Explicit solutions of certain exterior differential systems of degrees one and two are constructed on a star-shaped region of a differentiable manifold. A closed differential ideal that is generated by 1-forms that are solutions of an exterior differential is constructed. This ideal does not satisfy the Frobenius theorem because there are no solutions that satisfy the necessary independence condition, although a modified version of the Frobenius theorem is exhibited in this case. Solutions of certain exterior differential systems whose associated curvature 2-forms have compact support in a star-shaped region are also constructed.

Among all the homogeneous Riemannian graded metrics on the algebra of differential forms, those for which the exterior derivative is a Killing graded vector field are characterized. It is shown that all of them are odd, and are naturally associated to an underlying smooth Riemannian metric. It is also shown that all of them are Ricci-flat in the graded sense, and have a graded Laplacian operator that annihilates the whole algebra of differential forms.

In the Bargmann-Fock representation the coordinates $z^i$ act as bosonic creation operators while the partial derivatives $\partial_{z^j}$ act as annihilation operators on holomorphic $0$-forms as states of a $D$-dimensional bosonic oscillator. Considering also $p$-forms and further geometrical objects as the exterior derivative and Lie derivatives on a holomorphic ${\bf C}^D$, we end up with an analogous representation for the $D$-dimensional supersymmetric oscillator. In particular, the supersymmetry multiplet structure of the Hilbert space corresponds to the cohomology of the exterior derivative. In addition, a 1-complex parameter group emerges naturally and contains both time evolution and a homotopy related to cohomology. Emphasis is on calculus.

We study the basic Laplacian on Riemannian foliations by writing the basic Laplacian in terms of the orthogonal projection from square-integrable forms to basic square-integrable forms. Using a geometric interpretation of this projection, we relate the ordinary Laplacian to the basic Laplacian. Among other results, we show the existence of the basic heat kernel and establish estimates for the eigenvalues of the basic Laplacian. Introduction. Let M be a compact oriented manifold and let be a transver- sally oriented foliation on M. A foliation is a Riemannian foliation if there exists a Riemannian metric on M with the property that the distance from one leaf of to another is locally constant; such a metric is called a bundle-like metric for . Associated to are the space of basic forms: Ω B(M )= Ω B ( M , )= Ω ( M ): i(X) =0 , i ( X ) d = 0 for all X Γ(T ) , where i(X) is the interior product with the vector field X and Γ(T ) denotes the sections of the distribution T associated to . The exterior derivative d maps basic forms to basic forms; let dB denote d restricted to Ω B (M). The basic Laplacian is the operator ΔB = dB B + BdB on basic forms, where B is the adjoint of dB on Ω B (M). The analytic and geometric properties of this operator have been studied by several researchers. In (5), the basic Laplacian was studied as an operator on basic functions (i.e., functions that are constant on leaves of ), and the author proved the existence of the heat kernel in this case. In (13), the existence of the heat kernel on basic forms was proved for the case where the mean curvature form of the foliation is basic. There are also "basic" Hodge theorems, for example (6) and (10). However, the proof of the Hodge theorem in (6) does not yield various estimates that are important in applications, while the theorem proved in (10) has the same restriction as the results in (13), namely that the authors require the mean curvature form to be basic. In this paper, we study the basic Laplacian on forms, without any restriction on the mean curvature. We prove the existence and uniqueness of the heat kernel for ΔB on forms for any Riemannian foliation, and we write down an explicit formula for the heat kernel. We also present a proof of the Hodge theorem for

Over the years a third-rank antisymmetric tensor field, derived as the exterior derivative of a potential, has been considered as a part of string theory Lagrangians. It is shown that this antisymmetric field arises naturally in a theory of gravitation with a torsion potential in four dimensions. In this case, the physical significance of the antisymmetric field corresponds to a propagating spin field, and the material source is shown to consist of a loop, and not a point.

A supersymmetric quantum mechanics on SO(3) is examined. It was found that instanton solutions and the normal coordinate expansions around them allow the explicit path integration. The Witten complex is obtained by calculating the matrix elements of the generalized exterior derivative.