Hodge-de Rham Operator Research Articles

Effective actions for dual massive (super) p-forms

In d dimensions, the model for a massless p-form in curved space is known to be a reducible gauge theory for p > 1, and therefore its covariant quantisation cannot be carried out using the standard Faddeev-Popov scheme. However, adding a mass term and also introducing a Stueckelberg reformulation of the resulting p-form model, one ends up with an irreducible gauge theory which can be quantised à la Faddeev and Popov. We derive a compact expression for the massive p-form effective action, {Gamma}_p^{(m)} , in terms of the functional determinants of Hodge-de Rham operators. We then show that the effective actions {Gamma}_p^{(m)} and {Gamma}_{d-p-1}^{(m)} differ by a topological invariant. This is a generalisation of the known result in the massless case that the effective actions Γp and Γd−p−2 coincide modulo a topological term. Finally, our analysis is extended to the case of massive super p-forms coupled to background mathcal{N} = 1 supergravity in four dimensions. Specifically, we study the quantum dynamics of the following massive super p-forms: (i) vector multiplet; (ii) tensor multiplet; and (iii) three-form multiplet. It is demonstrated that the effective actions of the massive vector and tensor multiplets coincide. The effective action of the massive three-form is shown to be a sum of those corresponding to two massive scalar multiplets, modulo a topological term.

The $$S^1$$ S 1 -Equivariant Signature for Semi-free Actions as an Index Formula

Lott (Math Ann 316(4):617–657, 2000) defined an integer-valued signature $$\sigma _{S^1}(M)$$ for the orbit space of a compact orientable manifold with a semi-free $$S^1$$ -action but he did not construct a Dirac-type operator which has this signature as its index. We construct such operator on the orbit space and we show that it is essentially unique and that its index coincides with Lott’s signature, at least when the stratified space satisfies the so-called Witt condition. For the non-Witt case, this operator remains essentially self-adjoint (in contrast to the Hodge-de Rham operator) and it has a well-defined index which we conjecture will also compute $$\sigma _{S^1}(M)$$ .

Induced Dirac-Schrödinger operators on $S^1$-semi-free quotients

John Lott has computed an integer-valued signature for the orbit space of a compact orientable $(4k+1)$ manifold with a semi-free $S^1$-action, which is a homotopy invariant of that space, but he did not construct a Dirac type operator which has this signature as its index. In this Thesis, we construct such operator on the orbit space, a Thom-Mather stratified space with one singular stratum of positive dimension, and we show that it is essentially unique and that its index coincides with Lott's signature, at least when the stratified space satisfies the so called Witt condition. We call this operator the induced Dirac-Schrodinger operator. The strategy of the construction is to push down an appropriate $S^1$-invariant first order transversally elliptic operator to the quotient space. The Witt condition, a topological condition which in this case depends on the codimension of the fixed point set, has various analytic consequences. In particular, when not satisfied, the Hodge-de Rham operator on the quotient space does not need to be essentially self-adjoint and therefore a choice of boundary conditions is required. This choice freedom is not natural in view of the fact that Lott's signature is well defined independently of the Witt condition. The Dirac-Schrodinger operator constructed in this Thesis differs from the Hodge-de Rham operator by a zero order term which ensures it to be essentially self-adjoint. Moreover, this zero order term anti-commutes with the chirality involution allowing the whole operator to split so that the index can be computed even if the Witt condition is not satisfied.

On the Chern-Gauss-Bonnet Theorem and Conformally Twisted Spectral Triples for C<sup>*</sup>-Dynamical Systems

The analog of the Chern-Gauss-Bonnet theorem is studied for a $C^*$-dynamical system consisting of a $C^*$-algebra $A$ equipped with an ergodic action of a compact Lie group $G$. The structure of the Lie algebra $\mathfrak{g}$ of $G$ is used to interpret the Chevalley-Eilenberg complex with coefficients in the smooth subalgebra $\mathcal{A} \subset A$ as noncommutative differential forms on the dynamical system. We conformally perturb the standard metric, which is associated with the unique $G$-invariant state on $A$, by means of a Weyl conformal factor given by a positive invertible element of the algebra, and consider the Hermitian structure that it induces on the complex. A Hodge decomposition theorem is proved, which allows us to relate the Euler characteristic of the complex to the index properties of a Hodge-de Rham operator for the perturbed metric. This operator, which is shown to be selfadjoint, is a key ingredient in our construction of a spectral triple on $\mathcal{A}$ and a twisted spectral triple on its opposite algebra. The conformal invariance of the Euler characteristic is interpreted as an indication of the Chern-Gauss-Bonnet theorem in this setting. The spectral triples encoding the conformally perturbed metrics are shown to enjoy the same spectral summability properties as the unperturbed case.

Partial collapsing and the spectrum of the Hodge–de Rham operator

The goal of the present paper is to calculate the limit spectrum of the Hodge-de Rham operator under the perturbation of collapsing one part of a manifold obtained by gluing together two manifolds with the same boundary. It appears to take place in the general problem of blowing up conical singularities as introduced in \cite{Maz} and \cite{Row1,Row2}.

The spectral theory of the Yano rough Laplacian with some of its applications

J.H. Sampson has defined the Laplacian \(\triangle _\mathrm{sym}\) acting on the space of symmetric covariant tensors on Riemannian manifolds. This operator is an analogue of the well-known Hodge–de Rham Laplacian \(\triangle \) which acts on the space of skew-symmetric covariant tensors on Riemannian manifolds. In the present paper, we perform properties analysis of Sampson operator which acts on one-forms. We show that the Sampson operator is the Yano rough Laplacian. We also find the biggest lower bounds of spectra of the Yano and Hodge–de Rham operators and obtain estimates of their multiplicities for the space of one-forms on compact Riemannian manifolds with negative and positive Ricci curvatures, respectively.

Ricci-flow-conjugated initial data sets for Einstein equations

We discuss a natural form of Ricci-flow conjugation between two distinct general relativistic data sets given on a compact $n \ge 3$-dimensional manifold $\Sigma$. We establish the existence of the relevant entropy functionals for the matter and geometrical variables, their monotonicity properties, and the associated convergence in the appropriate sense. We show that in such a framework there is a natural mode expansion generated by the spectral resolution of the Ricci conjugate Hodge–DeRham operator. This mode expansion allows one to compare the two distinct data sets and gives rise to a computable heat kernel expansion of the fluctuations among the fields defining the data. In particular, this shows that Ricciflow conjugation entails a natural form of $L^2$ parabolic averaging of one data set with respect to the other with a number of desirable properties: (i) It preserves the dominant energy condition; (ii) It is localized by a heat kernel whose support sets the scale of averaging; (iii) It is characterized by a set of balance functionals, that allow the analysis of its entropic stability.

A distributional formulation of electromagnetic fields and sources in the presence of media discontinuities

A self-contained formulation of Maxwell's theory of electromagnetic fields and sources is presented in the language of distributional forms. Properties of a fundamental double-form of bi-degree (p,p) for p ≥ 0 are reviewed in order to establish a computational framework for analysing equations involving the Hodge-de Rham operator for p–forms on space or spacetime and singular sources. A constructive approach to Dirac distributions on (moving) submanifolds embedded in R3 is developed in terms of (Leray) forms generated by the geometry of the embedding and illustrative examples presented from problems in electrostatics, magnetostatics and radiating point charges. The formulation offers a straightforward analysis of the relativistic jump conditions across static and moving interfaces where certain fields become discontinuous and provides a general methodology for electromagnetic modeling where possibly time dependent sources of certain physical attributes, such as electric charge, electric current and polarization or magnetization, are concentrated on localized regions in space or spacetime.

Differential form valued forms and distributional electromagnetic sources

Properties of a fundamental double-form of bi-degree $(p,p)$ for $p\ge 0$ are reviewed in order to establish a distributional framework for analysing equations of the form $$\Delta \Phi + \lambda^2 \Phi = {\cal S} $$ where $\Delta$ is the Hodge-de Rham operator on $p-$forms $ \Phi $ on ${\bf R}^3$. Particular attention is devoted to singular distributional solutions that arise when the source $ {\cal S}$ is a singular $p-$form distribution. A constructive approach to Dirac distributions on (moving) submanifolds embedded in ${\bf R}^3$ is developed in terms of (Leray) forms generated by the geometry of the embedding. This framework offers a useful tool in electromagnetic modeling where the possibly time dependent sources of certain physical attributes, such as electric charge, electric current and polarization or magnetization, are concentrated on localized regions in space.

Gaps in the differential forms spectrum on cyclic coverings

We are interested in the spectrum of the Hodge–de Rham operator on a \({\mathbb{Z}}\) -covering X over a compact manifold M of dimension n + 1. Let Σ be a hypersurface in M which does not disconnect M and such that M − Σ is a fundamental domain of the covering. If the cohomology group H n/2(Σ) is trivial, we can construct for each \({N \in \mathbb{N}}\) a metric g = g N on M, such that the Hodge–de Rham operator on the covering (X, g) has at least N gaps in its (essential) spectrum. If \({H^{n/2}(\Sigma) \ne 0}\) , the same statement holds true for the Hodge–de Rham operators on p-forms provided \({p \notin \{n/2, n/2 + 1\}}\) .

The Plancherel measure for p-forms in real hyperbolic spaces

Abstract The Plancherel measure is calculated for antisymmetric tensor fields ( p -forms) on the real hyperbolic space H N . The Plancherel measure gives the spectral distribution of the eigenvalues ω λ of the Hodge-de Rham operator Δ = dδ + δd . The spectrum of Δ is purely continuous except for N even and p= 1 2 N . For N odd the Plancherel measure μ(λ) is a polynomial in λ 2 . For N even the continuous part μ(λ) of the Plancherel measure is a meromorphic function in the complex λ-plane with simple poles on the imaginary axis. A simple relation between the residues of μ(λ) at these poles and the (known) degeneracies of Δ on the N -sphere is obtained. A similar relation between μ(λ) at discrete imaginary values of λ and the degeneracies of Δ on S N is found for N odd. The p -form ζ-function, defined as a Mellin transform of the trace of the heat kernel, is considered. A relation between the ζ-functions on S N and H N is obtained by means of complex contours. We construct square-integrable harmonic k -forms on H 2k . These k -forms contribute a discrete part to the spectrum of Δ and are related to the discrete series of SO 0 (2 k , 1). We also give a group-theoretic derivation of μ(λ) based on the Plancherel formula for the Lorentz group SO 0 ( N , 1).

Quantized antisymmetric tensor fields and self-consistent dimensional reduction in higher-dimensional spacetimes

The problem of quantizing an antisymmetric tensor field on a higher-dimensional gravitational background of the form M 4 × S N (where M 4 is an arbitrary 4-dimensional spacetime) is examined. Terms appearing in the effective gravitational action are found for such a theory. This effective gravitational action is then used to find the necessary conditions for the existence of solutions to the higher-dimensional dield equations of the form R 4 × S N . The cases where the background antisymmetric tensor field vanishes or else is of the Freund-Rubin form are both considered. It is mentioned how this analysis is modified when gauge fields are included in a Kaluza-Klein metric on the higher-dimensional space. Two appendices contain details of our conventions concerning differential forms, and results for the eigenvalues (and their multiplicities) of the Hodge de Rham operator acting on exact and coexact forms on a sphere. In a third appendix it is shown how the vacuum energy for photons in the Einstein static universe may be obtained using the methods of this paper.