Hodge Star Research Articles

Spectral Numerical Exterior Calculus Methods for Differential Equations on Radial Manifolds

We develop exterior calculus approaches for partial differential equations on radial manifolds. We introduce numerical methods that approximate with spectral accuracy the exterior derivative $\mathbf{d}$, Hodge star $\star$, and their compositions. To achieve discretizations with high precision and symmetry, we develop hyperinterpolation methods based on spherical harmonics and Lebedev quadrature. We perform convergence studies of our numerical exterior derivative operator $\overline{\mathbf{d}}$ and Hodge star operator $\overline{\star}$ showing each converge spectrally to $\mathbf{d}$ and $\star$. We show how the numerical operators can be naturally composed to formulate general numerical approximations for solving differential equations on manifolds. We present results for the Laplace-Beltrami equations demonstrating our approach.

Hyperscaling violation, quasinormal modes and shear diffusion

We study quasinormal modes of shear gravitational perturbations for hyperscaling violating Lifshitz theories, with Lifshitz and hyperscaling violating exponents z and θ. The lowest quasinormal mode frequency yields a shear diffusion constant which is in agreement with that obtained in previous work by other methods. In particular for theories with z < di + 2 − θ where di is the boundary spatial dimension, the shear diffusion constant exhibits power-law scaling with temperature, while for z = di + 2 − θ, it exhibits logarithmic scaling. We then calculate certain 2-point functions of the dual energy-momentum tensor holographically for z ≤ di + 2 − θ, identifying the diffusive poles with the quasinormal modes above. This reveals universal behaviour η/s = 1/4π for the viscosity-to-entropy-density ratio for all z ≤ di + 2 − θ.

A note on the sphere free energy of p-form gauge theory and Hodge duality

We consider a free p-form gauge theory on a d-dimensional sphere of radius R and calculate its free energy. We perform the calculation for generic values of p and obtain the free energy as a function of and R. The result contains a term with a coefficient proportional to , which is consistent with lack of conformal invariance for p form theories in dimensions other than . We also compare the result for p-form and -form theory which are classically Hodge dual to each other in d dimensions and find that they agree for odd values of d. Also, for even d, we find that the results disagree by an amount that is consistent with the reported values in the literature.

Observables and dispersion relations in \u03ba-Minkowski spacetime

We revisit the notion of quantum Lie algebra of symmetries of a noncommutative spacetime, its elements are shown to be the generators of infinitesimal transformations and are naturally identified with physical observables. Wave equations on noncommutative spaces are derived from a quantum Hodge star operator.This general noncommutative geometry construction is then exemplified in the case of κ-Minkowski spacetime. The corresponding quantum Poincaré-Weyl Lie algebra of in-finitesimal translations, rotations and dilatations is obtained. The d’Alembert wave operator coincides with the quadratic Casimir of quantum translations and it is deformed as in Deformed Special Relativity theories. Also momenta (infinitesimal quantum translations) are deformed, and correspondingly the Einstein-Planck relation and the de Broglie one. The energy-momentum relations (dispersion relations) are consequently deduced. These results complement those of the phenomenological literature on the subject.

Black holes in vector-tensor theories

We study static and spherically symmetric black hole (BH) solutions in second-order generalized Proca theories with nonminimal vector field derivative couplings to the Ricci scalar, the Einstein tensor, and the double dual Riemann tensor. We find concrete Lagrangians which give rise to exact BH solutions by imposing two conditions of the two identical metric components and the constant norm of the vector field. These exact solutions are described by either Reissner-Nordström (RN), stealth Schwarzschild, or extremal RN solutions with a non-trivial longitudinal mode of the vector field. We then numerically construct BH solutions without imposing these conditions. For cubic and quartic Lagrangians with power-law couplings which encompass vector Galileons as the specific cases, we show the existence of BH solutions with the difference between two non-trivial metric components. The quintic-order power-law couplings do not give rise to non-trivial BH solutions regular throughout the horizon exterior. The sixth-order and intrinsic vector-mode couplings can lead to BH solutions with a secondary hair. For all the solutions, the vector field is regular at least at the future or past horizon. The deviation from General Relativity induced by the Proca hair can be potentially tested by future measurements of gravitational waves in the nonlinear regime of gravity.

A fluid/gravity prescription of the post-Newtonian parameter

The KSS bound for the shear viscosity-to-entropy density ratio is employed in the fluid/gravity correspondence setup, to provide a theoretical bound for the post-Newtonian parameter in the Casadio-Fabbri-Mazzacurati black string solution that matches the observational bound [1]. This derived bound is here shown to vary according to the black string temperature, and, although the post-Newtonian parameter reaches bigger values as the temperature increases for fixed black string masses, it is still into the observational/experimental bound range. The Casadio-Fabbri-Mazzacurati-AdS black brane is shown to converge to the Schwarzschild-AdS black brane, for sufficiently high order expansion of the dual boundary energy-momentum tensor.

Mirror symmetry in emergent gravity

Given a six-dimensional symplectic manifold (M,B), a nondegenerate, co-closed four-form C introduces a dual symplectic structure B˜=⁎C independent of B via the Hodge duality ⁎. We show that the doubling of symplectic structures due to the Hodge duality results in two independent classes of noncommutative U(1) gauge fields by considering the Seiberg–Witten map for each symplectic structure. As a result, emergent gravity suggests a beautiful picture that the variety of six-dimensional manifolds emergent from noncommutative U(1) gauge fields is doubled. In particular, the doubling for the variety of emergent Calabi–Yau manifolds allows us to arrange a pair of Calabi–Yau manifolds such that they are mirror to each other. Therefore, we argue that the mirror symmetry of Calabi–Yau manifolds is the Hodge theory for the deformation of symplectic and dual symplectic structures.

Extension of the correspondence to the infinite-dimensional chiral spinors and self dual tensors

The spinor representations of the orthosymplectic Lie superalgebras are considered and constructed. These are infinite-dimensional irreducible representations, of which the superdimension coincides with the dimension of the spinor representation of . Next, we consider the self dual tensor representations of and their generalizations: these are also infinite-dimensional and correspond to the highest irreducible component of the pth power of the spinor representation. We determine the character of these representations, and deduce a superdimension formula. From this, it follows that also for these representations the correspondence holds.

Hodge Star as Braided Fourier Transform

We study super-braided Hopf algebras Λ primitively generated by finite-dimensional right crossed (or Drinfeld-Radford-Yetter) modules Λ1 over a Hopf algebra A which are quotients of the augmentation ideal A+ as a crossed module by right multiplication and the adjoint coaction. Here super-bosonisation provides a bicovariant differential graded algebra on A. We introduce Λmax providing the maximal prolongation, while the canonical braided-exterior algebra Λmin = B−(Λ1) provides the Woronowicz exterior calculus. In this context we introduce a Hodge star operator ♯ by super-braided Fourier transform on B−(Λ1) and left and right interior products by braided partial derivatives. Our new approach to the Hodge star (a) differs from previous approaches in that it is canonically determined by the differential calculus and (b) differs on key examples, having order 3 in middle degree on k[S3] with its 3D calculus and obeying the q-Hecke relation ♯2 = 1 + (q − q−1)♯ in middle degree on kq[SL2] with its 4D calculus. Our work also provides a Hodge map on quantum plane calculi and a new starting point for calculi on coquasitriangular Hopf algebras A whereby any subcoalgebra mathcal {L}subseteq A defines a sub-braided Lie algebra and {Lambda }^{1}subseteq mathcal {L}^{*} provides the required data A+ → Λ1.

Nonminimal de Rham–Hodge operators and non-commutative residue

In this paper, we get a Kastler-Kalau-Walze type theorem associated to nonminimal de Rham-Hodge operators on compact manifolds with boundary. We give two kinds of operator-theoretic explanations of the gravitational action in the case of four dimensional compact manifolds with flat boundary.

Spin Gauge Interactions as a Topological Mechanism of Superconductivity

We talk about a low energy, effective, topological theory of superconductivity in which a topological mass term is radiatively induced in one loop effective action. In this field theoretic model, an antisymmetric tensor field couples with the vorticity current of charged Dirac fermions in the Lagrangian. The fermion loop generates a coupling between the gauge field and the antisymmetric tensor field below an ultraviolet cut-off. The spin interactions mediated by the antisymmetric tensor field induces a mass for the photon field indicating Meissner effect. The dual antisymmetric tensor field produces a current which satisfies the relativistic version of the London equations of superconductivity. In the non-relativistic limit, the static effective potential shows a linear, always attractive term between two electrons. Thus, the theory can be considered as an alternative, low energy, effective field theory of superconductivity without spontaneous symmetry breaking.

Calabi-Yau Manifolds, Hermitian Yang-Mills Instantons, and Mirror Symmetry

We address the issue of why Calabi-Yau manifolds exist with a mirror pair. We observe that the irreducible spinor representation of the Lorentz group Spin(6) requires us to consider the vector spaces of two forms and four forms on an equal footing. The doubling of the two-form vector space due to the Hodge duality doubles the variety of six-dimensional spin manifolds. We explore how the doubling is related to the mirror symmetry of Calabi-Yau manifolds. Via the gauge theory formulation of six-dimensional Riemannian manifolds, we show that the curvature tensor of a Calabi-Yau manifold satisfies the Hermitian Yang-Mills equations on the Calabi-Yau manifold. Therefore, the mirror symmetry of Calabi-Yau manifolds can be recast as the mirror pair of Hermitian Yang-Mills instantons. We discuss the mirror symmetry from the gauge theory perspective.

Global gravitational anomalies and transport

We investigate the constraints imposed by global gravitational anomalies on parity odd induced transport coefficients in even dimensions for theories with chiral fermions, gravitinos and self dual tensors. The eta-invariant for the large diffeomorphism corresponding to the T transformation on a torus constraints the coefficients in the thermal effective action up to mod 2. We show that the result obtained for the parity odd transport for gravitinos using global anomaly matching is consistent with the direct perturbative calculation. In d = 6 we see that the second Pontryagin class in the anomaly polynomial does not contribute to the eta-invariant which provides a topological explanation of this observation in the `replacement rule'. We then perform a direct perturbative calculation for the contribution of the self dual tensor in d = 6 to the parity odd transport coefficient using the Feynman rules proposed by Gaume and Witten. The result for the transport coefficient agrees with that obtained using matching of global anomalies.

Integral representations on supermanifolds: super Hodge duals, PCOs and Liouville forms

We present a few types of integral transforms and integral representations that are very useful for extending to supergeometry many familiar concepts of differential geometry. Among them we discuss the construction of the super Hodge dual, the integral representation of picture changing operators of string theories and the construction of the super-Liouville form of a symplectic supermanifold.

Reliable Dual Tensor Model Estimation in Single and Crossing Fibers Based on Jeffreys Prior.

PurposeThis paper presents and studies a framework for reliable modeling of diffusion MRI using a data-acquisition adaptive prior.MethodsAutomated relevance determination estimates the mean of the posterior distribution of a rank-2 dual tensor model exploiting Jeffreys prior (JARD). This data-acquisition prior is based on the Fisher information matrix and enables the assessment whether two tensors are mandatory to describe the data. The method is compared to Maximum Likelihood Estimation (MLE) of the dual tensor model and to FSL’s ball-and-stick approach.ResultsMonte Carlo experiments demonstrated that JARD’s volume fractions correlated well with the ground truth for single and crossing fiber configurations. In single fiber configurations JARD automatically reduced the volume fraction of one compartment to (almost) zero. The variance in fractional anisotropy (FA) of the main tensor component was thereby reduced compared to MLE. JARD and MLE gave a comparable outcome in data simulating crossing fibers. On brain data, JARD yielded a smaller spread in FA along the corpus callosum compared to MLE. Tract-based spatial statistics demonstrated a higher sensitivity in detecting age-related white matter atrophy using JARD compared to both MLE and the ball-and-stick approach.ConclusionsThe proposed framework offers accurate and precise estimation of diffusion properties in single and dual fiber regions.

Studying Dynamic Myofiber Aggregate Reorientation in Dilated Cardiomyopathy Using In Vivo Magnetic Resonance Diffusion Tensor Imaging.

The objective of this study is to assess the dynamic alterations of myocardial microstructure and strain between diastole and systole in patients with dilated cardiomyopathy relative to healthy controls using the magnetic resonance diffusion tensor imaging, myocardial tagging, and biomechanical modeling. Dual heart-phase diffusion tensor imaging was successfully performed in 9 patients and 9 controls. Tagging data were acquired for the diffusion tensor strain correction and cardiac motion analysis. Mean diffusivity, fractional anisotropy, and myocyte aggregate orientations were compared between both cohorts. Cardiac function was assessed by left ventricular ejection fraction, torsion, and strain. Computational modeling was used to study the impact of cardiac shape on fiber reorientation and how fiber orientations affect strain. In patients with dilated cardiomyopathy, a more longitudinal orientation of diastolic myofiber aggregates was measured compared with controls. Although a significant steepening of helix angles (HAs) during contraction was found in the controls, consistent change in HAs during contraction was absent in patients. Left ventricular ejection fraction, cardiac torsion, and strain were significantly lower in the patients compared with controls. Computational modeling revealed that the dilated heart results in reduced HA changes compared with a normal heart. Reduced torsion was found to be exacerbated by steeper HAs. Diffusion tensor imaging revealed reduced reorientation of myofiber aggregates during cardiac contraction in patients with dilated cardiomyopathy relative to controls. Left ventricular remodeling seems to be an important factor in the changes to myocyte orientation. Steeper HAs are coupled with a worsening in strain and torsion. Overall, the findings provide new insights into the structural alterations in patients with dilated cardiomyopathy.

Collocated electrodynamic FDTD schemes using overlapping Yee grids and higher-order Hodge duals

The collocated Lebedev grid has previously been proposed as an alternative to the Yee grid for electromagnetic finite-difference time-domain (FDTD) simulations. While it performs better in anisotropic media, it performs poorly in isotropic media because it is equivalent to four overlapping, uncoupled Yee grids. We propose to couple the four Yee grids and fix the Lebedev method using discrete exterior calculus (DEC) with higher-order Hodge duals. We find that higher-order Hodge duals do improve the performance of the Lebedev grid, but they also improve the Yee grid by a similar amount. The effectiveness of coupling overlapping Yee grids with a higher-order Hodge dual is thus questionable. However, the theoretical foundations developed to derive these methods may be of interest in other problems.

Perturbative construction of stationary Randall-Sundrum II black holes on a 5-brane

We numerically construct large Randall-Sundrum II brane black holes in 4 and 5 dimensions from associated AdS/CFT spacetimes. Our solutions are leading order perturbations of a representative of the boundary conformal structure of the AdS spacetime sourced by the dual CFT stress tensor. The 4-dimensional solutions are static perturbations of the Euclidean Schwarzschild metric, while the 5-dimensional solutions are perturbations of the Myers-Perry metric with equal angular momenta. We compare the former with previous numerical results for Randall-Sundrum bulk black holes and find good agreement down to a horizon radius of about rH ~30l. The latter are the first numerical results pertaining to rotating Randall-Sundrum black holes. They have the same entropy, but a larger horizon area than Myers-Perry black holes of the same mass and angular momentum.

Open/closed string duality and relativistic fluids

We propose an open/closed string duality in general backgrounds extending previous ideas about open string completeness by Ashoke Sen. Our proposal sets up a general version of holography that works in gravity as a tomographic principle. We argue, in particular, that previous expectations of a supergravity/Dirac-Born-Infeld (DBI) correspondence are naturally embedded in this conjecture and can be tested in a well-defined manner. As an example, we consider the correspondence between open string field theories on extremal D-brane setups in flat space in the large-N, large 't Hooft limit, and asymptotically flat solutions in ten-dimensional type II supergravity. We focus on a convenient long-wavelength regime, where specific effects of higher-spin open string modes can be traced explicitly in the dual supergravity computation. For instance, in this regime we show how the full abelian DBI action arises from supergravity as a straightforward reformulation of relativistic hydrodynamics. In the example of a (2+1)-dimensional open string theory this reformulation involves an abelian Hodge duality. We also point out how different deformations of the DBI action, related to higher-derivative corrections and non-abelian effects, can arise in this context as deformations in corresponding relativistic hydrodynamics.

Orthogonal dual tensor method for solving the AX = XB sensor calibration problem

The AX=XB sensor calibration problem is very important in multiple fields such as robotics, computer vision or machine vision. In this type of problem, A and B are parameterizations of rigid body motions, while X is an unknown transformation. This research proposes a new approach for solving the AX=XB sensor calibration problem by mapping the SE3 classic formulation into an orthogonal dual tensors set SO̲3representation. Using an isomorphism between SE3 and SO̲3, together with a set of properties from SO̲3, we formulate closed-form simultaneous solutions for data recorded from any number of displacements. These solutions are free of coordinates and can easily be put into practice. Implementation algorithms are presented and numerical results are discussed in order to evaluate the performance of the proposed solutions.