Harmonic space and quaternionic manifolds

We find a principle of harmonic analyticity underlying the quaternionic (quaternion-K\"ahler) geometry and solve the differential constraints which define this geometry. To this end the original $4n$-dimensional quaternionic manifold is extended to a bi-harmonic space. The latter includes additional harmonic coordinates associated with both the tangent local $Sp(1)$ group and an extra rigid $SU(2)$ group rotating the complex structures. Then the constraints can be rewritten as integrability conditions for the existence of an analytic subspace in the bi-harmonic space and solved in terms of two unconstrained potentials on the analytic subspace. Geometrically, the potentials have the meaning of vielbeins associated with the harmonic coordinates. We also establish a one-to-one correspondence between the quaternionic spaces and off-shell $N=2$ supersymmetric sigma-models coupled to $N=2$ supergravity. The general $N=2$ sigma-model Lagrangian when written in the harmonic superspace is composed of the quaternionic potentials. Coordinates of the analytic subspace are identified with superfields describing $N=2$ matter hypermultiplets and a compensating hypermultiplet of $N=2$ supergravity. As an illustration we present the potentials for the symmetric quaternionic spaces.Comment: 44 pages, LATEX, JHU-TIPAC-920023, ENSLAPP-L-405-92, MPI-Ph/92-8

Four Dimensional Integrable Theories

There exist many four dimensional integrable theories. They include self-dual gauge and gravity theories, all their extended supersymmetric generalisations, as well the full (non-self-dual) N=3 super Yang-Mills equations. We review the harmonic space formulation of the twistor transform for these theories which yields a method of producing explicit connections and metrics. This formulation uses the concept of harmonic space analyticity which is closely related to that of quaternionic analyticity. (Talk by V. Ogievetsky at the G\"ursey Memorial Conference I, Istanbul, June 1994)Comment: 11 pages, late

On the growth of von Neumann dimension of harmonic spaces of semipositive line bundles over covering manifolds

We study the harmonic space of line bundle valued forms over a covering manifold with a discrete group action $\Gamma$, and obtain an asymptotic estimate for the $\Gamma$-dimension of the harmonic space with respect to the tensor times $k$ in the holomorphic line bundle $L^{k}\otimes E$ and the type $(n,q)$ of the differential form, when $L$ is semipositive. In particular, we estimate the $\Gamma$-dimension of the corresponding reduced $L^2$-Dolbeault cohomology group. Essentially, we obtain a local estimate of the pointwise norm of harmonic forms with valued in semipositive line bundles over Hermitian manifolds

Protein signatures using electrostatic molecular surfaces in harmonic space

We developed a novel method based on the Fourier analysis of protein molecular surfaces to speed up the analysis of the vast structural data generated in the post-genomic era. This method computes the power spectrum of surfaces of the molecular electrostatic potential, whose three-dimensional coordinates have been either experimentally or theoretically determined. Thus we achieve a reduction of the initial three-dimensional information on the molecular surface to the one-dimensional information on pairs of points at a fixed scale apart. Consequently, the similarity search in our method is computationally less demanding and significantly faster than shape comparison methods. As proof of principle, we applied our method to a training set of viral proteins that are involved in major diseases such as Hepatitis C, Dengue fever, Yellow fever, Bovine viral diarrhea and West Nile fever. The training set contains proteins of four different protein families, as well as a mammalian representative enzyme. We found that the power spectrum successfully assigns a unique signature to each protein included in our training set, thus providing a direct probe of functional similarity among proteins. The results agree with established biological data from conventional structural biochemistry analyses.Comment: 9 pages, 10 figures Published in PeerJ (2013), https://peerj.com/articles/185

Local symmetry of harmonic spaces as determined by the spectra of small geodesic spheres

We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm $|\nabla R|$ of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determine whether the space is locally symmetric. For the proof we use the first few heat invariants and consider certain coefficients in the radial power series expansions of the curvature invariants $|R|^2$ and $|Ric|^2$ of the geodesic spheres. Moreover, we obtain analogous results for geodesic balls with either Dirichlet or Neumann boundary conditions.Comment: 18 pages, LaTeX. Added a few lines in the introduction, corrected a few typos. Final version. Accepted for publication in GAF

CMB Lensing Reconstruction in Real Space

We explore the reconstruction of the gravitational lensing field of the cosmic microwave background in real space showing that very little statistical information is lost when estimators of short range on the celestial sphere are used in place of the customary estimators in harmonic space, which are nonlocal and in principle require a simultaneous analysis of the entire sky without any cuts or excisions. Because virtually all the information relevant to lensing reconstruction lies on angular scales close to the resolution scale of the sky map, the gravitational lensing dilatation and shear fields (which unlike the deflection field or lensing potential are directly related to the observations in a local manner) may be reconstructed by means of quadratic combinations involving only very closely separated pixels. Even though harmonic space provides a more natural context for understanding lensing reconstruction theoretically, the real space methods developed here have the virtue of being faster to implement and are likely to prove useful for analyzing realistic maps containing a galactic cut and possibly numerous small excisions to exclude point sources that cannot be reliably subtracted.Comment: 21 pages, 8 figure