WITTEN DEFORMATION FOR NONCOMPACT MANIFOLDS WITH BOUNDED GEOMETRY

Motivated by the Landau–Ginzburg model, we study the Witten deformation on a noncompact manifold with bounded geometry, together with some tameness condition on the growth of the Morse function f near infinity. We prove that the cohomology of the Witten deformation $d_{Tf}$ acting on the complex of smooth $L^2$ forms is isomorphic to the cohomology of the Thom–Smale complex of f as well as the relative cohomology of a certain pair $(M, U)$ for sufficiently large T. We establish an Agmon estimate for eigenforms of the Witten Laplacian which plays an essential role in identifying these cohomologies via Witten’s instanton complex, defined in terms of eigenspaces of the Witten Laplacian for small eigenvalues. As an application, we obtain the strong Morse inequalities in this setting.

Accurate modeling of the strong and weak lensing profiles for the galaxy clusters Abell 1689 and 1835

An accurate, spherically symmetric description of the mass distribution is presented for two quite virialized galaxy clusters, Abell 1689 and Abell 1835. A suitable regularization of the small eigenvalues of the covariance matrices is introduced. A stretched exponential profile is assumed for the brightest cluster galaxy. A similar stretched exponential for the dark matter and halo galaxies combined, functions well, as do thermal fermions for the dark matter and a standard profile for the halo galaxies. To discriminate between them, sensitive tests have been identified and applied. A definite verdict can follow from sharp data near the cluster centers and beyond 1 Mpc.

Monopole Floer Homology, Eigenform Multiplicities, and the Seifert–Weber Dodecahedral Space

We show that the Seifert–Weber dodecahedral space $\textsf{SW}$ is a monopole Floer homology $L$-space. The proof relies on our approach to study Floer homology using hyperbolic geometry. Although $\textsf{SW}$ is significantly larger than previous manifolds studied with this technique, we overcome computational complexity issues inherent to our method by exploiting the many symmetries of $\textsf{SW}$. In particular, we prove that small eigenvalues on coexact $1$-forms on $\textsf{SW}$ have large multiplicity.

Subjecting Dark Matter Candidates to the Cluster Test

Galaxy clusters, employed by Zwicky to demonstrate the existence of dark matter (DM), pose new stringent tests. First, the possibility is considered that merging clusters demonstrate that DM is self-interacting with cross-section [Formula: see text] 2[Formula: see text]cm2/gr. In that case, MACHOs, primordial black holes (PBHs) and light axions that build MACHOs are ruled out as cluster DM, while GeV and TeV WIMPs and keV sterile neutrinos are challenged. Next, recent strong lensing and X-ray gas data of the quite relaxed and quite spherical cluster A1835 are analyzed. These lensing data involve a covariance matrix of which the small eigenvalues have to be regularized. This is achieved with a new, general, parameter-free method: binning with respect to a model fit, and accounting for intra-bin fluctuations. This method allows to test the cases of DM with Maxwell–Boltzmann, Bose–Einstein and Fermi–Dirac (FD) distribution, next to Navarro–Frenck–White profiles. Fits to all these profiles are formally rejected at over [Formula: see text], except in the fermionic situation. The interpretation in terms of pseudo-Dirac neutrinos with mass of [Formula: see text][Formula: see text]eV/[Formula: see text] is consistent with results on the cluster A1689, with the DM fractions from WMAP, Planck and DES, and with the non-detection of neutrinoless double [Formula: see text]-decay. The predicted mass will be tested in the KATRIN and PTOLEMY experiments.