On the nonexistence of Smith-Toda complexes

Let p be a prime. The Smith-Toda complex V(k) is a finite spectrum whose BP-homology is isomorphic to BP_*/(p,v_1,...,v_k). For example, V(-1) is the sphere spectrum and V(0) the mod p Moore spectrum. In this paper we show that if p > 5, then V((p+3)/2) does not exist and V((p+1)/2), if it exists, is not a ring spectrum. The proof uses the new homotopy fixed point spectral sequences of Hopkins and Miller.Comment: 10 pages, AMSLate

Geodesically Complete Metrics Induce Boundary Non-locality in Holography: Consequences for the Entanglement Entropy

We show explicitly that the full structure of IIB string theory is needed to remove the non-localities that arise in boundary conformal theories that border hyperbolic spaces on AdS$_5$. Specifically, using the Caffarelli/Silvestri\cite{caffarelli}, Graham/Zworski\cite{graham}, and Chang/Gonzalez\cite{chang:2010} extension theorems, we prove that the boundary operator conjugate to bulk p-forms with negative mass in geodesically complete metrics is inherently a non-local operator, specifically the fractional conformal Laplacian. The non-locality, which arises even in compact spaces, applies to any degree p-form such as a gauge field. We show that the boundary theory contains fractional derivatives of the longitudinal components of the gauge field if the gauge field in the bulk along the holographic direction acquires a mass via the Higgs mechanism. The non-locality is shown to vanish once the metric becomes incomplete, for example, either 1) asymptotically by adding N transversely stacked Dd-branes or 2) exactly by giving the boundary a brane structure and including a single transverse Dd-brane in the bulk. The original Maldacena conjecture within IIB string theory corresponds to the former. In either of these proposals, the location of the Dd-branes places an upper bound on the entanglement entropy because the minimal bulk surface in the AdS reduction is ill-defined at a brane interface. Since the brane singularities can be circumvented in the full 10-dimensional spacetime, we conjecture that the true entanglement entropy must be computed from the minimal surface in 10-dimensions, which is of course not minimal in the AdS$_5$ reduction.Comment: published versio

Wavelengths of the 3d6(5D)4s a6D - 3d6(5D)4p y6P Multiplet of Fe II (UV 8)

We investigate the wavenumber scale of Fe I and Fe II lines using new spectra recorded with Fourier transform spectroscopy and using a re-analysis of archival spectra. We find that standards in Ar II, Mg I, Mg II and Ge I give a consistent wavenumber calibration. We use the recalibrated spectra to derive accurate wavelengths for the a6D-y6P multiplet of Fe II (UV 8) using both directly measured lines and Ritz wavelengths. Lines from this multiplet are important for astronomical tests of the invariance of the fine structure constant on a cosmological time scale. We recommend a wavelength of 1608.45081 {\AA} with a one standard deviation uncertainty of 0.00007 {\AA} for the a6D9/2-y6P7/2 transition.Comment: 29 pages, 7 figures, 8 tables. Accepted for publication in J. Opt. Soc. Am

Ricci solitons - The equation point of view

We discuss some classification results for Ricci solitons, that is, self similar solutions of the Ricci Flow. Some simple proofs of known results will be presented. In detail, we will take the equation point of view, trying to avoid the tools provided by considering the dynamic properties of the Ricci flow.Comment: Revised Version - Corrected some mistakes and inaccuracie