Winding Number in String Field Theory

Motivated by the similarity between cubic string field theory (CSFT) and the Chern-Simons theory in three dimensions, we study the possibility of interpreting N=(\pi^2/3)\int(U Q_B U^{-1})^3 as a kind of winding number in CSFT taking quantized values. In particular, we focus on the expression of N as the integration of a BRST-exact quantity, N=\int Q_B A, which vanishes identically in naive treatments. For realizing non-trivial N, we need a regularization for divergences from the zero eigenvalue of the operator K in the KBc algebra. This regularization must at same time violate the BRST-exactness of the integrand of N. By adopting the regularization of shifting K by a positive infinitesimal, we obtain the desired value N[(U_tv)^{\pm 1}]=\mp 1 for U_tv corresponding to the tachyon vacuum. However, we find that N[(U_tv)^{\pm 2}] differs from \mp 2, the value expected from the additive law of N. This result may be understood from the fact that \Psi=U Q_B U^{-1} with U=(U_tv)^{\pm 2} does not satisfy the CSFT EOM in the strong sense and hence is not truly a pure-gauge in our regularization.Comment: 20 pages, no figures; v2: references added, minor change

Far-Ultraviolet Number Counts of Field Galaxies

The far-ultraviolet (FUV) number counts of galaxies constrain the evolution of the star-formation rate density of the universe. We report the FUV number counts computed from FUV imaging of several fields including the Hubble Ultra Deep Field, the Hubble Deep Field North, and small areas within the GOODS-North and -South fields. These data were obtained with the Hubble Space Telescope Solar Blind Channel of the Advance Camera for Surveys. The number counts sample a FUV AB magnitude range from 21-29 and cover a total area of 15.9 arcmin^2, ~4 times larger than the most recent HST FUV study. Our FUV counts intersect bright FUV GALEX counts at 22.5 mag and they show good agreement with recent semi-analytic models based on dark matter "merger trees" by Somerville et al. (2011). We show that the number counts are ~35% lower than in previous HST studies that use smaller areas. The differences between these studies are likely the result of cosmic variance; our new data cover more lines of sight and more area than previous HST FUV studies. The integrated light from field galaxies is found to contribute between 65.9 +/-8 - 82.6 +/-12 photons/s/cm^2/sr/angstrom to the FUV extragalactic background. These measurements set a lower limit for the total FUV background light.Comment: Accepted for publication in ApJ, including 34 pages, 6 figures, and 2 table

Far-Ultraviolet Number Counts on Field Galaxies

The far-ultraviolet (FUV) number counts of galaxies constrain the evolution of the star formation rate density of the universe. We report the FUV number counts computed from FUV imaging of several fields including the Hubble Ultra Deep Field, the Hubble Deep Field North, and small areas within the GOODS-North and South fields. These data were obtained with the Hubble Space Telescope (HST) Solar Blind Channel of the Advance Camera for Surveys. The number counts sample an FUV AB magnitude range from 21 to 29 and cover a total area of 15.9 arcmin^2, ~4 times larger than the most recent HST FUV study. Our FUV counts intersect bright FUV Galaxy Evolution Explorer counts at 22.5 mag and they show good agreement with recent semi-analytic models based on dark matter "merger trees" by R. S. Somerville et al. We show that the number counts are ~35% lower than in previous HST studies that use smaller areas. The differences between these studies are likely the result of cosmic variance; our new data cover more lines of sight and more area than previous HST FUV studies. The integrated light from field galaxies is found to contribute between 65.9^(+8)_(–8) and 82.6^(+12)_(–)12 photons s^(–1) cm^(–2) sr^(–1) Å^(–1) to the FUV extragalactic background. These measurements set a lower limit for the total FUV background light

Galois cohomology of a number field is Koszul

We prove that the Milnor ring of any (one-dimensional) local or global field K modulo a prime number l is a Koszul algebra over Z/l. Under mild assumptions that are only needed in the case l=2, we also prove various module Koszulity properties of this algebra. This provides evidence in support of Koszulity conjectures that were proposed in our previous papers. The proofs are based on the Class Field Theory and computations with quadratic commutative Groebner bases (commutative PBW-bases).Comment: LaTeX 2e, 25 pages; v.2: minor grammatic changes; v.3: classical references added, remark inserted in subsection 1.6, details of arguments added in subsections 1.4, 1.7 and sections 5 and 6; v.4: still more misprints corrected, acknowledgement updated, a sentence inserted in section 4, a reference added -- this is intended as the final versio

Quantitative Riemann existence theorem over a number field

Given a covering of the projective line with ramifications defined over a number field, we define a plain model of the algebraic curve realizing the Riemann existence theorem for this covering, and bound explicitly the defining equation of this curve and its definition field.Comment: 23 pages, version 4, minor change