The Low-Dimensional Algebraic Cohomology of the Virasoro Algebra

The main aim of this article is to prove the one-dimensionality of the third algebraic cohomology of the Virasoro algebra with values in the adjoint module. We announced this result in a previous publication with only a sketch of the proof. The detailed proof is provided in the present article. We also show that the third algebraic cohomology of the Witt and the Virasoro algebra with values in the trivial module is one-dimensional. We consider purely algebraic cohomology, i.e. our results are independent of any topology chosen. The vanishing of the third algebraic cohomology of the Witt algebra with values in the adjoint module has already been proven by Ecker and Schlichenmaier.Comment: 21 page

D6-Brane Model Building on Z(2)xZ(6): MSSM-like and Left-Right Symmetric Models

We perform a systematic search for globally defined MSSM-like and left-right symmetric models on D6-branes on the T6/Z(2)xZ(6)xOR orientifold with discrete torsion. Our search is exhaustive for models that are independent of the value of the one free complex structure modulus. Preliminary investigations suggest that there exists one prototype of visible sector for MSSM-like and another for left-right symmetric models with differences arising from various hidden sector completions to global models. For each prototype, we provide the full matter spectrum, as well as the Yukawa and other three-point couplings needed to render vector-like matter states massive. This provides us with tentative explanations for the mass hierarchies within the quark and lepton sectors. We also observe that the MSSM-like models correspond to explicit realisations of the supersymmetric DFSZ axion model, and that the left-right symmetric models allow for global completions with either completely decoupled hidden sectors or with some messenger states charged under both visible and hidden gauge groups.Comment: 1+95 pages, 5 figures, 40 table

The low-dimensional algebraic cohomology of the Witt and the Virasoro algebra with values in natural modules

The main aim of this contribution is to compute the low-dimensional algebraic cohomology of the Witt and the Virasoro algebra with values in the adjoint and the trivial module. The last section includes results for the general tensor densities modules, presented without proof. One of our main results is that the third algebraic cohomology of the Witt algebra with values in the adjoint module vanishes, while it is one-dimensional for the Virasoro algebra. The first and the second algebraic cohomology of the Witt and the Virasoro algebra with values in tensor densities modules vanish for almost all modules. In the case they do not vanish, we give explicit expressions for the generating cocycles. In our work, we consider algebraic cohomology and not only the sub-complex of continuous cohomology, meaning we do not put any continuity constraints on the cochains. Consequently, our results are independent of any choice of an underlying topology, and valid for any concrete realizations of the considered Lie algebras

The Vanishing of Low-Dimensional Cohomology Groups of the Witt and the Virasoro algebra

A proof of the vanishing of the first and the third cohomology groups of the Witt algebra with values in the adjoint module is given. The proofs given in the present article are completely algebraic and independent of any underlying topology. They are a generalization of the ones provided by Schlichenmaier, who proved the vanishing of the second cohomology group using purely algebraic methods. In the case of the third cohomology group though, extra difficulties arise and the involved proofs are distinctly more complicated

Expanded encyclopaedias of DNA elements in the human and mouse genomes

All data are available on the ENCODE data portal: www.encodeproject. org. All code is available on GitHub from the links provided in the methods section. Code related to the Registry of cCREs can be found at https:// github.com/weng-lab/ENCODE-cCREs. Code related to SCREEN can be found at https://github.com/weng-lab/SCREEN.© The Author(s) 2020. The human and mouse genomes contain instructions that specify RNAs and proteins and govern the timing, magnitude, and cellular context of their production. To better delineate these elements, phase III of the Encyclopedia of DNA Elements (ENCODE) Project has expanded analysis of the cell and tissue repertoires of RNA transcription, chromatin structure and modification, DNA methylation, chromatin looping, and occupancy by transcription factors and RNA-binding proteins. Here we summarize these efforts, which have produced 5,992 new experimental datasets, including systematic determinations across mouse fetal development. All data are available through the ENCODE data portal (https://www.encodeproject.org), including phase II ENCODE1 and Roadmap Epigenomics2 data. We have developed a registry of 926,535 human and 339,815 mouse candidate cis-regulatory elements, covering 7.9 and 3.4% of their respective genomes, by integrating selected datatypes associated with gene regulation, and constructed a web-based server (SCREEN; http://screen.encodeproject.org) to provide flexible, user-defined access to this resource. Collectively, the ENCODE data and registry provide an expansive resource for the scientific community to build a better understanding of the organization and function of the human and mouse genomes.This work was supported by grants from the NIH under U01HG007019, U01HG007033, U01HG007036, U01HG007037, U41HG006992, U41HG006993, U41HG006994, U41HG006995, U41HG006996, U41HG006997, U41HG006998, U41HG006999, U41HG007000, U41HG007001, U41HG007002, U41HG007003, U54HG006991, U54HG006997, U54HG006998, U54HG007004, U54HG007005, U54HG007010 and UM1HG009442