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    πNσ\pi N \sigma Term and Quark Spin Content of the Nucleon

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    We report results of our calculation on the πNσ\pi N\sigma term and quark spin content of the nucleon on the quenched 163×2416^3 \times 24 lattice at β=6.0\beta = 6.0. The disconnected insertions which involve contributions from the sea quarks are calculated with the stochastic Z2Z_2 noise algorithm. As a physical test of the algorithm, we show that the forward matrix elements of the vector and pseudoscalar currents for the disconnected insertions are indeed consistent with the known results of zero. We tried the Wuppertal smeared source and found it to be more noisy than the point source. With unrenormalized mq=4.42(17)m_q=4.42(17)MeV, we find the πNσ\pi N\sigma term to be 39.2±5.239.2\pm 5.2 MeV. The strange quark condensate in the nucleon is large, i.e. NsˉsN=1.16±0.54\langle N|\bar{s}s|N\rangle = 1.16 \pm 0.54. For the quark spin content, we find Δu=0.78±0.07\Delta u =0.78\pm 0.07, Δd=0.42±0.07\Delta d =-0.42\pm 0.07, and Δs=0.13±0.06\Delta s = -0.13\pm 0.06. The flavor-singlet axial charge gA1=ΔΣ=0.22±0.09g_A^1 = \Delta \Sigma =0.22\pm 0.09 .Comment: contribution to Lattice '94; 3 page uuencoded ps fil

    Tensor products of strongly graded vertex algebras and their modules

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    We study strongly graded vertex algebras and their strongly graded modules, which are conformal vertex algebras and their modules with a second, compatible grading by an abelian group satisfying certain grading restriction conditions. We consider a tensor product of strongly graded vertex algebras and its tensor product strongly graded modules. We prove that a tensor product of strongly graded irreducible modules for a tensor product of strongly graded vertex algebras is irreducible, and that such irreducible modules, up to equivalence, exhaust certain naturally defined strongly graded irreducible modules for a tensor product of strongly graded vertex algebras. We also prove that certain naturally defined strongly graded modules for the tensor product strongly graded vertex algebra are completely reducible if and only if every strongly graded module for each of the tensor product factors is completely reducible. These results generalize the corresponding known results for vertex operator algebras and their modules.Comment: 26 pages. For the sake of readability, I quote certain necessary technical definitions from earlier work of Y.-Z. Huang, J. Lepowsky and L. Zhang [arXiv:0710.2687, arXiv:1012.4193, arXiv:math/0609833
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