An explicit realization of logarithmic modules for the vertex operator algebra W_{p,p'}

By extending the methods used in our earlier work, in this paper, we present an explicit realization of logarithmic \mathcal{W}_{p,p'}-modules that have L(0) nilpotent rank three. This was achieved by combining the techniques developed in \cite{AdM-2009} with the theory of local systems of vertex operators \cite{LL}. In addition, we also construct a new type of extension of $\mathcal{W}_{p,p'}$, denoted by $\mathcal{V}$. Our results confirm several claims in the physics literature regarding the structure of projective covers of certain irreducible representations in the principal block. This approach can be applied to other models defined via a pair screenings.Comment: 18 pages, v2: one reference added, other minor change

Logarithmic intertwining operators and vertex operators

This is the first in a series of papers where we study logarithmic intertwining operators for various vertex subalgebras of Heisenberg vertex operator algebras. In this paper we examine logarithmic intertwining operators associated with rank one Heisenberg vertex operator algebra $M(1)_a$, of central charge $1-12a^2$. We classify these operators in terms of {\em depth} and provide explicit constructions in all cases. Furthermore, for $a=0$ we focus on the vertex operator subalgebra L(1,0) of $M(1)_0$ and obtain logarithmic intertwining operators among indecomposable Virasoro algebra modules. In particular, we construct explicitly a family of {\em hidden} logarithmic intertwining operators, i.e., those that operate among two ordinary and one genuine logarithmic L(1,0)-module.Comment: 32 pages. To appear in CM

The N=1 triplet vertex operator superalgebras

We introduce a new family of C_2-cofinite N=1 vertex operator superalgebras SW(m), $m \geq 1$, which are natural super analogs of the triplet vertex algebra family W(p), $p \geq 2$, important in logarithmic conformal field theory. We classify irreducible SW(m)-modules and discuss logarithmic modules. We also compute bosonic and fermionic formulas of irreducible SW(m) characters. Finally, we contemplate possible connections between the category of SW(m)-modules and the category of modules for the quantum group U^{small}_q(sl_2), q=e^{\frac{2 \pi i}{2m+1}}, by focusing primarily on properties of characters and the Zhu's algebra A(SW(m)). This paper is a continuation of arXiv:0707.1857.Comment: 53 pages; v2: references added; v3: a few changes; v4: final version, to appear in CM

Pivot-and-bond model explains microtubule bundle formation

During mitosis, microtubules form a spindle, which is responsible for proper segregation of the genetic material. A common structural element in a mitotic spindle is a parallel bundle, consisting of two or more microtubules growing from the same origin and held together by cross-linking proteins. An interesting question is what are the physical principles underlying the formation and stability of such microtubule bundles. Here we show, by introducing the pivot-and-bond model, that random angular movement of microtubules around the spindle pole and forces exerted by cross-linking proteins can explain the formation of microtubule bundles as observed in our experiments. The model predicts that stable parallel bundles can form in the presence of either passive crosslinkers or plus-end directed motors, but not minus-end directed motors. In the cases where bundles form, the time needed for their formation depends mainly on the concentration of cross-linking proteins and the angular diffusion of the microtubule. In conclusion, the angular motion drives the alignment of microtubules, which in turn allows the cross-linking proteins to connect the microtubules into a stable bundle

The logarithmic triplet theory with boundary

The boundary theory for the c=-2 triplet model is investigated in detail. In particular, we show that there are four different boundary conditions that preserve the triplet algebra, and check the consistency of the corresponding boundary operators by constructing their OPE coefficients explicitly. We also compute the correlation functions of two bulk fields in the presence of a boundary, and verify that they are consistent with factorisation.Comment: 43 pages, LaTeX; v2: references added, typos corrected, footnote 4 adde