Generalized twisted modules associated to general automorphisms of a vertex operator algebra

We introduce a notion of strongly C^{\times}-graded, or equivalently, C/Z-graded generalized g-twisted V-module associated to an automorphism g, not necessarily of finite order, of a vertex operator algebra. We also introduce a notion of strongly C-graded generalized g-twisted V-module if V admits an additional C-grading compatible with g. Let V=\coprod_{n\in \Z}V_{(n)} be a vertex operator algebra such that V_{(0)}=\C\one and V_{(n)}=0 for n<0 and let u be an element of V of weight 1 such that L(1)u=0. Then the exponential of 2\pi \sqrt{-1} Res_{x} Y(u, x) is an automorphism g_{u} of V. In this case, a strongly C-graded generalized g_{u}-twisted V-module is constructed from a strongly C-graded generalized V-module with a compatible action of g_{u} by modifying the vertex operator map for the generalized V-module using the exponential of the negative-power part of the vertex operator Y(u, x). In particular, we give examples of such generalized twisted modules associated to the exponentials of some screening operators on certain vertex operator algebras related to the triplet W-algebras. An important feature is that we have to work with generalized (twisted) V-modules which are doubly graded by the group C/Z or C and by generalized eigenspaces (not just eigenspaces) for L(0), and the twisted vertex operators in general involve the logarithm of the formal variable.Comment: Final version to appear in Comm. Math. Phys. 38 pages. References on triplet W-algebras added, misprints corrected, and expositions revise

W-extended Kac representations and integrable boundary conditions in the logarithmic minimal models WLM(1,p)

We construct new Yang-Baxter integrable boundary conditions in the lattice approach to the logarithmic minimal model WLM(1,p) giving rise to reducible yet indecomposable representations of rank 1 in the continuum scaling limit. We interpret these W-extended Kac representations as finitely-generated W-extended Feigin-Fuchs modules over the triplet W-algebra W(p). The W-extended fusion rules of these representations are inferred from the recently conjectured Virasoro fusion rules of the Kac representations in the underlying logarithmic minimal model LM(1,p). We also introduce the modules contragredient to the W-extended Kac modules and work out the correspondingly-extended fusion algebra. Our results are in accordance with the Kazhdan-Lusztig dual of tensor products of modules over the restricted quantum universal enveloping algebra $\bar{U}_q(sl_2)$ at $q=e^{\pi i/p}$. Finally, polynomial fusion rings isomorphic with the various fusion algebras are determined, and the corresponding Grothendieck ring of characters is identified.Comment: 28 page