Weak modules and logarithmic intertwining operators for vertex operator algebras.pdf

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Weak modules and logarithmic intertwining operators for vertex operator algebras.pdf

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Weak modules and logarithmic intertwining operators for vertex operator algebras

a r X i v : m a t h / 0 1 0 1 1 6 7 v 2 [ m a t h .Q A ] 8 A p r 2 0 0 2 Contemporary Mathematics Weak modules and logarithmic intertwining operators for vertex operator algebras Antun Milas Abstract. We consider a class of weak modules for vertex operator algebras that we call logarithmic modules. We also construct nontrivial examples of intertwining operators between certain logarithmic modules for the Virasoro vertex operator algebra. At the end we speculate about some possible loga- rithmic intertwiners at the level c = 0. Introduction This work is an attempt to explain an algebraic reformulation of “logarithmic conformal field theory” from the vertex operator algebra point of view. The theory of vertex operator algebras, introduced in works of Borcherds (cf. [Bo]), Frenkel, Huang, Lepowsky and Meurman ([FHL], [FLM]) , is the mathe- matical counterpart of conformal field theory, introduced in [BPZ]. One usually studies rational vertex operator algebras, which satisfy a certain semisimplicity con- dition for modules. If we want to go beyond rational vertex operator algebras we encounter several difficulties. First, we have to study indecomposable, reducible modules, for which there is no classification theory. Another problem is that the notion of intertwining operator, as defined in [FHL], is not the most general. We can illustrate this with the following example. Let L(c, 0) (cf. [FZ]) be a non–rational vertex operator algebra associated to a lowest weight representation of the Virasoro algebra and Y1,Y2(0.1) a pair of intertwining operators for certain triples of L(c, 0)–modules. It is well known that one can study matrix coefficients 〈w′3,Y1(w1, x)w2〉(0.2) and 〈v′4,Y1(v1, x1)Y2(v2, x2)v3〉,(0.3) where wi, vj , i = 1, 2, j = 1, 2, 3, v ′ 4 and w ′ 3 are some vectors, by solving ap- propriate differential equations (see [BPZ] or [H]). In several examples we find, 1991 Mathematics Subject Classification. Primary 17B69, 17B68; Secondary 17B10, 81R10. The a