# Direct Integral Theory by O. A. Nielsen

By O. A. Nielsen

Ebook through Nielsen, O. A.

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38 below for further discussion. Chapter 7 Duality for generalized vertex operator algebras This chapter is patterned after Section 3 of [FHL). We shall present three important properties of a generalized vertex operator algebra generalized rationality, generalized commutativity and generalized associativity - and we shall show that they may be used in place of the generalized Jacobi identity in the definition of generalized vertex operator algebra. These properties are aspects of "duality," in the terminology of conformal field theory.

L/A]. 14) t(Ab)t(Ad) = t(Abd). Chapter 4. 15) = wt t(b) = ~W, h'} for bEl. 16) vt(Ab) = t(vAb) and we have v(bA . 17) = v(b) . vt(Ad) v(t(Ad) . 18) for bEL, = v(b. t(Ad)) = v(bA) . h(Ad), = vt(Ad) . v(e) for b E Lo, d, eEL. 19) = (h', h}t(Ab) for h E h, bEL, so that v(h. t{Ab)) = v(h) . vt(Ab). 21 ) for hE h. 22) V(Zh . t(Ab)) = zv(h) . vt(Ab). 23) VL / A = M(1) ®C C{L/A} ~ S(h-) ® C[L/A] (linearly). Then L acts on the right on VL / A by acting on C{L/ A}, and Lo, L o/ A, hz, h, zh (h E h) act naturally on the left on VL / A by acting on either M(1) or C{L/A} as indicated above.

We begin with our definition of this notion, and then we give some basic elementary properties and related definitions. We identify the precise sense in which the notion of (ordinary) vertex operator algebra, in the sense of [FLM3] and [FHL], is a special case. Then we formulate a theorem which clarifies the sense in which the structures developed in the earlier chapters give examples of generalized vertex operator algebras. After this, we carry out the analogous discussion for modules for generalized vertex operator algebras, giving our definition of the notion, followed by a discussion of related concepts and a theorem asserting that certain structures constructed earlier are examples of modules.