ar X iv : q - a lg / 9 50 50 19 v 1 1 7 M ay 1 99 5 A theory of tensor products for module categories for a vertex operator algebra , IV ∗

This is the fourth part of a series of papers developing a tensor product theory of modules for a vertex operator algebra. In this paper, We establish the associativity of P (z)-tensor products for nonzero complex numbers z constructed in Part III of the present series under suitable conditions. The associativity isomorphisms constructed in this paper are analogous to associativity isomorphisms for vector space tensor products in the sense that it relates the tensor products of three elements in three modules taken in different ways. The main new feature is that they are controlled by the decompositions of certain spheres with four punctures into spheres with three punctures using a sewing operation. We also show that under certain conditions, the existence of the associativity isomorphisms is equivalent to the associativity (or (nonmeromorphic) operator product expansion in the language of physicists) for the intertwining operators (or chiral vertex operators). Thus the associativity of tensor products provides a means to establish the (nonmeromorphic) operator product expansion. The present paper (Part IV) is the fourth in a series of papers developing a theory of tensor products of modules for a vertex operator algebra. An ∗1991 Mathematics Subject Classification. Primary 17B69; Secondary 18D10, 81T40. 1 overview of the theory being developed has been given in [HL5] and the reader is referred to it for the motivation and the description of the main results. In Part I ([HL3]), the notions of P (z)and Q(z)-tensor product for any nonzero complex number z are introduced and two constructions of a Q(z)tensor product are given based on certain results proved in Part II ([HL4]). In Part III ([HL6]), the notion of P (z)-tensor product is discussed in the same way as in Section 4 of Part I for that of Q(z)-tensor product, and two constructions of a P (z)-tensor product are given using the results for the Q(z)-tensor product. Part III ([HL6]) also contains a brief description of the results in [HL3] and [HL4]. In the present paper, the associativity for P (·)-tensor products is formulated and the associativity isomorphisms are constructed under certain assumptions on the vertex operator algebra and on the products or iterates of two intertwining operators for the vertex operator algebra. These assumptions are satisfied by familiar examples and will be discussed in separate papers on applications of the theory of tensor products developed in the present series of papers. We also show that when the vertex operator algebra is rational and products of two intertwining operators are convergent in a suitable region, the existence of the associativity isomorphisms is equivalent to the associativity (or (nonmeromorphic) operator product expansion in the language of physicists) for the intertwining operators (or chiral vertex operators). See Theorems 14.11, 16.3 and 16.5 for the precise statements of the main results of the present paper. Our conventions in this paper is the same as those in [HL3], [HL4] and [HL6]. We also add y to our list of formal variables. The symbols z1, z2, . . . will also denote nonzero complex numbers. We fix a vertex operator algebra V . The numberings of sections, formulas, etc., continues those of Part I, Part II and Part III. Part IV is organized as follows: The associativity isomorphisms are constructed in Section 14, based on certain assumptions and some lemmas. We also prove in Section 14 that the existence of the associativity isomorphisms is equivalent to the associativity of the intertwining operators. The lemmas used in Section 14 are proved in Section 15. In Section 16, we give some conditions and show that they imply the assumptions used in the construction of the associativity isomorphisms. 2 Acknowledgments The present paper is one of the papers resulted from a long-term project jointly with James Lepowsky developing a tensor product theory of modules for a vertex operator algebra. I would like to express my gratitude to him for collaborations and many discussions. This work has been supported in part by NSF grants DMS-9104519 and DMS-9301020 and by DIMACS, an NSF Science and Technology Center funded under contract STC-88-09648. 14 Associativity isomorphisms for P (·)-tensor products In this section, we construct the associativity isomorphisms for P (·)-tensor products. To discuss the associativity we need to consider the compositions of one P (z1)-intertwining map and one P (z2)-intertwining map for suitable nonzero complex numbers z1 and z2. Since intertwining maps are maps from W1 ⊗W2 to W 3, not to W3, we have to give a precise meaning of the compositions of one P (z1)-intertwining maps and one P (z2)-intertwining map. Compositions of maps of this type have been defined precisely in [HL1] and [HL2]. Here we repeat the definition for the concrete examples of intertwining maps. Let F1 and F2 be P (z1)and P (z2)-intertwining maps of type ( W4 W1W5 ) and ( W5 W2W3 ) , respectively. If for any w(1) ∈ W1, w(2) ∈ W2, w(3) ∈ W3 and w (4) ∈ W ′ 4, the series