Logarithmic tensor category theory, II: Logarithmic formal calculus and properties of logarithmic intertwining operators

This is the second part in a series of papers in which we introduce and develop a natural, general tensor category theory for suitable module categories for a vertex (operator) algebra. In this paper (Part II), we develop logarithmic formal calculus and study logarithmic intertwining operators. In this paper, Part II of a series of eight papers on logarithmic tensor category theory, we develop logarithmic formal calculus and study logarithmic intertwining operators. The sections, equations, theorems and so on are numbered globally in the series of papers rather than within each paper, so that for example equation (a.b) is the b-th labeled equation in Section a, which is contained in the paper indicated as follows: In Part I [HLZ1], which contains Sections 1 and 2, we give a detailed overview of our theory, state our main results and introduce the basic objects that we shall study in this work. We include a brief discussion of some of the recent applications of this theory, and also a discussion of some recent literature. The present paper, Part II, contains Section 3. In Part III [HLZ2], which contains Section 4, we introduce and study intertwining maps and tensor product bifunctors. In Part IV [HLZ3], which contains Sections 5 and 6, we give constructions of the P (z)and Q(z)tensor product bifunctors using what we call “compatibility conditions” and certain other conditions. In Part V [HLZ4], which contains Sections 7 and 8, we study products and iterates of intertwining maps and of logarithmic intertwining operators and we begin the development of our analytic approach. In Part VI [HLZ5], which contains Sections 9 and 10, we construct the appropriate natural associativity isomorphisms between triple tensor product functors. In Part VII [HLZ6], which contains Section 11, we give sufficient conditions for the existence of the associativity isomorphisms. In Part VIII [HLZ7], which contains Section 12, we construct braided tensor category structure. Acknowledgments The authors gratefully acknowledge partial support from NSF grants DMS-0070800 and DMS-0401302. Y.-Z. H. is also grateful for partial support from NSF grant PHY-0901237 and for the hospitality of Institut des Hautes Études Scientifiques in the fall of 2007.