Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

10
The categories of this paper are: q-bio
5
The categories of this paper are: gr-qc
20
The categories of this paper are: acc-phys
19
The categories of this paper are: eess
0
The categories of this paper are: hep-ph
3
The categories of this paper are: physics
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
23
The categories of this paper are: solv-int
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
27
The categories of this paper are: comp-gas
35
The categories of this paper are: mtrl-th
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
13
The categories of this paper are: quant-ph